p-Adic distribution of CM points and Hecke orbits. II: Linnik equidistribution on the supersingular locus
aa r X i v : . [ m a t h . N T ] F e b p -ADIC DISTRIBUTION OF CM POINTS AND HECKE ORBITS. II: LINNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS
SEBASTI ´AN HERRERO, RICARDO MENARES, AND JUAN RIVERA-LETELIER
Abstract.
For a prime number p , we study the asymptotic distribution of CMpoints on the moduli space of elliptic curves over C p . In stark contrast to thecomplex case, in the p -adic setting there are infinitely many different measuresdescribing the asymptotic distribution of CM points. In this paper we identifyall of these measures. A key insight is to translate this problem into a p -adicversion of Linnik’s classical problem on the asymptotic distribution of integerpoints on spheres. To do this translation, we use the close relationship betweenthe deformation theories of elliptic curves and formal modules and then applyresults of Gross and Hopkins. We solve this p -adic Linnik problem using adeviation estimate extracted from the bounds for the Fourier coefficients ofcuspidal modular forms of Deligne, Iwaniec and Duke. We also identify allaccumulation measures of an arbitrary Hecke orbit. Contents
1. Introduction 12. Preliminaries 93. Asymptotic distribution of integer points on p -adic spheres 194. CM points formulae 295. Asymptotic distribution of CM points of fundamental discriminant 416. Equidistribution of partial Hecke orbits 537. Equidistribution of CM points along a p -adic discriminant 63Appendix A. Quadratic extensions of Q p and p -adic discriminants 74References 771. Introduction
For every prime number p , in this paper we give a complete description of theasymptotic distribution of CM points on the moduli space of elliptic curves over C p .A special case is treated in the companion paper [HMRL20] and all the remainingcases are treated in this paper. This is motivated by arithmetic applications inthe companion paper [HMR21], and by the results of Linnik, Duke and Clozeland Ullmo in the complex setting, see [Lin68, Duk88, CU04] and the proceedingsarticle [MV06]. To describe our results more precisely, we introduce some notation.Throughout the rest of this paper, fix a prime number p and a completion( C p , | · | p ) of an algebraic closure of the field of p -adic numbers Q p . The endo-morphism ring of an elliptic curve over C p is isomorphic to Z or to an order in aquadratic imaginary extension of Q . In the latter case, the order only depends onthe class E in the moduli space Y ( C p ) of elliptic curves over C p . The class E is then said to have complex multiplication or to be a CM point . The discriminantof a CM point is the discriminant of the endomorphism ring of a representativeelliptic curve. In this paper, a discriminant is the discriminant of an order in aquadratic imaginary extension of Q . For every discriminant D , the setΛ D := { E ∈ Y ( C p ) : CM point of discriminant D } is finite and nonempty. So, if for each x in Y ( C p ) we denote by δ x the Dirac measureon Y ( C p ) at x , then δ D := 1 D X E ∈ Λ D δ E is a Borel probability measure on Y ( C p ).In this paper we identify all accumulation measures of(1.1) (cid:8) δ D : D discriminant (cid:9) , in the weak topology on the space of Borel measures on the Berkovich space as-sociated to Y ( C p ). In stark contrast to the complex case where the limit exists[Duk88, CU04], there are infinitely many different accumulation measures of (1.1).In the companion paper [HMRL20], we identify all subsequences of (1.1) convergingto the Dirac measure at the “Gauss” or “canonical” point. They correspond to thesequences of CM points that are either in the ordinary reduction locus, or that arein the supersingular reduction locus and the p -adic norms of their discriminantstend to 0 [HMRL20, Theorem A]. In this paper we treat the remaining case, ofsequences of CM points in the supersingular locus whose discriminants have p -adicnorm bounded from below by a strictly positive constant.A key special case is that of a sequence of discriminants ( D n ) ∞ n =1 tending to −∞ ,such that for every n the conductor of D n is a p -adic unit and Q ( √ D n ) embeds in-side a fixed quadratic extension of Q p . The corresponding CM points are naturallyrelated to points in certain Gross lattices, and each of these lattices is embeddedinside a three dimensional subspace of a p -adic quaternion algebra. Figuratively, foreach n the set of CM points Λ D n corresponds to the integer points in the sphere ofradius | D n | of a three dimensional p -adic space. Thus, the problem of determiningthe accumulation measures of ( δ D n ) ∞ n =1 translates to a p -adic version of Linnik’sclassical problem on the asymptotic distribution of integer points on spheres (The-orem D in Section 1.3). We solve this p -adic Linnik problem using a deviationestimate extracted from the bounds for the Fourier coefficients of cuspidal modularforms of Deligne [Del74], Iwaniec [Iwa87] and Duke [Duk88]. The end result is thatin this key special case the sequence of measures ( δ D n ) ∞ n =1 converges, except in apeculiar case where there are precisely two accumulation measures (Theorems Aand B in Section 1.1). The peculiar case is that of a fixed fundamental discriminantsuch that p is the only prime number dividing it and a varying conductor tendingto ∞ . Genus theory elucidates the phenomenon, somewhat reminiscent of symme-try breaking, that is responsible for the emergence of two accumulation measuresin this case. To pass from the key special case to the general case, we prove ananalogous equidistribution result for Hecke orbits (Theorem C in Section 1.2) thatwe also deduce from the p -adic Linnik equidistribution result shown in this paper.In the companion paper [HMR21], we use results in this paper and in [HMRL20]to prove that for every finite set of prime numbers S there are at most finitely many INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 3 singular moduli that are S -units. This extends a result of Habegger in [Hab15] inthe case where S = ∅ .We proceed to describe our results more precisely.1.1. Equidistribution of CM points along a p -adic discriminant. A fun-damental discriminant is the discriminant of the ring of integers of a quadraticimaginary extension of Q . The fundamental discriminant of a discriminant D , isthe discriminant d of Q ( √ D ). It divides D and the quotient Dd is the square ofan integer in N := { , , . . . } that is called the conductor of D . A discriminant is prime , if it is fundamental and divisible by only one prime number. Note that, if d is a prime discriminant divisible by p , then p ≡ − d = − p, or p = 2 and d = − d = − p -adic quadratic order is a Z p -order in a quadratic extension of Q p , and a p -adic discriminant is a set formed by the discriminants of all Z p -bases of a p -adicquadratic order. Every p -adic discriminant is thus a coset in Q × p / ( Z × p ) containedin Z p . Moreover, the p -adic discriminant is a complete isomorphism invariant of a p -adic quadratic order (Lemma A.1( ii ) in Appendix A).Denote by Q p the algebraic closure of Q p inside C p , and by O p and O Q p the ringof integers of C p and Q p , respectively. For E in Y ( C p ) represented by a Weierstrassequation with coefficients in O Q p having smooth reduction, denote by F E its formalgroup and by End( F E ) the ring of endomorphisms of F E that are defined over O Q p .Then End( F E ) is either isomorphic to Z p , or to a p -adic quadratic order, see, e.g. ,[Fr¨o68, Chapter IV, Section 1, Theorem 1( iii )]. In the latter case, E is said to have formal complex multiplication or to be a formal CM point . Moreover, the p -adicdiscriminant of E is the p -adic discriminant of the p -adic quadratic order End( F E ),and for a p -adic discriminant D we putΛ D := { E ∈ Y ( C p ) : formal CM point of p -adic discriminant D } . Theorem A.
For every p -adic discriminant D , the set Λ D is a compact sub-set of Y ( C p ) and there is a Borel probability measure ν D on Y ( C p ) whose sup-port is equal to Λ D , and such that the following equidistribution property holds.Let ( D n ) ∞ n =1 be a sequence of discriminants in D tending to −∞ , such that forevery n the fundamental discriminant of D n is either not divisible by p , or not aprime discriminant. Then we have the weak convergence of measures (1.2) δ D n → ν D as n → ∞ . Our next result addresses the case left out in the theorem above. Namely, that forsome integer n ≥ d of D n is a prime discriminantdivisible by p . Passing to a subsequence if necessary, we can restrict to the casewhere for every n the fundamental discriminant of D n is equal to d . In the followingresult, (cid:0) ·· (cid:1) denotes the Kronecker symbol. Theorem B.
Let d be a prime discriminant that is divisible by p . Moreover, let m ≥ be a given integer, put D := dp m , and denote by D the p -adic discriminantcontaining D . Then there is a partition of Λ D into disjoint compact sets Λ D = Λ + D ⊔ Λ − D , such that ν + D := 2 ν D | Λ + D and ν − D := 2 ν D | Λ − D SEBASTI´AN HERRERO, RICARDO MENARES, AND JUAN RIVERA-LETELIER are both probability measures and such that the following equidistribution propertyholds. For every sequence ( f n ) ∞ n =0 in N tending to ∞ such that for every n wehave (cid:16) df n (cid:17) = 1 (resp. (cid:16) df n (cid:17) = − ), we have the weak convergence of measures δ D ( f n ) → ν + D (resp. δ D ( f n ) → ν − D ) as n → ∞ . In what follows, consider Y ( C p ) as a subspace of the Berkovich affine line A over C p , using the j -invariant to identify Y ( C p ) with the subspace C p of A .Moreover, denote by x can the “canonical” or “Gauss point” of A .Theorems A and B together with [HMRL20, Theorem A] identify all accumu-lation measures of (1.1), see Corollary 1.2 below. We consider first the importantspecial case of fundamental discriminants, which is simpler. A p -adic discriminantis fundamental , if it is the p -adic discriminant of the ring of integers of a quadraticextension of Q p . Note that there are three fundamental p -adic discriminants if p isodd and seven if p = 2, see, e.g. , Lemma A.1( iii ) in Appendix A. Corollary 1.1.
The set of all accumulation measures of (1.3) (cid:8) δ d : d fundamental discriminant (cid:9) in the space of Borel measures on A , is equal to { ν d : d fundamental p -adic discriminant } ∪ { δ x can } . Note that for distinct p -adic discriminants D and D ′ , the compact sets Λ D and Λ D ′ are disjoint by definition, so the measures ν D and ν D ′ are different. Thus,Corollary 1.1 implies that (1.3) has precisely four accumulation measures if p is oddand eight if p = 2. This is in contrast to Duke’s result that in the complex settingthe limit exists [Duk88].To explain how Corollary 1.1 follows from Theorem A and [HMRL20, The-orem A], we recall a consequence of this last result. An elliptic curve class E in Y ( C p ) has supersingular reduction , if there is a representative Weierstrass equa-tion with coefficients in O p whose reduction is smooth and supersingular. Denoteby Y sups ( C p ) the set of all elliptic curve classes in Y ( C p ) with supersingular reduc-tion. For a sequence of discriminants ( D j ) ∞ j =1 tending to −∞ , [HMRL20, Theo-rem A] implies the convergence of measures δ D j → δ x can as j → ∞ in each of thefollowing situations:( i ) For every j the set Λ D j is disjoint from Y sups ( C p );( ii ) For every j the set Λ D j is contained in Y sups ( C p ) and | D j | p → j → ∞ .Corollary 1.1 is a direct consequence of this property, Theorem A and the factthat a CM point of fundamental discriminant d is contained in Y sups ( C p ) if andonly if d is in a fundamental p -adic discriminant (Lemma 2.1 in Section 2.1). Onthe other hand, the consequence of [HMRL20, Theorem A] above, combined withTheorems A and B, and with the fact that a CM point is in Y sups ( C p ) if and onlyif its discriminant is contained in a p -adic discriminant (Lemma 2.1 in Section 2.1),implies the following corollary as an immediate consequence. Corollary 1.2.
In the case where p ≡ − , denote by b d the p -adic discrim-inant containing − p . In the case where p = 2 , denote by b d (resp. b d ′ ) the p -adicdiscriminant containing − (resp. − ). Then the set of all accumulation measuresof (1.1) in the space of Borel measures on A , is equal to { ν D : D p -adic discriminant } ∪ { δ x can } , INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 5 { ν D : D p -adic discriminant } ∪ n ν + b d p m , ν − b d p m : m ≥ o ∪ { δ x can } , or { ν D : D p -adic discriminant } ∪ n ν + b d p m , ν − b d p m , ν + b d ′ p m , ν − b d ′ p m : m ≥ o ∪ { δ x can } , depending on whether p ≡ , p ≡ − , or p = 2 , respectively.In particular, in all the cases the set of all accumulation measures of (1.1) iscountably infinite. This is in stark contrast to the complex setting where the limit exists [Duk88,CU04].In the companion paper [HMR21], we also prove that every accumulation mea-sure of (1.1) different from δ x can is nonatomic [HMR21, Theorem B]. This is one ofthe main ingredients in showing that for every finite set of prime numbers S , thereare at most finitely many singular moduli that are S -units [HMR21, Theorem A].Every p -adic discriminant D contains a dense subset of discriminants, so thereare plenty of sequences ( D n ) ∞ n =1 as in Theorem A. Moreover, a CM point of dis-criminant D is contained in Λ D if and only if D belongs to D (Corollary 4.12( ii )).In particular, for every discriminant D in D the set Λ D is contained in Λ D . Thus,the following corollary is an immediate consequence of Theorem A. Corollary 1.3.
The CM points in Y sups ( C p ) form a dense subset of the set offormal CM points. Coleman and McMurdy proved the first result of this type [CM06, Theorem 4.1],for p ≥ E such that End( F E ) is the ringof integers of a ramified quadratic extension of Q p , is approximated by CM points.1.2. Equidistribution of partial Hecke orbits.
To state our next main result,we recall the definition of Hecke correspondences, see Section 2.8 for background.A divisor on Y ( C p ) is an element of the free abelian groupDiv( Y ( C p )) := M E ∈ Y ( C p ) Z E. For a divisor D = P E ∈ Y ( C p ) n E E in Div( Y ( C p )), the degree and support of D aredeg( D ) := X E ∈ Y ( C p ) n E and supp( D ) := { E ∈ Y ( C p ) : n E = 0 } , respectively. If in addition deg( D ) ≥ E in Y ( C p ) we have n E ≥ δ D := 1deg( D ) X E ∈ Y ( C p ) n E δ E is a Borel probability measure on Y ( C p ).For n in N , the n -th Hecke correspondence is the linear map T n : Div( Y ( C p )) → Div( Y ( C p ))defined for E in Y ( C p ) by T n ( E ) := X C ≤ E of order n E/C,
SEBASTI´AN HERRERO, RICARDO MENARES, AND JUAN RIVERA-LETELIER where the sum runs over all subgroups C of E of order n . Note that supp( T n ( E ))is the set of all E ′ in Y ( C p ) for which there is an isogeny E → E ′ of degree n .For E in Y ( C p ), but not in Y sups ( C p ), the asymptotic distribution of the Heckeorbit ( T n ( E )) ∞ n =1 is described by [HMRL20, Theorem C]. Our next main resultaddresses the more difficult case where E is in Y sups ( C p ). The description dependson a subgroup Nr E of Z × p that we proceed to define. If E is not a formal CM point,then Nr E := ( Z × p ) . In the case where E is a formal CM point, denote by Aut( F E )the group of isomorphisms of F E defined over O Q p , and by nr the norm map of thefield of fractions of End( F E ) to Q p . Then, Nr E := { nr ( ϕ ) : ϕ ∈ Aut( F E ) } . In all the cases Nr E is a multiplicative subgroup of Z × p containing ( Z × p ) . Inparticular, the index of Nr E in Z × p is at most two if p is odd, and at most fourif p = 2. Theorem C (Equidistribution of partial Hecke orbits) . Let E be in Y sups ( C p ) ,let N be a coset in Q × p / Nr E contained in Z p , and consider the partial Hecke orbit(1.4) Orb N ( E ) := [ n ∈ N ∩ N supp( T n ( E )) . Then the closure
Orb N ( E ) in Y sups ( C p ) of this set is compact. Moreover, there isa Borel probability measure µ E N on Y ( C p ) whose support is equal to Orb N ( E ) , andsuch that for every sequence ( n j ) ∞ j =1 in N ∩ N tending to ∞ , we have the weakconvergence of measures δ T nj ( E ) → µ E N as j → ∞ . See Theorem C’ in Section 6 for a quantitative version of this result.Together with [HMRL20, Theorem C], Theorem C identifies all limits of Heckeorbits in Y ( C p ). In fact, [HMRL20, Theorem C] implies that for E in Y ( C p )and a sequence ( n j ) ∞ j =1 in N tending to ∞ , we have the convergence of measures δ T nj ( E ) → δ x can as j → ∞ in each of the following situations:( i ) E is not in Y sups ( C p );( ii ) E is in Y sups ( C p ) and | n j | p → j → ∞ .Combined with Theorem C, this implies the following as an immediate consequence. Corollary 1.4.
For each E in Y ( C p ) , the set of all accumulation measures of ( δ T n ( E ) ) ∞ n =1 in the space of Borel probability measures on A , is equal to (1.5) (cid:8) µ E N : N ∈ Q × p / Nr E , N ⊂ Z p (cid:9) ∪ { δ x can } . We also show that for distinct cosets N and N ′ in Q × p / Nr E contained in Z p , themeasures µ E N and µ E N ′ are different (Proposition 6.9( ii ) in Section 6.4). In particular,the set of accumulation measures (1.5) is countably infinite. This is in stark contrastto the complex setting where the limit exists, see [CU04, COU01, EO06]. We alsoprove that the measure µ E N is nonatomic in the companion paper [HMR21].1.3. Asymptotic distribution of integer points on p -adic spheres. Theproofs of Theorems A, B and C rely on the p -adic equidistribution result stated be-low, which is inspired by Linnik’s classical problem on the asymptotic distribution INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 7 of integer points on spheres. See [Duk88, EMV13] for refinements and a historicalperspective.Fix an integer n ≥ Q in Z [ X , X , . . . , X n ].For m in N put V m ( Q ) := { x ∈ Z n : Q ( x ) = m } and for ℓ in Z p define the sphere S ℓ ( Q ) := { x ∈ Z np : Q ( x ) = ℓ } . Note that the orthogonal group of Q with coefficients in Z p , defined asO Q ( Z p ) := { T ∈ GL n ( Z p ) : Q ( T · X ) = Q ( X ) } , is compact, acts on Z np and for every ℓ in Z p it preserves the sphere S ℓ ( Q ).In our following result, we fix ℓ in Z p r { } for which the sphere S ℓ ( Q ) is nonemptyand such that the compact group O Q ( Z p ) acts transitively on it. In this case, thereis a unique Borel probability measure on S ℓ ( Q ) that is invariant under the actionof O Q ( Z p ), see, e.g. , Lemma 5.3. For every u in Z × p denote by M u the elementof GL n ( Z p ) defined by M u ( X , . . . , X n ) := ( uX , . . . , uX n ) . Note that for every ℓ in Z p we have M u ( S ℓ ( Q )) = S ℓu ( Q ). Theorem D ( p -Adic Linnik equidistribution) . Let κ n be equal to if n is even andto if n is odd and fix c > n − κ n . Let ℓ in Z p r { } be such that S ℓ ( Q ) is nonemptyand O Q ( Z p ) acts transitively on S ℓ ( Q ) and denote by µ ℓ be the unique Borel prob-ability measure on S ℓ ( Q ) that is invariant under the action of O Q ( Z p ) . Moreover,let ( m j ) ∞ j =1 be a sequence in N tending to ∞ that is contained in the coset ℓ ( Z × p ) of Q × p / ( Z × p ) and such that for every sufficiently large j we have V m j ( Q ) ≥ m cj .For each j ≥ , let u j in Z × p be such that m j = ℓu j . If n = 3 , then assume inaddition that there is S ≥ such that for each j the largest square diving m j is lessthan or equal to S . Then we have the weak convergence of measures V m j ( Q ) X x ∈ V mj ( Q ) δ M − uj ( x ) → µ ℓ as j → ∞ . See also Theorem 3.1 and Corollary 3.2 for quantitative variants of this result.The circle method can be used to show that V m ( Q ) grows at least like m n ,provided n ≥ q the equation Q ( x ) = m issolvable in Z nq . For n = 4, the circle method yields that for every ε > V m ( Q ) grows at least like m − ε , provided that for every prime number q the equation Q ( x ) = m has a solution x in Z nq for which ∇ Q ( x ) is a unit in Z q , see, e.g. , [HB96, Theorem 4 and Corollary 1]. For n ≥
3, the quantity V m ( Q ) can alsobe estimated in some situations using the theory of modular forms, see for examplethe introduction of [DSP90] and references therein. In our applications the growthof V m ( Q ) in m is well understood, so we do not use these general results.1.4. Notes and references.
For a prime number ℓ different from p , Goren andKassaei study in [GK19] the p -adic dynamical system generated by the Hecke cor-respondence of index ℓ acting on the moduli space Y ( N ), of elliptic curves witha marked torsion point of exact order N coprime to pℓ . They study the case of SEBASTI´AN HERRERO, RICARDO MENARES, AND JUAN RIVERA-LETELIER ordinary and supersingular reduction. To describe the results of Goren and Kas-saei in the latter case, we use the notation in Section 2.6. For the former, seealso [HMRL20]. For each e in Y sups ( F p ), they restrict to the action on the ideal disc { x ∈ X e ( O Q p ) : ord p ( x ) ≥ } of a certain subgroup H N of G e that depends on ℓ and on a point in Y ( N )( F p ) above e . Goren and Kassaei then use that the Gross–Hopkins period map restricts to an equivariant rigid analytic isomorphism from theideal disc onto its image, and apply general results about random walks on groups.As explained in [GK19, Section 5.10], one of the difficulties in this approach is totranslate the results back to the action of H N on the ideal disc. Moreover, thisstrategy breaks down beyond ideal discs because the period map is not injective.In contrast, our approach makes no use of the period map and applies to the Heckeorbit of every point in Y sups ( C p ). For a comparison, let E be in Y sups ( C p ) andlet ℓ ≥ Nr E (resp. Z × p r Nr E ). Then, Theorem C implies thatthe closure of the forward orbit of E under T ℓ equals(1.6) Orb Nr E ( E ) (resp. Orb Nr E ( E ) ∪ Orb ℓ Nr E ( E )).In the particular case that ℓ is a prime number and that E corresponds to the imageof a point in the ideal disc by the period map, the associated H -minimal set inthe sense of [GK19, Section 5.10] corresponds to the intersection of (1.6) with D e under the composition of the period map with Π − e .The p -adic asymptotic distribution of CM points is also studied by Disegniin [Dis19]. The main result of [Dis19] is stated for Shimura curves. When appliedto the modular curve of level one it is a particular case of [HMRL20, Theorem A].There is no intersection between the results in [Dis19] and those in this paper.1.5. Strategy and organization.
In this section we explain the strategy of proofof our main results and simultaneously describe the organization of the paper.After some preliminaries in Section 2, in Section 3 we prove Theorem D on theasymptotic distribution of integer points on p -adic spheres. We deduce this resultfrom a deviation estimate modulo large powers of p (Theorem 3.1). The mainingredient in the proof is the construction of an auxiliary modular form that iscuspidal (Proposition 3.3 in Section 3.1). We derive the deviation estimate fromthe bounds for the Fourier coefficients of cuspidal modular forms of Deligne [Del74],Iwaniec [Iwa87] and Duke [Duk88]. The proof of Theorem D is given in Section 3.2.In Section 4 we give several formulae for (formal) CM points having supersingularreduction. In the first formula we use the Gross–Hopkins group action on the Lubin–Tate deformation space [HG94], which we recall in Section 2.6. It interprets CMpoints with fundamental discriminant as (projections of) fixed points of certainelements of this action (Theorem 4.2 in Section 4.1). The remaining formulae usethe canonical branch t of T p to relate (formal) CM points whose conductors differby a power of p (Theorems 4.6 and 4.11 in Sections 4.2 and 4.3, respectively).In Section 5 we describe the asymptotic distribution of CM points of funda-mental discriminant in a quantitative form (Theorem 5.1). It is one of the mainingredients in the proof of Theorem A. To explain the strategy of proof, fix a super-singular elliptic curve class e in the moduli space Y ( F p ) of elliptic curves over F p and denote by R e the p -adic space of endomorphisms of the formal Z p -module of e .We start by defining the “zero-trace spheres” of R e and by showing that each ofthese sets carries a natural homogeneous measure (Proposition 5.2 in Section 5.1).A key step in the proof of Theorem 5.1 is showing that for every fundamental p -adic INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 9 discriminant d , the set of formal CM points in Λ d in the residue disc associated to e is naturally parametrized by a zero-trace sphere (Propositions 5.4 and 5.6 in Sec-tions 5.2 and 5.3, respectively). Using this parametrization, we deduce Theorem 5.1in Section 5.5 from our results on the asymptotic distribution of integer points on p -adic spheres in Section 3 and the equidistribution of CM points on supersingularresidue discs (Theorem 5.7 in Section 5.4).Our results on the asymptotic distribution of Hecke orbits (Theorem C in Sec-tion 1.2) are proved in Section 6. We give a quantitative version of this resultwith a convergence rate that is uniform on the initial point. It is stated as Theo-rem C’ and is used to prove Theorems A and B. To explain the strategy of proofof Theorem C’, fix supersingular elliptic curve classes e and e ′ in Y ( F p ) and denoteby R e,e ′ the p -adic space of morphisms from the formal Z p -module of e to that of e ′ .We start by introducing the “supersingular spheres” of R e,e ′ and by showing thateach supersingular sphere carries a natural homogeneous measure (Proposition 6.2in Section 6.1). A key step is to show that each partial Hecke orbit restricted tothe residue disc associated to e ′ is parametrized by a supersingular sphere (Propo-sition 6.4 in Section 6.2). After these considerations, we deduce Theorem C’ inSection 6.3 from our results on the asymptotic distribution of integer points on p -adic spheres in Section 3. We also show that distinct partial Hecke orbits havedifferent limit measures (Proposition 6.9 in Section 6.4).Our results on the asymptotic distribution of CM points (Theorems A and B inSection 1.2) are proved in Section 7. We reduce the proofs to the case of fundamental p -adic discriminants using the (formal) CM points formulae in Sections 4.2 and 4.3.For a fundamental p -adic discriminant d , we first study how Λ d is decomposedinto closures of partial Hecke orbits. The set Λ d coincides with the closure ofa partial Hecke orbit if Q p ( √ d ) is unramified over Q p and if Q p ( √ d ) is ramifiedover Q p , then Λ d is partitioned into precisely two closures of partial Hecke orbits(Proposition 7.1 in Section 7.1). In the latter case we use genus theory to determinefor each discriminant D in d , how Λ D is distributed between these closures of partialHecke orbits (Proposition 7.4 in Section 7.2). Here is where prime discriminantsdivisible by p play a special role. In Section 7.3 we use these results to deduceTheorems A and B from Theorems 5.1 and C’.For the reader’s convenience, in Appendix A we gather some basic facts aboutquadratic field extensions of Q p and p -adic discriminants. Acknowledgments.
The first named author was supported by ANID/CONICYT,FONDECYT Postdoctorado Nacional grant 3190086. The second named authorwas partially supported by FONDECYT grant 1171329. The third named au-thor acknowledges partial support from NSF grant DMS-1700291. The authorswould like to thank Pontificia U. Cat´olica de Valpara´ıso, U. of Rochester and U. deBarcelona for hospitality during the preparation of this work.2.
Preliminaries
Recall that N = { , , . . . } . Given n in N , put d ( n ) := X d> ,d | n σ ( n ) := X d> ,d | n d. We use several times that for every n in N , we have(2.1) σ ( n ) ≥ n, and the fact that for every ε > d ( n ) = o ( n ε ) . Given an algebraically closed field K , denote by Y ( K ) the moduli space of ellipticcurves over K . It is the space of all isomorphism classes of elliptic curves over K ,for isomorphisms over K . For a class E in Y ( K ), the j -invariant j ( E ) of E is anelement of K determining E completely and the map j : Y ( K ) → K is a bijection.Given a field extension K of Q p , denote by O K its ring of integers and by M K the maximal ideal of O K . In the case where K = C p , denote O K and M K by O p and M p , respectively. Moreover, identify the residue field of C p with an algebraicclosure F p of the field with p elements F p and denote by π : O p → F p the reductionmap. For every finite extension K of Q p inside C p , we have M K = M p ∩ O K .For a quadratic extension K of Q p , denote by x x the unique field automor-phism of K over Q p different from the identity. Moreover, for x in K puttr( x ) = x + x, nr( x ) := xx and ∆( x ) := ( x − x ) = tr( x ) − x ) , all of which are elements of Q p .Denote by Q p the unique unramified quadratic extension of Q p inside C p . More-over, denote by Z p the ring of integers of Q p and by F p its residue field. Foreach ∆ in Q p , denote by Q p ( √ ∆) the smallest extension of Q p inside C p containinga root of X − ∆. An explicit description of the set of all quadratic extensions of Q p inside C p is given in Lemma A.2( i ) in Appendix A.The endomorphism ring of an elliptic curve over F p is isomorphic to an order ineither a quadratic imaginary extension of Q or a quaternion algebra over Q . In thelatter case the corresponding elliptic curve class is supersingular .An elliptic curve class E has good reduction , if it is represented by a Weierstrassequation with coefficients in O p whose reduction is smooth. In this case the re-duction is an elliptic curve over F p , whose class e E only depends on E and is the reduction of E .A divisor on a set X ∗ is a formal finite sum P x ∈ X n x x in L x ∈ X Z x . In thespecial case where for some x in X we have n x = 1 and n x = 0 for every x = x ,we use [ x ] to denote this divisor. When there is no danger of confusion, sometimeswe use x to denote [ x ]. For a divisor D = P x ∈ X n x [ x ] on X , the degree deg( D )and support supp( D ) are defined bydeg( D ) := X x ∈ X n x and supp( D ) := { x ∈ X : n x = 0 } . For a set X ′ and a map f : X → X ′ , the push-forward action of f on divi-sors f ∗ : Div( X ) → Div( X ′ ) is the linear extension of the action of f on points. ∗ We only use this definition in the case where X is one of several types of one-dimensionalobjects. For such X , the notion of divisor introduced here can be seen as a natural extension ofthe usual notion of Weil divisor. INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 11
Discriminants and their p -adic counterparts. Recall that a fundamentaldiscriminant d is the discriminant of the ring of integers of a quadratic imaginaryextension K of Q . If d is the unique square-free integer such that K = Q ( √ d ),then(2.3) d = ( d if d ≡ d if d ≡ − , . Recall that a discriminant D is the discriminant of an order in a quadratic imag-inary extension of Q . Moreover, the fundamental discriminant of D is the discrim-inant d of Q ( √ D ), it divides D and the quotient Dd is the square of the conductorof D . Conversely, for every fundamental discriminant d and every integer f ≥ D := df is the unique discriminant of fundamental discriminant d andconductor f . Moreover, there is a unique order O d,f of discriminant D in thequadratic imaginary extension Q ( √ d ) of Q , and it is given by O d,f := Z + f O d, = Z h D + √ D i . Conversely, every order in Q ( √ d ) is of this form, see, e.g. , [Lan87, Chapter 8,Section 1, Theorem 3]. In particular, the index of O d,f in O d, is equal to f ,and O d, is the unique maximal order in Q ( √ d ). Note that O d, is also the ring ofintegers of Q ( √ d ).A discriminant D is p -supersingular , if the reduction of some CM point of dis-criminant D is supersingular. In this case, the reduction of every CM point ofdiscriminant D is supersingular. Equivalently, a discriminant D is p -supersingularif p is ramified or inert in Q ( √ D ), see [Deu41] or [Lan87, Chapter 13, Section 4,Theorem 12]. Note that a discriminant is p -supersingular if and only if its funda-mental discriminant is. A fundamental discriminant d is p -supersingular if and onlyif (cid:16) dp (cid:17) = 1 if p is odd and d p = 2.Recall that a p -adic quadratic order is a Z p -order in a quadratic extension of Q p .For a quadratic extension K of Q p , the ring of integers O K is the unique maximal Z p -order in K . Moreover, for every integer m ≥ Z p + p m O K is a Z p -orderin K and every Z p -order in K is of this form.Recall that a p -adic discriminant is a coset in Q × p / ( Z × p ) formed by the discrimi-nants of all Z p -bases of a p -adic quadratic order. Furthermore, a p -adic discriminantis fundamental , if it is the p -adic discriminant of the ring of integers of a quadraticextension of Q p . The p -adic discriminant is an isomorphism invariant of p -adicquadratic orders. An explicit description of all p -adic quadratic orders and p -adicdiscriminants is given in Lemma A.1 in Appendix A. For a p -adic discriminant D and ∆ in D , the field Q p ( √ ∆) is a quadratic extension of Q p inside C p that dependsonly on D , but not on ∆. Denote it by Q p ( √ D ).The following basic facts are important in what follows. For the reader’s conve-nience, we give a proof in Appendix A. Lemma 2.1.
A discriminant (resp. a discriminant whose conductor is not divisibleby p ) belongs to a p -adic discriminant (resp. fundamental p -adic discriminant) ifand only if it is p -supersingular. Moreover, for each p -adic discriminant (resp.fundamental p -adic discriminant) D , the set of discriminants (resp. fundamentaldiscriminants) contained in D is dense in D . p -Adic division quaternion algebras. Recall that there is a unique divisionquaternion algebra over Q p up to isomorphism. For the rest of this paper we fixsuch an algebra B p . We refer to [Vig80] for background on quaternion algebras.Let B be an algebra over Q p isomorphic to B p . Denote by 1 B its multiplicativeidentity, and identify Q p with its image in B by the map ℓ ℓ · B . Moreover,denote by g g the involution of B , and for g in B denote bytr( g ) := g + g, nr( g ) := gg, and ∆( g ) := tr( g ) − g ) , the reduced trace , the reduced norm , and the discriminant of g , respectively. Notethat each of these functions takes images in Q p . On the other hand, the functionord B : B → Z ∪ {∞} defined for g in B by ord B ( g ) := ord p (nr( g )), is the uniquevaluation extending the valuation 2 ord p on Q p . The valuation ring of B , R := { g ∈ B : ord B ( g ) ≥ } is the unique maximal Z p -order in B , and it coincides with the set of elements of B that are integral over Z p . The function dist B : B × B → R defined for g and g ′ in B by dist B ( g, g ′ ) := p − ord B ( g − g ′ ) , defines an ultrametric distance on B that makes B into a topological algebraover Q p . Note that G := { g ∈ B : ord B ( g ) = 0 } is the group of units of R , and that each right (resp. left) multiplication map on B by an element of G is an isometry.The following consequence of the Skolem–Noether theorem is used in Section 7. Lemma 2.2.
Let B be an algebra over Q p isomorphic to B p , and let ϕ in B r Q p and θ in Z × p r nr( O × Q p ( ϕ ) ) be given. Then there is γ in G such that (2.4) γϕγ − = ϕ and γ = θ. In the proof of Lemma 2.2 given below, we use the following basic lemma. Forthe reader’s convenience, we give a proof in Appendix A of a more detailed versionof this lemma that is stated as Lemma A.2( ii ). Lemma 2.3.
Let K be a quadratic extension of Q p . Then the subgroup nr( O ×K ) of Z × p is equal to Z × p if K is unramified over Q p , and has index two in Z × p if K isramified over Q p .Proof of Lemma 2.2. If Q p ( ϕ ) is unramified over Q p , then nr( O × Q p ( ϕ ) ) = Z × p byLemma 2.3 and there is nothing to prove. Assume that Q p ( ϕ ) is ramified over Q p ,and let ̟ be a uniformizer of O Q p ( ϕ ) . Then ord p (nr( ̟ )) = 1, and nr( O × Q p ( ϕ ) ) hasindex two in Z × p by Lemma 2.3.By [Vig80, Chapitre I, Corollaire
Corollaire γ of B , such that γ ϕγ − = ϕ and γ ∈ Q × p r nr( Q p ( ϕ ) × ) . Note in particular that tr( γ ) = 0 and nr( γ ) = − γ . Let θ in Z × p and n in Z besuch that γ = nr( ̟ ) n θ . Then θ is not in nr( O × Q p ( ϕ ) ), and since nr( O × Q p ( ϕ ) ) hasindex two in Z × p , we conclude that the quotient θ/θ belongs to nr( O × Q p ( ϕ ) ). Let ρ INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 13 in O Q p ( ϕ ) be such that nr( ρ ) = θ/θ , and put γ := γ ρ̟ − n . Then for every ϕ ′ in Q p ( ϕ ) we have γϕ ′ γ − = γ ( ρ̟ − n ϕ ′ ( ρ̟ − n ) − ) γ − = γ ϕ ′ γ − = ϕ ′ . This applies in particular to ϕ ′ = ρ̟ − n , so we havetr( γ ) = γ ρ̟ − n + ρ̟ − n γ = γ ρ̟ − n − ( γ ρ̟ − n γ − ) γ = 0 , and therefore γ = − nr( γ ) = − nr( γ ) nr( ρ ) nr( ̟ ) − n = γ ( θ/θ ) nr( ̟ ) − n = θ. In particular, nr( γ ) = − θ belongs to Z × p and therefore γ belongs to G . The lemmais thus proved. (cid:3) Supersingular elliptic curves.
Denote by Y sups ( F p ) the finite subset of Y ( F p )of supersingular elliptic curves classes. Using j : Y ( F p ) → F p to identify Y ( F p )with F p , note that Y sups ( F p ) is contained in F p and that Y sups ( F p ) can be writtenas the zero set of a polynomial with coefficients in F p , see, e.g. , [Deu41] and [Sil09,Chapter V, Theorems 3.1 and 4.1]. In particular, the Frobenius map Frob: F p → F p maps Y sups ( F p ) onto itself and it induces an involution on this set.For e in Y sups ( F p ), denote by End( e ) and Aut( e ) the ring of endomorphisms andthe group of automorphisms of e defined over F p , respectively. We use several timesthe mass formula of Deuring and Eichler,(2.5) X e ∈ Y sups ( F p ) e ) = p − , see, e.g. , [Eic55] or [Sil09, Exercise 5.9].Given e and e ′ in Y sups ( F p ) and an integer m ≥
1, denote by Hom m ( e, e ′ ) theset of all isogenies from e to e ′ of degree m . If e is supersingular, then the ringEnd( e ) ⊗ Q p is isomorphic to B p . Note that for g in End( e ), viewed as an elementof End( e ) ⊗ Z p , the discriminant ∆( g ) belongs to Z p .For e in Y sups ( F p ), denote by D e the set of all E in Y ( C p ) having good reduction,and such that the reduced elliptic curve is isomorphic to e . The set D e is a residuedisc in Y ( C p ).2.4. Formal Z p -modules. In this section we make a brief review of formal Z p -modules.We refer to [Fr¨o68, Haz78] for background.Fix a complete, local, Noetherian Z p -algebra R with structural map π : Z p → R ,maximal ideal M , and residue field isomorphic to a subfield k of F p . Endow R with its natural M -adic topology and fix a reduction morphism R → k , whichwe denote by z e z . We are mainly interested in the special case where R is asubfield of F p , or the ring of integers of a finite extension of Q p inside C p , wherewe take the inclusion map, or the restriction of π , as the corresponding reductionmorphism. We stick to the general case for convenience.For formal groups F and F ′ defined over a ring R , denote by Hom R ( F , F ′ )the set of morphisms F → F ′ defined over R and put End R ( F ) := Hom R ( F , F ).Denote by Iso R ( F , F ′ ) the set of all isomorphisms F → F ′ defined over R andput Aut R ( F ) := Iso R ( F , F ).Given rings R and R ′ , a ring morphism σ : R → R ′ and a formal power series f with coefficients in R , define σf as the power series with coefficients in R ′ obtained by applying σ to the coefficients of f . We refer to σf as the base change of f under σ .For a formal group F over R , denote by e F its reduction, which is the formalgroup over k obtained as base change of F under the reduction map R → k .In this paper, a formal Z p -module over R (resp. k ) is a formal group F over R (resp. k ) of dimension 1, together with a ring homomorphism θ : Z p → End R ( F )(resp. θ : Z p → End k ( F )) such that, in coordinates, for every ℓ in Z p we have θ ( ℓ )( X ) ≡ π ( ℓ ) X mod X (resp. ] π ( ℓ ) X mod X ) . Every formal group F over R admits a unique structure of formal Z p -moduleover R , such that the structural ring homomorphism θ is continuous with respectto the p -adic filtration on Z p and the height filtration on End R ( F ), see [Fr¨o68,Chapter IV, Section 1, proof of Theorem 1 and Chapter III, Section 2, Corollary ofProposition 2].If R is another complete, local, Noetherian Z p -algebra, F is a formal Z p -moduleover R and σ : R → R is a morphism of Z p -algebras, then σ F has a canonicalstructure of formal Z p -module over R .2.5. Deformation spaces of formal Z p -modules. In this section we make abrief review of deformation theory of formal Z p -modules. We refer to [Dd74, Haz78,HG94] for background.Let R , π , M , and k be as in the previous section, let k be a subfield of k andlet F be a formal Z p -module over k . A deformation of F over R is a pair ( F , α ),where F is a formal Z p -module over R and α : e F → F is an isomorphism offormal Z p -modules defined over k . Two such deformations ( F , α ) and ( F ′ , α ′ ) are isomorphic , if there exists an isomorphism ϕ in Iso R ( F , F ′ ) such that α ′ ◦ e ϕ = α .Denote by X ( F , R ) the set of isomorphism classes of deformations of F over R .From the work of Gross and Hopkins in [HG94, Section 12], there exists a formal Z p -module F ⋆ ( t ) over Z p [[ t ]] satisfying the following properties:( i ) The reduction ^ F ⋆ (0) is a formal Z p -module over F p of height two.( ii ) The p -th power Frobenius endomorphism ϕ on ^ F ⋆ (0), given in coordinatesby ϕ ( X ) = X p , satisfies the relation ϕ = − p in End F p ( ^ F ⋆ (0)).( iii ) Denoting by Id the identity automorphism of ^ F ⋆ ( x ) = ^ F ⋆ (0), the map(2.6) M → X ( ^ F ⋆ (0) , R ) x ( F ⋆ ( x ) , Id) , is a bijection.Moreover, the bijection (2.6) is functorial on R , see [Haz78, Theorem 21.5.6]. Werefer to F ⋆ ( t ) as a universal formal Z p -module of height two. As a consequenceof ( iii ) the set X ( ^ F ⋆ (0) , R ) is parametrized by the open disc M of R . We usethis parametrization to endow X ( ^ F ⋆ (0) , R ) with the topology coming from thetopology on M inherited from R .Given Z p -formal modules F and F ′ defined over a subfield of k , we have thenatural map Iso k ( F , F ′ ) × X ( F , R ) → X ( F ′ , R )( β, ( F , α )) β · ( F , α ) := ( F , β ◦ α ) . INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 15
In particular, the group Aut k ( F ) acts on X ( F , R ). By fixing the parametriza-tion (2.6), we obtain an action of the group Aut k ( ^ F ⋆ (0)) on M .The following lemma is used several times. Lemma 2.4 ([HG94, Proposition 14.13]) . An element g of Aut k ( F ) fixes thepoint ( F , α ) in X ( F , R ) if and only if g belongs to the image of the injectivegroup homomorphism Aut R ( F ) → Aut k ( F ) given by ϕ α ◦ e ϕ ◦ α − . From elliptic curves to formal Z p -modules. Let R be either a subfieldof F p or the ring of integers of a finite extension of Q p inside C p . For an ellipticcurve e given by a Weierstrass equation with coefficients in R , and having smoothreduction if R has characteristic zero, denote by F e the formal group of e thatwe consider as a formal Z p -module, see, e.g. , [Blu98, Section 4]. Given e and e ′ as before, denote by φ b φ the natural morphism Hom R ( e, e ′ ) → Hom R ( F e , F e ′ ).This morphism is injective and compatible with addition and composition, see[Blu98, Proposition 5.1]. In the case where e ′ = e , it is a ring homomorphismEnd R ( e ) → End R ( F e ).Fix a universal formal Z p -module F ⋆ ( t ) as in Section 2.5. Recall that if e is asupersingular elliptic curve over F p , then the height of F e is two, see, e.g. , [Sil09,Chapter V, Theorem 3.1]. Moreover, Q p denotes the unique unramified quadraticextension of Q p inside C p , and Z p and F p the ring of integers and the residue fieldof Q p , respectively. Lemma 2.5.
Let e in Y sups ( F p ) be given. Then there is an elliptic curve e definedover F p representing e and such that there exists an isomorphism φ : ^ F ⋆ (0) → F e of formal Z p -modules defined over F p .Proof. Consider an elliptic curve e representing e that is given by a Weierstrassequation with coefficients in F p , such that the p -th power Frobenius endomor-phism Frob of e , defined in affine coordinates by Frob ( x, y ) = ( x p , y p ), satisfiesthe equation Frob = − p in End( e ), see, e.g. , [BGJGP05, Lemma 3.21]. Thus, theminimal polynomial of Frob over Z p is X + p . Since the minimal polynomial ofthe p -th Frobenius endomorphism of ^ F ⋆ (0) over Z p is the same, the existence of φ follows from [Haz78, Proposition 24.2.9]. (cid:3) For each e in Y sups ( F p ), fix e be as in the previous lemma and put F e := F e and φ e := φ . Then the algebra over Q p B e := End F p ( F e ) ⊗ Q p , is isomorphic to B p . Endow B e with its natural metric, as in Section 2.2. Moreover,identify R e := End F p ( F e ) and G e := Aut F p ( F e )with the unique maximal Z p -order in B e and with the group of units of this order,respectively, see, e.g. , [Fr¨o68, Chapter III, Section 2, Theorem 3]. In particular,both of these sets are metric subspaces of B e and therefore each right (resp. left)multiplication map on R e by an element of G e is an isometry.Since End( e ) is a maximal order in End( e ) ⊗ Q , and being a maximal order isa local property, see, e.g. , [Vig80, Chapitre
III, Section 5.A], it follows that thenatural map End( e ) ⊗ Z p → R e is an isomorphism. This natural map extends toan isomorphism End( e ) ⊗ Q p → B e . Given a finite extension K of Q p inside C p , put X e ( O K ) := X ( F e , O K ). Thenthe map X ( ^ F ⋆ (0) , O K ) → X e ( O K )( F , α ) φ e · ( F , α ) = ( F , φ e ◦ α ) . is a bijection. If K ′ is a finite extension of K inside C p , then each deformationof F e over O K can be considered as a deformation of F e over O K ′ , and this inducesa natural map X e ( O K ) → X e ( O K ′ ) that is injective [HG94, Proposition 12.10].Consider the direct limit X e ( O Q p ) := lim −→ X e ( O K ) , over the directed set of all finite extensions K of Q p inside C p , ordered by inclusion.As K runs through the finite extensions of Q p inside C p , the parametrizationof X e ( O K ) by M K given by (2.6) and the action of Aut π ( O K ) ( F e ) on X e ( O K )defined in Section 2.5, induce a parametrization of X e ( O Q p ) by M Q p and a groupaction of G e on X e ( O Q p ). The field of definition of an element of X e ( O Q p ) isdetermined by the corresponding parameter in M Q p , in the following sense: Forevery x in M Q p the associated deformation of F e can be defined over O Q p ( x ) andthis is the smallest extension of Q p inside C p where this deformation can be defined.Fix a completion b D e of X e ( O Q p ) and note that the parametrization of X e ( O Q p )by M Q p extends to a parametrization of b D e by M p . The following lemma impliesthat the action of G e on X e ( O Q p ) extends to a continuous map G e × b D e → b D e that is analytic in the second variable, see [HG94, Section 14, Proposition 19.2 andLemma 19.3]. Lemma 2.6.
For every e in Y sups ( F p ) , the following properties hold. ( i ) Each element of G e acts on b D e as an analytic automorphism with coeffi-cients in Z p . In particular, G e acts by isometries on b D e . ( ii ) For all integers N ≥ and r ≥ , every element g of G e in p N R e , andevery x in b D e satisfying ord p ( x ) ≥ r , we have ord p ( x − g · x ) ≥ N + 1 r . From formal Z p -modules to elliptic curves. Let R , π , M , k and k beas in Section 2.5, and let e be an elliptic curve defined over k . Denote by Y ( e, R )the space of isomorphism classes of pairs ( E, α ) formed by an elliptic curve E givenby a Weierstrass equation with coefficients in R and having smooth reduction,and an isomorphism α : e E → e defined over k , where two pairs ( E, α ) and ( E ′ , α ′ )are isomorphic if there exists an isomorphism ψ : E → E ′ defined over k such that α ′ ◦ e ψ = α . There is a natural action of Aut k ( e ) on Y ( e, R ) given for φ in Aut k ( e )by φ · ( E, α ) = (
E, φ ◦ α ).There is a natural map Y ( e, R ) → X ( F e , R )that associates to a class in Y ( e, R ) represented by a pair ( E, α ), the class in X ( F e , R )represented by the deformation ( F E , b α ). This map is known to be a bijection thanksto the so-called Woods-Hole Theory, see [LST64, Section 6] or [MC10, Theorem 4.1]. INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 17
Using this bijection together with the group homomorphism Aut k ( e ) → Aut k ( F e )given by φ b φ , we get an identificationAut k ( e ) \ Y ( e, R ) ∼ −→ Aut k ( e ) \ X ( F e , R ) . Choosing R = O K , with K a finite extension of Q p inside C p , and taking directlimits over all such field extensions, we obtain an identification(2.7) { E ∈ Y sups ( Q p ) : e E isomorphic to e over F p } → Aut( e ) \ X e ( O Q p ) . Identifying the left-hand side with D e ∩ Q p , we obtain a map(2.8) Π e : X e ( O Q p ) → Y sups ( Q p ) ∩ D e by composing the natural projection from X e ( O Q p ) to Aut( e ) \ X e ( O Q p ), with theinverse of (2.7).In the following theorem, δ e := e ) /
2. Note that δ e = 1 if j ( e ) = 0 , ≤ δ e ≤
12, see, e.g. , [Sil09, Appendix A, Propo-sition 1.2(c)].
Theorem 2.7.
Fix e in Y sups ( F p ) . Then, (2.8) extends to a map Π e : b D e → D e such that j ◦ Π e is represented by a power series with coefficients in Z p that is aramified covering of degree δ e . Moreover, for every x in b D e and every E in D e wehave (2.9) min {| x − x ′ | p : x ′ ∈ Π − e ( E ) } δ e ≤ | j (Π e ( x )) − j ( E ) | p ≤ min {| x − x ′ | p : x ′ ∈ Π − e ( E ) } . In particular, j ◦ Π e is an isometry if j ( e ) = 0 , .Proof. To prove the first assertion, note that the ring Z p [[ t ]] is a complete, localand Noetherian Z p -algebra whose residue field is isomorphic to F p . Using theparametrization of X ( F e , Z p [[ t ]]) by the maximal ideal of Z p [[ t ]], the element t corresponds to the deformation ( F ⋆ ( t ) , φ e ) of F e . Denote by E ( t ) the ellipticcurve class in Y ( e, Z p [[ t ]]) corresponding to the element Aut( e ) · ( F ⋆ ( t ) , φ e ) ofAut( e ) \ X ( F e , Z p [[ t ]]). Since the j -invariant j ( E ( t )) of E ( t ) is an element of Z p [[ t ]],to prove the first assertion of the theorem it is enough to prove that for ev-ery x in M Q p we have Π e ( F ⋆ ( x ) , φ e ) = E ( x ). Consider the evaluation map ψ : Z p [[ t ]] → O Q p defined by ψ ( f ( t )) := f ( x ), which is a continuous ring homomor-phism. Moreover, denote by e ψ the induced morphism on residue fields. Then ( ψ F ⋆ ( t ) , e ψφ e ) =( F ⋆ ( x ) , e ψφ e ) and the orbit Aut( e ) · ( ψ F ⋆ ( t ) , e ψφ e ) corresponds to the base changeof E ( t ) under ψ , which is E ( x ). This proves the equality Π e ( F ⋆ ( x ) , φ e ) = E ( x )and completes the proof of the first assertion of the theorem.To prove that j ◦ Π e is a ramified covering of degree δ e , it is sufficient toshow that for every E in an uncountable subset of D e we have − e ( E ) = δ e .Let E in Y sups ( Q p ) ∩ D e be such that j ( E ) = 0 , e the iden-tity in Aut( e ) and note that e and − e act trivially on X e ( O Q p ). Thus, foreach x in Π − e ( E ) the stabilizer of x for the action of Aut( e ) on X e ( O Q p ) con-tains { e , − e } . Let φ in Aut( e ) be in the stabilizer of x and let α : e E → e be anisomorphism such that ( F E , b α ) represents x . By Lemma 2.4 there is ϕ in Aut( E ) such that α ◦ e ϕ ◦ α − = φ . Together with our assumption j ( E ) = 0 , ϕ or − ϕ is the identity, see, e.g. , [Sil09, Appendix A, Proposition 1.2(c)].It follows that φ is in { e , − e } . This proves that the stabilizer of each elementof Π − e ( E ) is equal to { e , − e } . In particular we have − e ( E ) = δ e , as wanted.To prove (2.9), let E in D e be given and let x , . . . , x δ e be the zeros of j ◦ Π e − j ( E ), repeated according to multiplicity. Then, there is h ( t ) in Z p [[ t ]]such that | h | p is constant equal to 1 on O p and such that j ◦ Π e ( t ) − j ( E ) = h ( t ) · δ e Y i =1 ( t − x i ) , see, e.g. , [FvdP04, Exercise 3.2.2(1)]. Together with the fact that for every i in { , . . . , δ e } and x in b D e we havemin {| x − x ′ | p : x ′ ∈ Π − e ( E ) } ≤ | x − x i | p ≤ . This implies (2.9) and completes the proof of the theorem. (cid:3)
Hecke correspondences.
In this section we recall the construction and mainproperties of the Hecke correspondences. For details we refer the reader to [Shi71,Sections 7.2 and 7.3] for the general theory, or to the survey [DI95, Part II].Let K be an algebraically closed field of characteristic 0. First, note that forevery integer n ≥ D in Div( Y ( K )), we have(2.10) deg( T n ( D )) = σ ( n ) deg( D ) . Moreover, for n = 1 the correspondence T is by definition the identity on Div( Y ( K )).We also consider the linear extension of Hecke correspondences to Div( Y ( K )) ⊗ Q .For an integer N ≥
1, denote by Y ( N ) the modular curve of level N . It is aquasi-projective variety defined over Q . The points of Y ( N ) over K parametrizethe moduli space of equivalence classes of pairs ( E, C ), where E is an elliptic curveover K and C is a cyclic subgroup of E of order N . Here, two such pairs ( E, C )and ( E ′ , C ′ ) are equivalent if there exists an isomorphism φ : E → E ′ over K tak-ing C to C ′ . In particular, when N = 1, for every algebraically closed field K wecan parametrize Y ( K ) by Y (1)( K ), and Y (1) is isomorphic to the affine line A Q .For N >
1, denote by Φ N ( X, Y ) the modular polynomial of level N , which is asymmetric polynomial in Z [ X, Y ] that is monic in both X and Y , see, e.g. , [Lan87,Chapter 5, Sections 2 and 3]. This polynomial is characterized by the equality(2.11) Φ N ( j ( E ) , Y ) = Y C ≤ E cyclic of order N ( Y − j ( E/C )) for every E in Y ( K ) . This implies that a birational model for Y ( N ) is provided by the plane algebraiccurve(2.12) Φ N ( X, Y ) = 0 . For each prime number q , let α q , β q : Y ( q ) → Y (1) be the rational maps definedover Q given in terms of moduli spaces by α q ( E, C ) := E and β q ( E, C ) :=
E/C.
In terms of the model (2.12) with N = q , the rational maps α q and β q correspond tothe projections on the X and Y coordinate, respectively. Denote by ( α q ) ∗ and ( β q ) ∗ INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 19 the push-forward action of α q and β q on divisors, respectively. Denote also by α ∗ q the pull-back action of α q on divisors, defined at x in Y (1)( K ) by α ∗ q ( x ) := X y ∈ Y ( q )( K ) α q ( y )= x deg α q ( y )[ y ] , where deg α q ( y ) is the local degree of α q at y . This definition is extended by linearityto arbitrary divisors. The pull-back action β ∗ q of β q is defined in a similar way. Thenthe Hecke correspondence T q : Div( Y ( K )) → Div( Y ( K )) is recovered as(2.13) T q = ( α q ) ∗ ◦ β ∗ q = ( β q ) ∗ ◦ α ∗ q , where the second equality follows from the first and from the symmetry of T q .For an arbitrary integer n ≥
2, the correspondence T n can be recovered fromdifferent T q ’s, for q running over prime divisors of n , by using the identities(2.14) T ℓ ◦ T m = T ℓm for coprime ℓ and m in N ;(2.15) T q r = T q ◦ T q r − − q · T q r − for every prime number q and r ≥ . We consider the following action of Hecke correspondences on sets and compactlysupported measures. For each n in N and every subset A of C p , put T n ( A ) := [ a ∈ A supp( T n ( a )) . This defines an action of T n on sets that is compatible with the action on effective di-visors: T n (supp( D )) = supp( T n D ). To state further properties of this action, recallthat T n acts on the space C b ( C p ) of continuous and bounded functions F : C p → R ,by T n F ( E ) := F ( T n ( E )), see, e.g. , [HMRL20, Lemma 2.1]. Standard approxima-tion arguments show that the image of an open (resp. closed, compact) set by T n is a set of the same nature. To define the action of Hecke correspondences oncompactly supported measures, note that for each n in N the action of the Heckecorrespondence T n on C b ( C p ) is continuous. Then for each Borel measure µ on C p whose support is compact, the linear functional F R T n F d µ is continuous, andtherefore defines a Borel measure on C p supported on the compact set T n (supp( µ )).It is the push-forward of µ by T n that we denote by ( T n ) ∗ µ . Note that the sup-port of ( T n ) ∗ µ is equal to T n (supp( µ )), and that the total mass of ( T n ) ∗ µ is equalto σ ( n ) times the total mass of µ .Finally, note that for every x in X e ( O Q p ), every n in N that is not divisible by p ,and every isogeny φ in Hom n ( e, e ′ ), the isomorphism b φ belongs to Iso F p ( F e , F e ′ ),and(2.16) T n (Π e ( x )) | D ( e ′ ) = 1 e ′ ) X φ ∈ Hom n ( e,e ′ ) Π e ′ ( b φ · x ) . By continuity of T n , this holds for every x in b D e , see, e.g. , [HMRL20, Lemma 2.1].3. Asymptotic distribution of integer points on p -adic spheres The goal of this section is to prove the following result, from which we deduce ourresult on the asymptotic distribution of integer points on p -adic spheres (Theorem Din Section 1.3). Let n , Q , V m ( Q ), S ℓ ( Q ) and O Q ( Z p ) be as in Section 1.3. Given an integer r ≥ red r : Z np → ( Z /p r Z ) n the reduction map and by O Q ( Z /p r Z ) the corre-sponding orthogonal group of Q . This group is finite, acts on the finite set ( Z /p r Z ) n and for every ℓ in Z × p it leaves red r ( S ℓ ( Q )) invariant. Theorem 3.1 (Modular deviation estimate) . Let κ n be as in Theorem D and fixan integer r ≥ . Then for every ε > if n ≥ , and for every ε > and S ≥ if n = 3 , there is a constant C > such that the following property holds. Let Σ bean orbit of O Q ( Z /p r Z ) in ( Z /p r Z ) n and let m in N be such that V m ( Q ) = ∅ and red r ( V m ( Q )) ⊆ Σ . If n = 3 , then assume in addition that the largest square diving m is less than S .Then, for every σ in Σ we have (cid:12)(cid:12)(cid:12)(cid:12) { x ∈ V m ( Q ) : red r ( x ) = σ } V m ( Q ) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ C m n − κ n + ε V m ( Q ) . The following corollary is obtained from an application of Hensel’s lemma andit is proved in Section 3.2. Endow Z np with a distance compatible with its producttopology. Assume that for some ℓ in Z p r { } the set S ℓ ( Q ) is nonempty and thecompact group O Q ( Z p ) acts transitively on it. As in the statement of Theorem D,denote by µ ℓ the unique Borel probability measure on S ℓ ( Q ) that is invariant underthe action of O Q ( Z p ), see, e.g. , Lemma 5.3. This measure is uniquely determinedby the property that for every integer r ≥ σ in red r ( S ℓ ( Q )), we have(3.1) µ ℓ ( S ℓ ( Q ) ∩ red − r ( σ )) = 1 red r ( S ℓ ( Q )) . Corollary 3.2.
Let κ n be as in Theorem D and let δ > be given. Then forevery ε > if n ≥ , and for every ε > and S ≥ if n = 3 , there is aconstant C > and an integer N ≥ , such that the following property holds. Let ℓ in Z p r { } be such that S ℓ ( Q ) is nonempty and O Q ( Z p ) acts transitively on S ℓ ( Q ) .Moreover, let m in N be such that m ≡ ℓ mod p N and V m ( Q ) = ∅ . If n = 3 , then assume in addition that the largest square diving m is less than orequal to S . Then for every function F : Z np → R that is constant on every ball ofradius δ , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) V m ( Q ) X x ∈ V m ( Q ) F ( x ) − Z F d µ ℓ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C sup Z np | F | ! m n − κ n + ε V m ( Q ) . To prove it, we rephrase Theorem 3.1 in terms of a norm estimate on a cer-tain finite dimensional L function space (Lemma 3.6), as described for examplein [EMV13, Section 11.3]. The main ingredient to prove this L -norm estimateis the construction of an auxiliary modular form that has the key property of be-ing cuspidal (Proposition 3.3). The L -norm estimate is then deduced from thebounds for the Fourier coefficients of cuspidal modular forms shown by Delignefor n even [Del74], by Iwaniec [Iwa87] for n ≥ n = 3.The modular form is defined in Section 3.1, where we also show it is cuspidal.The proof of Theorems D and 3.1 are derived from this in Section 3.2. The proofof Corollary 3.2 is also given in Section 3.2. INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 21
Auxiliary modular form.
For a row vector x , we use x ⊺ to denote its trans-pose. Let n ≥ Q a positive definite quadratic form in Z [ X , X , . . . , X n ]and A Q the symmetric matrix in M n ( Z ) such that Q ( x ) = x A Q x ⊺ . Note thateach of the diagonal entries of A Q is even. The level N Q of A Q is the smallestpositive integer N such that N A − Q belongs to M n ( Z ).We consider the usual action of SL(2 , Z ) on the upper half-plane H := { τ ∈ C : ℑ ( τ ) > } , defined for γ = (cid:0) a bc d (cid:1) by γ ( τ ) := aτ + bcτ + d . For an integer N ≥
1, considerthe congruence subgroupsΓ ( N ) := (cid:8)(cid:0) a bc d (cid:1) ∈ SL(2 , Z ) : c ≡ N (cid:9) and Γ ( N ) := (cid:8)(cid:0) a bc d (cid:1) ∈ Γ ( N ) : a, d ≡ N (cid:9) . Given a nonempty finite set Σ, denote by F (Σ) the vector space of complexvalued functions defined on Σ, endowed with the inner product h f, g i Σ := X σ ∈ Σ f ( σ ) g ( σ ) . Denote by k · k Σ the corresponding norm and by Σ the constant function in F (Σ)equal to 1.The following is the main ingredient in the proof of Theorem 3.1. Proposition 3.3.
Fix an integer r ≥ and put N := lcm (cid:8) p r N Q , det( A Q )2 n +2 (cid:9) . Moreover, let Σ be a nonempty subset of ( Z /p r Z ) n r { } and let f be a functionin F (Σ) . Then, for τ in H , the series ϑ f ( τ ) := ∞ X m =0 X x ∈ V m ( Q ) red r ( x ) ∈ Σ f ( red r ( x )) exp(2 πimτ ) defines a modular form of weight n for the group Γ ( N ) in the sense of Shimura [Shi73] .If this modular form is cuspidal, then f is orthogonal to Σ in F (Σ) . If in addi-tion Σ is an orbit of O Q ( Z /p r Z ) , then this condition is also sufficient for ϑ f to becuspidal. After recalling basic properties of theta functions in Section 3.1.1, we prove themodularity of ϑ f ( τ ) in Section 3.1.2, which is obtained from work of Shimura [Shi73],following Hecke, Pfetzer and Schoeneberg. We complete the proof of Proposition 3.3in Section 3.1.3 by showing the cuspidality criterion.3.1.1. Preliminaries on theta series.
For an odd integer d , put ε d := 1 if d ≡ ε d := i if d ≡ − . Moreover, for an integer a denote by (cid:0) ad (cid:1) the extended quadratic residue symbol asdefined in [Shi73, p. 442], see also [Iwa97, p. 46].For M in N we consider the elements of ( Z /M Z ) n as row vectors. For ξ in ( Z /M Z ) n and N in N , denote by N · ξ the vector in ( Z / ( M N ) Z ) n that is equalto N x mod M N for every x in Z n such that x mod M = ξ . For ξ in ( Z /N Q Z ) n satisfying A Q ξ ⊺ = , define the theta function Θ( τ ; Q, ξ )for τ in H by Θ( τ ; Q, ξ ) := X x ∈ Z n x mod N Q = ξ exp (cid:0) πiQ ( x ) τ /N Q (cid:1) . It satisfies lim τ → i ∞ Θ( τ ; Q, ξ ) = ( ξ = ;0 if ξ = , (3.2) Θ( τ + 1; Q, ξ ) = exp (cid:0) πiQ ( ξ ) /N Q (cid:1) Θ( τ ; Q, ξ )(3.3)and for every c in N , Θ( τ ; Q, ξ ) = X ξ ′ ∈ ( Z /cN Q Z ) n ξ ′ mod N Q = ξ Θ( cτ ; cQ, ξ ′ ) . (3.4)Moreover, if for z in C r { } we denote by arg( z ) the argument of z taking valuesin ( − π, π ] and for r in R we put z r := | z | r exp( r arg( z ) i ), then(3.5) Θ (cid:18) − τ ; Q, ξ (cid:19) = ( − iτ ) n det( A Q ) X ξ ′ ∈ ( Z /N Q Z ) n A Q ( ξ ′ ) ⊺ = exp (cid:0) πi ( ξ ′ A Q ξ ⊺ ) /N Q (cid:1) Θ( τ ; Q, ξ ′ ) , see [Shi73, Section 2] or [Iwa97, Proposition 10.4]. Noting that for γ = (cid:0) a bc d (cid:1) in Γ (2 N Q ) the number d is odd, the properties above imply(3.6) Θ( γ ( τ ); Q, ξ )= exp (cid:0) πiabQ ( ξ ) /N Q (cid:1) (cid:18) det( A Q ) d (cid:19) (cid:18) cd (cid:19) n ε − nd ( cτ + d ) n Θ( τ ; Q, aξ ) , see [Shi73, Proposition 2.1 and comment (i) below it] or [Iwa97, Proposition 10.6( ii )]. Lemma 3.4.
For every γ = (cid:0) a bc d (cid:1) in SL(2 , Z ) with c > , we have lim τ → i ∞ Θ( γ ( τ ); Q, ξ )( − iτ ) n = 1det( A Q ) X ξ ′ ∈ ( Z /cN Q Z ) n ξ ′ mod N Q = ξ exp(2 πiaQ ( ξ ′ ) / ( cN Q )) . Proof.
By (3.3), (3.4) and the formula cγ ( τ ) = a − cτ + d , we haveΘ( γ ( τ ); Q, ξ ) = X ξ ′ ∈ ( Z /cN Q Z ) n ξ ′ mod N Q = ξ Θ( cγ ( τ ); cQ, ξ ′ )= X ξ ′ ∈ ( Z /cN Q Z ) n ξ ′ mod N Q = ξ exp(2 πiaQ ( ξ ′ ) / ( cN Q ))Θ (cid:18) − cτ + d ; cQ, ξ ′ (cid:19) . INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 23
On the other hand, by (3.5) for every ξ ′ in ( Z /cN Q Z ) n with ξ ′ mod N Q = ξ wehaveΘ (cid:18) − cτ + d ; cQ, ξ ′ (cid:19) = ( − i ( cτ + d )) n ( c n det( A Q )) X b ξ ∈ ( Z /cN Q Z ) n cA Q b ξ ⊺ = exp(2 πi b ξA Q ( ξ ′ ) ⊺ / ( cN Q ))Θ( cτ + d ; cQ, b ξ ) . Using (3.2) it follows thatlim τ → i ∞ Θ (cid:16) − cτ + d ; cQ, ξ ′ (cid:17) ( − i ( cτ + d )) n = 1( c n det( A Q )) and thereforelim τ → i ∞ Θ( γ ( τ ); Q, ξ )( − i ( cτ + d )) n = 1( c n det( A Q )) X ξ ′ ∈ ( Z /cN Q Z ) n ξ ′ mod N Q ≡ ξ exp(2 πiaQ ( ξ ′ ) / ( cN Q )) . Using that ( − i ( cτ + d )) n ∼ c n ( − iτ ) n as τ → i ∞ , we obtain the desired result. (cid:3) Modularity.
To prove that the series ϑ f ( τ ) in Proposition 3.3 is modular, wefirst observe that A p r Q = p r A Q , N p r Q = p r N Q and that for every σ in ( Z /p r Z ) n we have A p r Q ( N Q · σ ) ⊺ = in ( Z /p r N Q Z ) n , so the theta series Θ( τ ; p r Q, N Q · σ )is well defined.A direct computation shows that for every σ in ( Z /p r Z ) n , we haveΘ( p r τ ; p r Q, N Q · σ ) = X x ∈ Z n red r ( x )= σ exp(2 πiQ ( x ) τ )and therefore(3.7) ϑ f ( τ ) = X σ ∈ Σ f ( σ )Θ( p r τ ; p r Q, N Q · σ ) . Thus, to prove that ϑ f ( τ ) is modular for Γ ( N ) it is enough to show that for every σ in ( Z /p r Z ) n , the theta series Θ( p r τ ; p r Q, N Q · σ ) is modular for Γ ( N ).Let γ = (cid:0) a bc d (cid:1) in Γ (2 p r N Q ) ∩ Γ ( p r ) be given. Then a ( N Q · σ ) = N Q · σ andby (3.6) applied with γ replaced by (cid:16) a p r bc/p r d (cid:17) , we haveΘ( p r γ ( τ ); p r Q, N Q · σ )= Θ (cid:18) a ( p r τ ) + p r b ( c/p r )( p r τ ) + d ; p r Q, N Q · σ (cid:19) = (cid:18) det( p r A Q ) d (cid:19) (cid:18) c/p r ) d (cid:19) n ε − nd ( cτ + d ) n Θ( p r τ ; p r Q, N Q · σ ) . Note that the map m (cid:0) md (cid:1) is a completely multiplicative function, see, e.g. ,[Shi73, 3.(iii) and the last line in p. 442]. So we have (cid:18) det( p r A Q ) d (cid:19) (cid:18) c/p r ) d (cid:19) n = (cid:18) n det( A Q ) d (cid:19) (cid:16) cd (cid:17) n . Using that m (cid:16) n det( A Q ) m (cid:17) is a character modulo a divisor of 2 n +2 det( A Q ),it follows that if we assume in addition that d ≡ n +2 det( A Q ), then (cid:16) n det( A Q ) d (cid:17) = 1. Thus, if γ belongs to Γ (2 p r N Q ) ∩ Γ ( p r ) ∩ Γ (2 n +2 det( A Q )),then we haveΘ( p r γ ( τ ); p r Q, N Q · σ ) = (cid:16) cd (cid:17) n ε − nd ( cτ + d ) n Θ( p r τ ; p r Q, N Q · σ ) . This implies that Θ( p r τ ; p r Q, N Q · σ ) is a modular form of weight n for Γ ( N ) andthat the same holds for ϑ f ( τ ).3.1.3. Cuspidality.
In this section we complete the proof of Proposition 3.3, byproving the cuspidality criterion.For the cusp i ∞ , note that for every σ in ( Z /p r Z ) n r { } we have by (3.2)lim τ → i ∞ Θ( τ ; p r Q, N Q · σ ) = 0 , so by (3.7) and our assumption that is not in Σ, we have lim τ → i ∞ ϑ f ( τ ) = 0.To study the behavior of ϑ f ( τ ) at a different cusp, let σ be in ( Z /p r Z ) n r { } and take γ = (cid:0) a bc d (cid:1) in SL(2 , Z ) with c >
0. Let s be the largest integer in { , . . . , r } such that p s divides c . We have gcd ( p − s c, p r − s ) = 1, hence we can find j in Z suchthat jp − s c ≡ d mod p r − s . Note that e γ := (cid:18) p r − s a p s b − jap − s c p − r ( p s d − cj ) (cid:19) belongs to SL(2 , Z ) and (cid:18) p r
00 1 (cid:19) γ = e γ (cid:18) p s j p r − s (cid:19) . By Lemma 3.4 we havelim τ → i ∞ Θ( p r γ ( τ ); p r Q, N Q · σ )( − iτ ) n = lim τ → i ∞ Θ (cid:16)e γ (cid:16) p s τ + jp r − s (cid:17) ; p r Q, N Q · σ (cid:17) ( − iτ ) n = p (2 s − r ) n lim τ → i ∞ Θ ( e γ ( τ ) ; p r Q, N Q · σ )( − iτ ) n = p (2 s − r ) n det( A p r Q )) X ξ ∈ ( Z /p r − s cN Q Z ) n ξ mod p r N Q = N Q · σ exp(2 πiaQ ( ξ ) / ( cN Q ))= 1 p ( r − s ) n det( A Q ) X σ ′ ∈ ( Z /p r − s c Z ) n σ ′ mod p r = σ exp(2 πiaQ ( σ ′ ) /c ) . INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 25
Together with (3.7) this implies(3.8) lim τ → i ∞ ϑ f ( γ ( τ ))( − iτ ) n = 1 p ( r − s ) n det( A Q ) X σ ∈ Σ f ( σ ) X σ ′ ∈ ( Z /p r − s c Z ) n σ ′ mod p r = σ exp(2 πiaQ ( σ ′ ) /c ) . If ϑ f ( τ ) is cuspidal, then (3.8) with γ = (cid:0) −
11 0 (cid:1) implies that h f, Σ i Σ = 0. Thisproves the statement in Proposition 3.3 about the necessary condition for cuspidal-ity.The statement in Proposition 3.3 about the sufficient condition for cuspidality isa direct consequence of (3.8) and the following lemma, which completes the proofof Proposition 3.3. Lemma 3.5.
Let Σ be an orbit of O Q ( Z /p r Z ) in ( Z /p r Z ) n different from { } .Then for every a in Z and every c , r and t in N such that p r | p t c , the function E a,c : Σ → C σ E a,c ( σ ) := P σ ′ ∈ ( Z /p t c Z ) n σ ′ mod p r = σ exp(2 πiaQ ( σ ′ ) /c ) is constant.Proof. Write c = p ℓ c with ℓ ≥ p ∤ c . Choosing A and B in Z with Ap t + ℓ + Bc = 1 gives an isomorphism( Z /c Z ) n × ( Z /p t + ℓ Z ) n → ( Z /p t c Z ) n ( µ, ν ) Ap t + ℓ · µ + Bc · ν. This implies that for every σ in Σ we have E a,c ( σ ) = X µ ∈ ( Z /c Z ) n X ν ∈ ( Z /p t + ℓ Z ) n ν mod p r = σ exp (cid:0) πiaQ ( Ap t + ℓ · µ + Bc · ν ) / ( p ℓ c ) (cid:1) = X µ ∈ ( Z /c Z ) n exp (cid:0) πiaA p t + ℓ Q ( µ ) /c (cid:1) · X ν ∈ ( Z /p t + ℓ Z ) n ν mod p r = σ exp (cid:0) πiaB c Q ( ν ) /p ℓ (cid:1) = X µ ∈ ( Z /c Z ) n exp (cid:0) πiaA p t + ℓ Q ( µ ) /c (cid:1) E aB c ,p ℓ ( σ ) . Hence, we can assume c = p ℓ .Let σ and b σ in Σ be given. Our hypothesis that Σ is an orbit of O Q ( Z /p r Z )implies that there is T in O Q ( Z /p t + ℓ Z ) such that T ( σ ) = b σ . Noting that { ν ∈ ( Z /p t + ℓ Z ) n : ν mod p r = σ } → { b ν ∈ ( Z /p t + ℓ Z ) n : b ν mod p r = b σ } ν T ( ν ) is a bijective map, we obtain E a,p ℓ ( b σ ) = X ν ∈ ( Z /p t + ℓ Z ) n ν mod p r = σ exp (cid:0) πiaQ ( T ( ν )) /p ℓ (cid:1) = X ν ∈ ( Z /p t + ℓ Z ) n ν mod p r = σ exp (cid:0) πiaQ ( ν ) /p ℓ (cid:1) = E a,p ℓ ( σ ) . This completes the proof of the lemma. (cid:3)
Proof of Theorem D.
The proofs of Theorems D and 3.1 are given after thefollowing lemma and that of Corollary 3.2 is given at the end of this section.
Lemma 3.6.
Fix an integer r ≥ , let m in N be such that V m ( Q ) is nonempty andlet Σ be a subset of ( Z /p r Z ) n containing red r ( V m ( Q )) . Then, for every orthonormalbasis B of the orthogonal complement of Σ in F (Σ) we have Var( m, Σ) := X σ ∈ Σ (cid:18) { x ∈ V m ( Q ) : red r ( x ) = σ } V m ( Q ) − (cid:19) = 1 V m ( Q ) X f ∈B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x ∈ V m ( Q ) f ( red r ( x )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Proof.
Consider the function F : Σ → C given by F ( σ ) := { x ∈ V m ( Q ) : red r ( x ) = σ } . We have h F, Σ i Σ = V m ( Q ) and(3.9) (cid:13)(cid:13)(cid:13)(cid:13) F − h F, Σ i Σ Σ (cid:13)(cid:13)(cid:13)(cid:13) = X σ ∈ Σ (cid:18) { x ∈ V m ( Q ) : red r ( x ) = σ } − V m ( Q ) (cid:19) = V m ( Q ) · Var( m, Σ) . One the other hand, since B is an orthonormal basis for the orthogonal complementof Σ in F (Σ), we have F − h F, Σ i Σ Σ = X f ∈B h F, f i Σ f and therefore (cid:13)(cid:13)(cid:13)(cid:13) F − h F, Σ i Σ Σ (cid:13)(cid:13)(cid:13)(cid:13) = X f ∈B |h F, f i Σ | = X f ∈B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X σ ∈ Σ { x ∈ V m ( Q ) : red r ( x ) = σ } f ( σ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = X f ∈B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x ∈ V m ( Q ) f ( red r ( x )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Together with (3.9) this implies the desired identity. (cid:3)
INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 27
Proof of Theorem 3.1.
Since ( Z /p r Z ) n is finite, it is sufficient to prove the desiredestimate for a given orbit Σ of O Q ( Z /p r Z ). The case Σ = { } being trivial,assume Σ ⊆ ( Z /p r Z ) n r { } .Let B be an orthonormal basis of the orthogonal complement of Σ in F (Σ).By Lemma 3.6, for every σ in Σ we have(3.10) (cid:12)(cid:12)(cid:12)(cid:12) { x ∈ V m ( Q ) : red r ( x ) = σ } V m ( Q ) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ p Var( m, Σ) = 1 V m ( Q ) X f ∈B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x ∈ V m ( Q ) f ( red r ( x )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Since each f in B is orthogonal to Σ , by Proposition 3.3 the modular form ϑ f is cuspidal of weight n for Γ ( N ). When n ≥
4, for every ε >
C > f and ε , such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x ∈ V m ( Q ) f ( red r ( x )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = | m -th Fourier coefficient of ϑ f | ≤ Cm n − κ n + ε , by Deligne’s bound [Del74, Th´eor`eme n is even and by Iwaniec’s bound[Iwa87, Theorem 1] if n is odd. When n = 3 the same estimate holds for a con-stant C that also depends on S , by Duke’s [Duk88, Theorem 5] and Blomer’s [Blo04,Lemma 4.4] bounds. This implies the result. (cid:3) Remark . The bounds for the Fourier coefficients of cuspidal modular forms usedin the above proof are usually stated in the literature for cuspidal modular formsfor Γ ( N ) with characters. It is known that these bounds also hold for cuspidalmodular forms for Γ ( N ) since every such form can be written as a finite sum ofcuspidal modular forms for Γ ( N ) with characters, see, e.g. , [Miy89, Lemma 4.3.1](the proof given there extends to the case of half-integral weight modular forms). Proof of Theorem D.
Since the set of locally constant functions Z np → R is densein the space of continuous functions Z np → R , it is sufficient to show that for everylocally constant function F : Z np → R we have1 V m j ( Q ) X x ∈ V mj ( Q ) F ( M − u j ( x )) → Z F d µ ℓ as j → ∞ . Let r ≥ σ in Z /p r Z the function F is constant on red − r ( σ ) and let f : ( Z /p r Z ) n → R be the function determinedby F = f ◦ red r . Let ε > δ := c − ( n − κ n + ε ) > C be theconstant given by Theorem 3.1Our hypotheses that S ℓ ( Q ) is nonempty and that O Q ( Z p ) acts transitively on S ℓ ( Q ),imply that for every ℓ ′ in ℓ ( Z × p ) the set red r ( S ℓ ′ ( Q )) is nonempty and O Q ( Z /p r Z )acts transitively on red r ( S ℓ ′ ( Q )). In particular, for each j in N this appliesto ℓ ′ = m j and M − u j maps S m j ( Q ) to S ℓ ( Q ) and µ m j to µ ℓ . Note that M u j definesby reduction modulo p r an element of GL n ( Z /p r Z ) that we denote by M u j ,r . Apply-ing for each sufficiently large j Theorem 3.1 with Σ = red r ( S m j ( Q )) and m = m j , we obtain that for every σ in red r ( S m j ( Q ))(3.11) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:8) x ∈ V m j ( Q ) : red r ( x ) = σ (cid:9) V m j ( Q ) − red r ( S m j ( Q )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C m n − κ n + εj V m j ( Q ) ≤ Cm − δj . On the other hand, by the change of variables formula and (3.1) we have Z F d µ ℓ = Z F d( M − u j ) ∗ µ m j = Z F ◦ M − u j d µ m j = X σ ∈ red r ( S mj ( Q )) f ( M − u j ,r ( σ )) red r ( S m j ( Q )) . Together with (3.11), this implies (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) V m j ( Q ) X x ∈ V mj ( Q ) F ( M − u j ( x )) − Z F d µ ℓ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C X σ ′ ∈ red r ( S ℓ ( Q )) | f ( σ ′ ) | m − δj , from which the desired assertion follows. (cid:3) The proof of Corollary 3.2 is given after the following lemma.
Lemma 3.8.
Let r ≥ be an integer and let ℓ and m in Z p r { } be such that (3.12) | m − ℓ | p < | ℓ | p and | m − ℓ | p ≤ | ℓ | p p − r . Then red r ( S ℓ ( Q )) = red r ( S m ( Q )) . In particular, if in addition m belongs to N ,then red r ( V m ( Q )) ⊆ red r ( S ℓ ( Q )) .Proof. For a given x = ( x , . . . , x n ) in S m ( Q ), the relation2 Q ( x ) = n X i =1 x i · ∂ X i Q ( x )implies | ℓ | p = | m | p = | Q ( x ) | p ≤ | | − p · max i ∈{ ,...,n } {| ∂ X i Q ( x )) | p } , and therefore | Q ( x ) − ℓ | p = | m − ℓ | p < | ℓ | p ≤ max i ∈{ ,...,n } {| ∂ X i Q ( x ) | p } . Hence we can apply Hensel’s Lemma and find x ′ = ( x ′ , . . . , x ′ n ) in S ℓ ( Q ) such thatmax i ∈{ ,...,n } {| x ′ i − x i | p } ≤ | Q ( x ) − ℓ | p max i ∈{ ,...,n } {| ∂ X i Q ( x ) | p } ≤ | m − ℓ | p | ℓ | p ≤ p − r . In particular, x ′ ≡ x mod p r . This proves that x belongs to red r ( S ℓ ( Q )) andtherefore that red r ( S m ( Q )) ⊆ red r ( S ℓ ( Q )).The reverse inclusion is obtained by symmetry. (cid:3) INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 29
Proof of Corollary 3.2.
Let r ≥ σ in Z /p r Z the set red − r ( σ ) is contained in a ball of radius δ . Then Σ := red r ( S ℓ ( Q ))is an orbit of O Q ( Z /p r Z ). Moreover, if we put N := max { p (2 ℓ ) + 1 , ord p (2 ℓ ) + r } and if m is as in the statement of the corollary, then (3.12) is satisfied and byLemma 3.8 we have red r ( V m ( Q )) ⊆ Σ. Then the desired estimate follows fromTheorem 3.1. (cid:3) CM points formulae
In this section we give several formulae for (formal) CM points having super-singular reduction. The first formula is for CM points whose discriminant is fun-damental. We write such CM points as (projections of) fixed points of certainelements of the group action described in Section 2.6 (Theorem 4.2 in Section 4.1).For an integer r ≥ D whose conductor is not divisible by p ,the second formula relates Λ Dp r to Λ D using the canonical branch t of T p (Theo-rem 4.6 in Section 4.2). Finally, we give analogous formulae for formal CM pointsand describe the relation between CM and formal CM points (Theorem 4.11 andCorollary 4.12 in Section 4.3).In this section and for the rest of the paper, for every discriminant D we con-sider Λ D as a divisor.4.1. CM points as fixed points.
Throughout this section, fix e in Y sups ( F p ) andlet B e , R e , G e , b D e be as in Section 2.6. The Gross lattice associated to e is the Z -lattice of dimension three L ( e ) := { φ ∈ Z + 2 End( e ) : tr( φ ) = 0 } . It plays a central role in this section. Define for each integer m ≥ V m ( e ) := { φ ∈ L ( e ) : nr( φ ) = m } . Given a p -supersingular fundamental discriminant d , the goal of this section isto write every CM point of discriminant d in D e as the projection of a fixed pointof a certain element of the group action of G e on b D e . This is done in two steps.First, we define for each φ in V | d | ( e ) a certain unit U e ( b φ ) in the ring of integers ofthe subalgebra Q p ( b φ ) of B e (Lemma 4.1). The second step is to show that as φ varies over V | d | ( e ), the projections of the fixed points of U e ( b φ ) in b D e run throughall CM points in D e of discriminant d (Theorem 4.2).To state these results, we introduce some notation. The image of L ( e ) ⊗ Z p bythe natural isomorphism End( e ) ⊗ Z p → R e , is given by(4.1) L e := { ϕ ∈ Z p + 2 R e : tr( ϕ ) = 0 } . This set is compact because R e is compact and the reduced trace function is contin-uous. Note also that for every nonzero ϕ in L e , the p -adic number − nr( ϕ ) belongsto a p -adic discriminant. This motivates the definition, L e, f := { ϕ ∈ L e : − nr( ϕ ) belongs to a fundamental p -adic discriminant } . This set coincides with the set of all elements ϕ of B e such that ϕ belongs to a fun-damental p -adic discriminant. Moreover, for every p -supersingular discriminant D whose conductor is not divisible by p , the set V | D | ( e ) is mapped inside L e, f by themap φ b φ , see Lemma 2.1. Lemma 4.1 (Unit function) . Let U e : L e, f → B e be the function defined by U e ( ϕ ) := ( ϕ + ϕ if ϕ + ϕ belongs to G e ;1 + ϕ + ϕ otherwise . Then U e takes values in G e and for every ϕ in L e, f the following properties hold. ( i ) The subalgebra Q p ( ϕ ) of B e is a field extension of Q p that is isomorphic tothe subfield Q p ( p ϕ ) of C p . ( ii ) We have O Q p ( ϕ ) = Z p [ U e ( ϕ )] , U e ( ϕ ) is a unit in O Q p ( ϕ ) and ∆( U e ( ϕ )) be-longs to a fundamental p -adic discriminant.Proof. Since ϕ = − nr( ϕ ) and − nr( ϕ ) belongs to a fundamental p -adic discrimi-nant, we conclude that ϕ is not in ( Q p ) and obtain item ( i ). On the other hand,(A.6) in Lemma A.2( ii ) implies that O Q p ( ϕ ) = Z p h ϕ + ϕ i = Z p [ U e ( ϕ )] . In particular, ϕ + ϕ belongs to R e and therefore U e ( ϕ ) belongs to G e and it isa unit in Z p [ U e ( ϕ )]. Finally, noting that ∆( U e ( ϕ )) = − nr( ϕ ), we also obtainthat ∆( U e ( ϕ )) belongs to a fundamental p -adic discriminant. This completes theproof of item ( ii ) and of the lemma. (cid:3) For each ϕ in L e, f , defineFix e ( ϕ ) := n x ∈ b D e : U e ( ϕ ) · x = x o . Given a fundamental discriminant d and an integer f ≥
1, put w d,f := (cid:16) O × d,f / Z × (cid:17) = ( O × d,f ) / . Note that w − , = 3, w − , = 2 and that in all the remaining cases w d,f = 1. Theorem 4.2 (Fixed points formula) . Let d be a p -supersingular fundamentaldiscriminant. Then for every e in Y sups ( F p ) , we have (4.2) Λ d | D e = w d, e ) X φ ∈ V | d | ( e ) X x ∈ Fix e ( b φ ) Π e ( x ) . The proof of this theorem is at the end of this section. It is based on a versionof Deuring’s lifting theorem for formal Z p -modules, in the spirit of [Gro86, Propo-sition 2.1]. To state it, we introduce the following notation. For a formal group F over a ring R , denote by D F : End R ( F ) → R the ring homomorphism such thatfor every ϕ in End R ( F ) we have in coordinates ϕ ( X ) ≡ D F ( ϕ ) X mod X . Moreover, for a ring homomorphism δ : R → O Q p , denote by e δ : R → F p the com-position of δ with the reduction morphism O Q p → F p . INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 31
Proposition 4.3 (Lifting formal modules) . Let e be in Y sups ( F p ) . Let g in G e be such that the subalgebra Q p ( g ) of B e is a field extension of Q p of degree twowith ring of integers Z p [ g ] . Then there is a bijection between the fixed points of g in X e ( O Q p ) and the continuous ring homomorphisms δ : Z p [ g ] → O Q p satisfy-ing D F e | Z p [ g ] = e δ . For such a δ , the corresponding fixed point ( F , α ) of g is definedover the ring of integers O K of K := Q p ( δ ( g )) and it is uniquely determined by theproperty that the unique automorphism ϕ in Aut O K ( F ) such that g = α ◦ e ϕ ◦ α − satisfies D F ( ϕ ) = δ ( g ) .Proof. We first show how to assign to each fixed point ( F , α ) of g in X e ( O Q p ) acontinuous ring homomorphism δ : Z p [ g ] → O Q p as in the statement. Consider thering homomorphism ι : End( F ) → R e defined by ι ( ϕ ) := α ◦ e ϕ ◦ α − , which is continuous, see, e.g. , [Fr¨o68, Chapter IV, Section 1, Proposition 3]. Let ϕ in End( F ) be such that ι ( ϕ ) = g (Lemma 2.4). Then ι induces a continuous ringisomorphism ι : Z p [ ϕ ] → Z p [ g ]. Then the ring homomorphism δ := D F ◦ ι − : Z p [ g ] → O Q p , is such that δ ( g ) = D F ( ι − ( g )) = D F ( ϕ )and such that for every g in Z p [ g ] we have e δ ( g ) = ^ D F ( ι − ( g )) = D e F ( ^ ι − ( g )) = D F e ( g ) . Note that δ is continuous because D F is continuous, see [Fr¨o68, Chapter IV, Sec-tion 1, Corollary 3]. This proves that δ satisfies the desired properties.Let δ : Z p [ g ] → Q p be a continuous ring homomorphism satisfying D F e | Z p [ g ] = e δ and put K := Q p ( δ ( g )) ⊂ Q p . We now show that there is a fixed point ( F , α ) in X e ( O K ) whose correspondingring homomorphism is δ . Endow O K and F p with the structure of a Z p [ g ]-modulewith structural map δ and D F e | Z p [ g ] , respectively. Then the inclusion map of Z p [ g ]in R e gives F e the structure of a formal Z p [ g ]-module over F p in the sense ofDrinfel’d, see [Dd74, Section 1]. This formal Z p [ g ]-module is of height one, see, e.g. , [Dd74, Remark, p. 566]. Then there is a unique deformation ( F , α ) of theformal Z p [ g ]-module F e and this deformation is defined over O K , see [HG94, Propo-sition 12.10]. Denote by ϕ the image of g in End O K ( F ) by the structural map.Then by definition we have D F ( ϕ ) = δ ( g ). On the other hand, since α : e F → F e is an isomorphism of formal Z p [ g ]-modules, we have g = α ◦ e ϕ ◦ α − . By Lemma 2.4this proves that ( F , α ), seen as a formal Z p -module over O K that is a deformationof F e , is a fixed point of g .It remains to prove the uniqueness statement. Let ( F , α ) in X e ( O Q p ) be anotherfixed point of g , let ϕ be given by Lemma 2.4 and suppose that(4.3) D F ( ϕ ) = δ ( g ) . Let K ′ be a finite extension of K contained in Q p such that ( F , α ) is in X e ( O K ′ ).Consider O K ′ as a Z p [ g ]-module with structural map δ and consider the ringisomorphism ι : Z p [ ϕ ] → Z p [ g ], as above. Then the equality in (4.3) ensures that the ring homomorphim ι − : Z p [ g ] → End O K′ ( F ) endows F with a structure offormal Z p [ g ]-module over O K ′ . Finally, since the deformation space of the formal Z p [ g ]-module F e consists of a single point, ( F , α ) and ( F , α ) are both isomorphicas deformations of F e as a formal Z p [ g ]-module. It follows that they are isomorphicas deformations of F e as a formal Z p -module. This proves the uniqueness statementand completes the proof of the proposition. (cid:3) Remark . Proposition 4.3 is related to [Gro86, Proposition 2.1] as follows. Let g and δ be as in the above proposition, and put K := Q p ( δ ( g )). The inverse of δ givesan embedding ι † : O K → R e that is normalized in the sense of [Gro86, Section 2],and the unique fixed point of g in X e ( O Q p ) attached to δ is the canonical lifting of the pair ( F e , ι † ) in the sense of [Gro86, Section 3]. Lemma 4.5.
For every e in Y sups ( F p ) , the following properties hold. ( i ) For each element g of G e r Z × p , every fixed point of g in b D e is in X e ( O Q p ) . ( ii ) Let ϕ be in L e, f . If Q p ( ϕ ) is ramified (resp. unramified) over Q p , then Fix e ( ϕ ) has precisely two elements (resp. one element). ( iii ) Let g in G e r Z × p be such that Z p [ g ] = O Q p ( g ) . Then an element g ′ of G e r Z × p has a common fixed point with g in b D e if and only if g ′ is in Q p ( g ) . ( iv ) For ϕ and ϕ ′ in L e, f the sets Fix e ( ϕ ′ ) and Fix e ( ϕ ) coincide if ϕ ′ belongsto Q p ( ϕ ) and they are disjoint if ϕ ′ is not in Q p ( ϕ ) .Proof. Item ( i ) is a direct consequence of the fact that g acts as a power series f with coefficients in Z p (Lemma 2.6( i )), applying, e.g. , [FvdP04, Exercise 3.2.2(1)]to the restriction of the power series f ( z ) − z to an affinoid subdomain of b D e containing a given fixed point of g .To prove item ( ii ), note that the number of continuous ring homomorphisms O Q p ( ϕ ) → O Q p that reduce to D F e | O Q p ( ϕ ) is equal to two (resp. one) if Q p ( ϕ ) is ram-ified (resp. unramified) over Q p . Since by Lemma 4.1 we have Z p [ U e ( ϕ )] = O Q p ( ϕ ) ,the desired assertion is given by Proposition 4.3 with g = U e ( ϕ ).To prove item ( iii ), consider a fixed point of g in b D e . By item ( i ) this pointis in X e ( O Q p ) and therefore it is represented by a pair ( F , α ). If g ′ fixes ( F , α ),then by Lemma 2.4 both g and g ′ are in the image of the map Aut( F ) → G e given by φ α ◦ e φ ◦ α − . By our assumption that g is not in Z × p and [Fr¨o68,Chapter IV, Section 1, Theorem 1( iii )], this implies that g ′ is in Q p ( g ). Conversely,every element of ( Z p [ g ]) × = G e ∩ Q p ( g ) is in the image of the map φ α ◦ e φ ◦ α − and therefore it fixes ( F , α ) by Lemma 2.4. This completes the proof of item ( ii ).To prove item ( iv ), suppose that U e ( ϕ ) and U e ( ϕ ′ ) have a common fixed point.By item ( iii ) we have Q p ( ϕ ) = Q p ( ϕ ′ ). Consider an arbitrary element x of Fix e ( ϕ ).By item ( i ) the point x is in X e ( O Q p ) and therefore it is represented by a pair ( F , α ).By Lemma 2.4, the image of the map Aut( F ) → G e given by φ α ◦ e φ ◦ α − is equalto O × Q p ( ϕ ) and therefore to O × Q p ( ϕ ′ ) . Using Lemma 2.4 again, we conclude that ( F , α )is in Fix e ( ϕ ′ ). This proves that Fix e ( ϕ ) is contained in Fix e ( ϕ ′ ). Reversing theroles of ϕ and ϕ ′ , we conclude that these sets are equal. This completes the proofof item ( iv ) and of the lemma. (cid:3) INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 33
Let d be a p -supersingular fundamental discriminant and put ǫ d := ( p ramifies in Q ( √ d );1 / p is inert in Q ( √ d ) . For each e in Y sups ( F p ) and each discriminant D of the form D = df with f ≥ h ( D, e ) the number of conjugacy classes of optimal embeddings O d,f → End( e ). Then(4.4) deg(Λ d | D e ) = ǫ d h ( d, e ) , which is a consequence of the work of Deuring [Deu41], see [EOY05, Lemma 3.3]. Proof of Theorem 4.2.
By [Gro87, (12.8) and Proposition 12.9] we have V | d | ( e ) = e )2 w d, h ( d, e ) . The right-hand side of (4.2) has integer coefficients and by Lemma 4.5( ii ) and (4.4)its degree is equal to w d, e ) V | d | ( e )2 ǫ d = ǫ d h ( d, e ) = deg(Λ d | D e ) . Hence, it is enough to prove that supp(Λ d | D e ) is contained in the support of thedivisor at the right-hand side of (4.2). To do this, let E be in supp(Λ d | D e ) andlet α : e E → e be an isomorphism. Since E is a CM point, it is defined over Q p and therefore E = Π e (( F E , b α )). On the other hand, since End( E ) is isomorphicto O d, = Z h d + √ d i , there exists an element φ in Z + 2 End( E ) satisfying theequation X − d = 0. This implies that the endomorphism φ := α ◦ e φ ◦ α − of e belongs to L ( e ) and satisfiestr( φ ) = 0 and nr( φ ) = | d | . That is, φ belongs to V | d | ( e ). On the other hand, note that the element b φ of R e is the image of b φ by the ring homomorphism ι : End( F E ) → R e ϕ ι ( ϕ ) := b α ◦ e ϕ ◦ b α − . Since End( E ) contains d + φ , it follows that Z p h d + b φ i is contained in the imageof ι . Noting that Z p h U e ( b φ ) i = Z p h d + b φ i , Lemma 4.1( ii ) with ϕ = b φ impliesthat U e ( b φ ) is a unit in Z p h d + b φ i . It follows that U e ( b φ ) is in the image of Aut( F E )by ι . Then Lemma 2.4 implies that ( F E , b α ) is a fixed point of U e ( b φ ). This provesthat E = Π e (( F E , b α )) is contained in the support of the right-hand side of (4.2)and completes the proof of the theorem. (cid:3) CM points and the canonical branch of T p . The goal of this section is toprove the following formulae for CM points in Y sups ( C p ) for which the conductor ofits discriminant is divisible by p . This formula is stated in terms of the canonicalbranch of T p that we proceed to recall. Consider Katz’ valuation v p on Y sups ( C p ),as defined in [HMRL20, Section 4.1] and put N p := (cid:26) E ∈ Y sups ( C p ) : v p ( E ) < pp + 1 (cid:27) . For E in N p , denote by H ( E ) the canonical subgroup of E [Kat73, Theorem 3.10.7].The canonical branch of T p is the map t : N p → Y sups ( C p ) defined by t ( E ) := E/H ( E ). Theorem 4.6.
Let d be a p -supersingular fundamental discriminant. Then forevery integer r ≥ and every integer f ≥ that is not divisible by p , we have Λ d ( fp r ) = t ∗ (cid:16) Λ df w d,f (cid:17) (cid:12)(cid:12) v − p ( p ) if r = 1 and p ramifies in Q ( √ d );( t ∗ ) r − (Λ d ( fp ) ) if r ≥ and p ramifies in Q ( √ d );( t ∗ ) r (cid:16) Λ df w d,f (cid:17) if r ≥ and p is inert in Q ( √ d ) . The proof of this theorem is at the end of this section.
Lemma 4.7 ([HMRL20, Theorem B.1]) . The canonical branch t of T p is given bya finite sum of Laurent series, each of which converges on all of N p . Furthermore,for every E in Y sups ( C p ) we have (4.5) T p ( E ) = ( t ∗ ( E ) + [ t ( E )] if v p ( E ) ≤ p +1 ; t ∗ ( E ) if v p ( E ) > p +1 . The following is [HMRL20, Lemma 4.6], which is a reformulation in our con-text of [Kat73, Theorems 3.1 and 3.10.7], see also [Buz03, Theorem 3.3]. Let b v p : Y sups ( C p ) → h , pp +1 i be the map defined by b v p := min (cid:26) v p , pp + 1 (cid:27) . Lemma 4.8.
For every E in N p we have (4.6) b v p ( t ( E )) = pv p ( E ) if v p ( E ) ∈ i , p +1 i ;1 − v p ( E ) if v p ( E ) ∈ i p +1 , pp +1 h , and for every subgroup C of E of order p that is different from H ( E ) we have (4.7) v p ( E/C ) = p − v p ( E ) . Furthermore, the following properties hold. ( i ) Let E be in Y sups ( C p ) and let C be a subgroup of E of order p . In the casewhere v p ( E ) < pp +1 , assume in addition that C = H ( E ) . Then v p ( E/C ) = p − b v p ( E ) and t ( E/C ) = E. ( ii ) For E in Y sups ( C p ) satisfying p +1 < v p ( E ) < pp +1 , we have t ( E ) = E . The following lemma is [HMRL20, Lemma 4.9], see also [CM06, Lemma 4.8] and[Gro86, Proposition 5.3].
Lemma 4.9.
Let D be a p -supersingular discriminant and m ≥ the largest integersuch that p m divides the conductor of D . Then for every E in supp(Λ D ) we have b v p ( E ) = ( · p − m if p ramifies in Q ( √ D ); pp +1 · p − m if p is inert in Q ( √ D ) . †† When m = 0 and p is inert in Q ( √ D ), we have v p ( E ) ≥ b v p by the valuation v p . Compare with[CM06, Lemma 4.8]. INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 35
The following lemma gathers some variants of a formula of Zhang in [Zha01,Proposition 4.2.1], see also [CU04,
Lemme
Dirichlet convolution of two functions g, e g : N → C , is defined by( g ∗ e g )( n ) := X d ∈ N ,d | n g ( d ) e g (cid:16) nd (cid:17) . Denote by the constant function defined on N and taking the value 1. Given afundamental discriminant d , denote by ψ d : N → {− , , } the arithmetic functiongiven by the Kronecker symbol (cid:0) d · (cid:1) , put R d := ∗ ψ d and denote by R − d the inverseof R d with respect to the Dirichlet convolution. Lemma 4.10.
For every fundamental discriminant d and every pair of coprimeintegers f ≥ and e f ≥ , we have the relations (4.8) Λ d ( f e f ) w d,f e f = X f ∈ N ,f | f R − d (cid:18) ff (cid:19) T f Λ d e f w d, e f ! . If in addition f is not divisible by p , then we have (4.9) Λ d ( pf ) = T p (cid:16) Λ df w d,f (cid:17) − Λ df w d,f if p ramifies in Q ( √ d ); T p (cid:16) Λ df w d,f (cid:17) if p is inert in Q ( √ d ) , and for every integer m ≥ we have (4.10) Λ d ( p m f ) = T p m (cid:16) Λ df w d,f (cid:17) − T p m − (cid:16) Λ df w d,f (cid:17) if p ramifies in Q ( √ d ); T p m (cid:16) Λ df w d,f (cid:17) − T p m − (cid:16) Λ df w d,f (cid:17) if p is inert in Q ( √ d ) . Proof of Theorem 4.6.
First we show that if p ramifies (resp. is inert) in Q ( √ d ),then for every r ≥ r ≥
2) we have(4.11) T p (Λ d ( fp r ) ) = Λ d ( fp ( r +1) ) + p Λ d ( fp ( r − ) w d,fp r − . We use several times the recursive relation (2.15) and the formulae (4.9) and (4.10)in Lemma 4.10. If p ramifies in Q ( √ d ), then w d,f T p (Λ d ( fp ) ) = T p ( T p (Λ df ) − Λ df ) = T p (Λ df ) + p Λ df − T p (Λ df )= w d,f Λ d ( fp ) + p Λ df and for every r ≥ w d,f T p (Λ d ( fp r ) ) = T p ( T p r (Λ df ) − T p r − (Λ df ))= T p r +1 (Λ df ) + pT p r − (Λ df ) − T p r (Λ df ) − pT p r − (Λ df ))= w d,f (Λ d ( fp ( r +1) ) + p Λ d ( fp ( r − ) ) . On the other hand, if p is inert in Q ( √ d ), then w d,f T p (Λ d ( fp ) ) = T p ( T p (Λ df ) − Λ df ) = T p (Λ df ) + pT p (Λ df ) − T p (Λ df )= w d,f (Λ d ( fp ) + p Λ d ( fp ) ) , and for every r ≥ w d,f T p (Λ d ( fp r ) ) = T p ( T p r (Λ df ) − T p r − (Λ df ))= T p r +1 (Λ df ) + pT p r − (Λ df ) − T p r − (Λ df ) − pT p r − (Λ df )= w d,f (Λ d ( fp ( r +1) ) + p Λ d ( fp ( r − ) ) . This completes the proof of (4.11).By (4.5) in Lemma 4.7 and Lemma 4.9, for every r ≥ T p (Λ d ( fp r ) ) = t ∗ (Λ d ( fp r ) ) + t ∗ (Λ d ( fp r ) ) . Using Lemmas 4.8 and 4.9 to compare the support of this divisor with that in (4.11),we conclude that if p ramifies (resp. is inert) in Q ( √ d ), then for every r ≥ r ≥
2) we have(4.13) t ∗ (Λ d ( fp r ) ) = Λ d ( fp ( r +1) ) . Suppose p ramifies in Q ( √ d ). Then, by (4.13) for every r ≥ d ( fp r ) = ( t ∗ ) r − Λ d ( fp ) . Moreover, by (4.9) in Lemma 4.10, (4.5) in Lemma 4.7 and Lemma 4.9 we have w d,f Λ d ( fp ) = T p (Λ df ) − Λ df = t ∗ (Λ df ) − Λ df , so by Lemma 4.9 we have w d,f Λ d ( fp ) = t ∗ (Λ df ) | v − p ( p ). This completes the proofof the theorem in the case where p ramifies in Q ( √ d ).Assume p is inert in Q ( √ d ). Then by (4.9) and (4.10) in Lemma 4.10, (4.5) inLemma 4.7 and Lemma 4.9, we have(4.14) w d,f Λ d ( fp ) = T p (Λ df ) = t ∗ (Λ df )and w d,f T p (Λ d ( fp ) ) = T p (cid:0) T p (Λ df ) (cid:1) = T p (Λ df ) + p Λ df = w d,f Λ d ( fp ) + ( p + 1)Λ df . Using Lemmas 4.8 and 4.9 to compare the support of this last divisor with thatof (4.12) with r = 1, we conclude that t ∗ (Λ d ( fp ) ) = Λ d ( fp ) . Combined with (4.13)and (4.14), this implies that for every r ≥ d ( fp r ) = ( t ∗ ) r (cid:16) Λ df w d,f (cid:17) . Thiscompletes the proof of the theorem. (cid:3) Formal CM points formulae.
The goal of this section is to prove the fol-lowing formulae for formal CM points. We use the canonical branch t of T p andKatz’ valuation v p , as in Section 4.2. Given a fundamental p -adic discriminant d and an integer m ≥
0, define the affinoid(4.15) A d p m := v − p ( · p − m ) if Q p ( √ d ) is ramified over Q p ; v − p ([1 , ∞ ]) if Q p ( √ d ) is unramified over Q p and m = 0; v − p ( pp +1 · p − m ) if Q p ( √ d ) is unramified over Q p and m ≥ . Theorem 4.11.
Every formal CM point has supersingular reduction. Furthermore,for every fundamental p -adic discriminant d the following properties hold. ( i ) The set Λ d is contained in A d and we have t (Λ d ) = Λ d if Q p ( √ d ) is ramifiedover Q p and T p (Λ d ) = Λ d p if Q p ( √ d ) is unramified over Q p . ( ii ) For every integer m ≥ , we have Λ d p m = ( t m (cid:12)(cid:12) A d p m ) − (Λ d ) . INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 37
The proof of Theorem 4.11 and of the following corollary are given at the end ofthis section. Recall that every p -supersingular discriminant is contained in a unique p -adic discriminant (Lemma 2.1). Corollary 4.12.
The following properties hold. ( i ) A CM point E is a formal CM point if and only if it has supersingularreduction. In this case, the p -adic discriminant of E is the unique p -adicdiscriminant containing the discriminant of E . ( ii ) Let D be a discriminant and D a p -adic discriminant. Then supp(Λ D ) is contained in Λ D if D is in D , and if D is not in D then supp(Λ D ) isdisjoint from Λ D . Note that for every formal CM point E the height of F E must be at least two,see, e.g. , [Fr¨o68, Chapter IV, Section 1, Theorem 1( iii )] and therefore E has su-persingular reduction, see, e.g. , [Sil09, Chapter V, Theorem 3.1]. In particular,for every integer m ≥ E [ p m ] of E is contained in thekernel of the reduction morphism E ( Q p ) → e E ( F p ). In what follows we considereach endomorphism ϕ in End( F E ) as acting on the kernel of the reduction mor-phism E ( Q p ) → e E ( F p ), see, e.g. , [Sil09, Chapter VII, Propositions 2.1 and 2.2]. Inparticular, Ker( ϕ ) is a subgroup of E ( Q p ) and for every integer m ≥ ϕ is defined on E [ p m ].Let d be a fundamental p -adic discriminant. If Q p ( √ d ) is ramified over Q p ,then fix a uniformizer υ d of O Q p ( √ d ) . If Q p ( √ d ) is unramified over Q p , then fixan element υ d of O Q p ( √ d ) whose reduction is not in F p . In all the cases, for everyinteger m ≥ Z p [ v d p m ] = Z p + p m O Q p ( √ d ) . Fix m and for each E in Λ d p m let ϕ E be an element of End( F E ) with the sametrace and norm as υ d p m . Note that ϕ E is not in p End( F E ). Conversely, if E isin Y ( Q p ) and there is an element of End( F E ) r p End( F E ) with the same traceand norm as υ d p m , then End( F E ) is a p -adic quadratic order isomorphic to (4.16)and therefore E is in Λ d p m . Lemma 4.13.
Let d be a fundamental p -adic discriminant, let m ≥ be an integerand let E in Λ d p m be given. In the case where Q p ( √ d ) is unramified over Q p ,assume in addition that m ≥ . Then there is a unique subgroup C E of Ker( ϕ E ) oforder p and the following properties hold. ( i ) The quotient
E/C E is in Λ d p m − if m ≥ and in Λ d if m = 0 . ( ii ) If C is a subgroup of order p of E different from C E , then E/C is in Λ d p m +1) . The proof of this lemma is given after the following one. Recall that for E and E ′ in Y ( C p ) and every isogeny φ : E → E ′ , we denote by b φ : F E → F E ′ themap induced by φ . Lemma 4.14.
Let E be a formal CM point and let ϕ in End( F E ) r p End( F E ) besuch that ϕ ◦ ϕ is in p End( F E ) . Then the following properties hold. ( i ) There is a unique subgroup C of E of order p contained in Ker( ϕ ) . More-over, ϕ ( E [ p ]) = C . ( ii ) Let C be a subgroup of E of order p , put E ′ := E/C and let φ : E → E ′ bean isogeny whose kernel is equal to C . Then b φ ◦ ϕ ◦ b φ is in p End( F E ′ ) ifand only if C = C . ( iii ) Suppose in addition that nr( ϕ ) is in p Z p , put E := E/C and let φ : E → E be an isogeny whose kernel is equal to C . Then, there is ϕ in End( F E ) r p End( F E ) such that ϕ = b φ ◦ ϕ ◦ b φ .Proof. We use several times that if ˇ E is a formal CM point and m ≥ ψ of End( F ˇ E ) is in p m End( F ˇ E ) if and only if Ker( ψ )contains ˇ E [ p m ]. In fact, if we denote by [ p m ] ˇ E the morphism of multiplicationby p m on ˇ E , then for every element ψ in p m End( F ˇ E ) there is ψ ′ in End( F ˇ E ) suchthat ψ = ψ ′ ◦ d [ p m ] ˇ E , so Ker( ψ ) contains Ker( d [ p m ] ˇ E ) = ˇ E [ p m ]. On the other hand,if ψ is in End( F ˇ E ) and Ker( ψ ) contains ˇ E [ p m ], then we can find ψ ′ in End( F ˇ E ) suchthat ψ = ψ ′ ◦ d [ p m ] ˇ E , see [Lub67, Theorem 1.5]. So in this case ψ is in p m End( F ˇ E ).To prove item ( i ), note that Ker( ϕ ) cannot contain two distinct subgroups oforder p of E . Otherwise, Ker( ϕ ) would contain E [ p ] and therefore ϕ would bein p End( F E ), contradicting our hypothesis. On the other hand, our hypothesisthat ϕ ◦ ϕ is in p End( F E ) implies that Ker( ϕ ◦ ϕ ) contains E [ p ] and thereforethat Ker( ϕ ) contains ϕ ( E [ p ]). The group ϕ ( E [ p ]) cannot be reduced to the neutralelement of E because Ker( ϕ ) does not contain E [ p ]. We also have ϕ ( E [ p ]) = E [ p ]since Ker( ϕ ◦ ϕ ) contains E [ p ]. This implies that C := ϕ ( E [ p ]) is the uniquesubgroup of order p of Ker( ϕ ), which proves item ( i ).To prove item ( ii ), note that in the case where C = C we have b φ ( E ′ [ p ]) = C ⊆ Ker( ϕ ) , so Ker( b φ ◦ ϕ ◦ b φ ) contains E ′ [ p ] and therefore b φ ◦ ϕ ◦ b φ is in p End( F E ′ ). If C = C ,then by item ( i ) we have( ϕ ◦ b φ )( E ′ [ p ]) = ϕ ( C ) = ϕ ( E [ p ]) = C . This group is not contained in Ker( φ ), so Ker( b φ ◦ ϕ ◦ b φ ) does not contain E ′ [ p ]. Thisproves that b φ ◦ ϕ ◦ b φ is not in p End( F E ′ ) and completes the proof of item ( ii ).To prove item ( iii ), note that our additional hypothesis implies that Ker( ϕ ◦ ϕ )contains E (cid:2) p (cid:3) . It thus follows that Ker( ϕ ) contains ϕ ( E (cid:2) p (cid:3) ). By item ( i ) appliedto ϕ the group ϕ ( E [ p ]) has order p . Since pϕ ( E (cid:2) p (cid:3) ) = ϕ ( pE (cid:2) p (cid:3) ) = ϕ ( E [ p ])we deduce that ϕ ( E (cid:2) p (cid:3) ) contains a cyclic subgroup b C of E of order p . On theother hand, C is the unique group of order p contained in Ker( ϕ ), so p b C = C = φ ( E [ p ]) = pφ ( E (cid:2) p (cid:3) ) . Combined with the fact that φ ( E [ p ]) contains φ ◦ φ ( E (cid:2) p (cid:3) ) = pE (cid:2) p (cid:3) = E [ p ] , this implies that b C is contained in φ ( E (cid:2) p (cid:3) ). Since φ ( E [ p ]) also contains E [ p ]and is of order p , we conclude that(4.17) φ (cid:0) E (cid:2) p (cid:3)(cid:1) = E [ p ] + b C. On the other hand, note that b C ⊆ Ker( ϕ ) hence by item ( i ) ϕ ( E [ p ] + b C ) = ϕ ( E [ p ]) = C = Ker( φ ) . INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 39
Together with (4.17) this implies that E (cid:2) p (cid:3) is contained in Ker( b φ ◦ ϕ ◦ b φ ). So, thereis ϕ in End( F E ) such that p ϕ = b φ ◦ ϕ ◦ b φ and therefore b φ ◦ ϕ ◦ b φ = ϕ . Finally,note that ϕ cannot be in p End( F E ), for otherwise ϕ would be in p End( F E ). Thiscompletes the proof of item ( iii ) and of the lemma. (cid:3) Proof of Lemma 4.13.
Our hypotheses imply that ϕ E ◦ ϕ E is in p End( F E ), so thefirst assertion is given by Lemma 4.14( i ) with ϕ = ϕ E .To prove item ( i ), put E := E/C E and let φ : E → E be an isogeny whose kernelis equal to C E . Assume m ≥ ϕ E ) is in p Z p . Thenthe element ϕ of End( F E ) r p End( F E ) given by Lemma 4.14( iii ) with ϕ = ϕ E has the same trace and norm as υ d p m − . This implies that E is in Λ d p m − .It remains to consider the case m = 0. By assumption, in this case Q p ( √ d ) isramified over Q p . By Lemma 4.14( ii ) with ϕ = ϕ E there is ϕ ′ in End( F E ) suchthat pϕ ′ = b φ ◦ ϕ E ◦ b φ . It follows that ϕ ′ has the same trace and norm as υ d . Thisimplies that E is in Λ d and completes the proof of item ( i ).To prove item ( ii ), put E ′ := E/C and let φ : E → E ′ be an isogeny whosekernel is equal to C . Then the endomorphism b φ ◦ ϕ E ◦ b φ of F E ′ has the same normand trace as υ d p m +1 . By Lemma 4.14( ii ) with ϕ = ϕ E this endomorphism is notin p End( F E ′ ), so E ′ is in Λ d p m +1) . This completes the proof of item ( ii ) and ofthe lemma. (cid:3) Given a fundamental p -adic discriminant d denote by Q p ( √ d ) the compositumof Q p and Q p ( √ d ). Lemma 4.15.
Let e be in Y sups ( F p ) . Then, for every fundamental p -adic discrim-inant d the set Π − e (Λ d ∩ D e ) is contained in X e ( O Q p ( √ d ) ) .Proof. Let ( F , α ) be a point in Π − e (Λ d ∩ D e ). Denote by O the image of End( F E )by ϕ α ◦ e ϕ ◦ α − and let g in O × be such that O = Z p [ g ]. Then O is isomorphicto O Q p ( √ d ) and ( F , α ) is a fixed point of g by Lemma 2.4. It follows that the ringhomomorphism δ : Z p [ g ] → O Q p given by Proposition 4.3 takes values in Q p ( √ d )and therefore that ( F , α ) is in X e ( O Q p ( √ d ) ). (cid:3) Lemma 4.16.
For every e in Y sups ( F p ) , we have min { v p ◦ Π e , } = min { ord p , } . In particular, the map b v p = min { v p , pp +1 } satisfies b v p ◦ Π e = min n ord p , pp +1 o .Proof. Let δ e be as in Section 2.7 and j e as in [HMRL20, Proposition 4.3], so thatfor every E in D e we have v p ( E ) = δ e ord p ( j ( E ) − j e ). Using that j e is in Z p [HMRL20, Remark 4.4] and Theorem 2.7, the difference j ◦ Π e − j e is representedby a power series with coefficients in Z p that is a ramified covering of degree δ e from b D e to M p . Thus, if we denote by x , . . . , x δ e the zeros of j ◦ Π e − j e , repeatedaccording to multiplicity, then there is h in Z p [[ t ]] such that | h | p is constant equalto 1 and such that j ◦ Π e ( t ) − j e = h ( t ) · δ e Y i =1 ( t − x i ) , see, e.g. , [FvdP04, Exercise 3.2.2(1)]. Thus, for every x in b D e we have(4.18) v p ◦ Π e ( x ) = 1 δ e ord p ( j ◦ Π e ( x ) − j e ) = 1 δ e δ e X i =1 ord p ( x − x i ) . On other hand, if we denote by d the p -adic discriminant of Z p , then j e is in Λ d by [HMRL20, Remark 4.4]. Thus, for each i in { , . . . , δ e } the point x i is in X e ( Z p )by Lemma 4.15. That is, seen as an element of M p , the point x i is in p Z p . Inparticular, for every x in b D e we havemin { ord p ( x − x i ) , } = min { ord p ( x ) , } . Together with (4.18) this implies the lemma. (cid:3)
Proof of Theorem 4.11.
The first assertion is proved in the paragraph right afterCorollary 4.12.While proving item ( i ), we also show that if Q p ( √ d ) is ramified (resp. unram-ified) over Q p , then every E in Λ d (resp. Λ d p ) is not too supersingular and thegroup C E given in Lemma 4.13 is the canonical subgroup H ( E ) of E . Assumefirst that Q p ( √ d ) is ramified over Q p and let E be in Λ d . By Lemma 4.13( i ) thequotient E := E/C E is in Λ d . Thus, by Lemmas 4.15 and 4.16 we have b v p ( E ) ≥ and b v p ( E ) ≥ . Using Lemma 4.8 several times, we conclude that b v p ( E ) = b v p ( E ) = 12 , C E = H ( E ) , t ( E ) = E and t ( E ) = E. This implies item ( i ) in the case where Q p ( √ d ) is ramified over Q p . Assumethat Q p ( √ d ) is unramified over Q p . Using Lemmas 4.15 and 4.16, we obtain that Λ d is contained in A d . To prove that Λ d p is contained in T p (Λ d ), let E in Λ d p begiven. Then the quotient E := E/C E is in Λ d by Lemma 4.13( i ), so E is insupp( T p ( E )) ⊆ T p (Λ d ). By Lemma 4.8 we also obtain that E is not too supersin-gular and that C E = H ( E ). It remains to prove that T p (Λ d ) is contained in Λ d p .To do this, let E ′ in Λ d and E ′′ in supp( T p ( E ′ )) be given and let φ : E ′ → E ′′ bean isogeny of degree p . Note that b v p ( E ′′ ) = p +1 by the first assertion of item ( i )and Lemma 4.8, so E ′′ is not in Λ d . The endomorphism b φ ◦ ϕ E ′ ◦ b φ of F E ′′ hasthe same trace and norm as υ d p . It follows that E ′′ is in Λ d or Λ d p . But wealready established that E ′′ is not in Λ d , so E ′′ is in Λ d p . This completes the proofof T p (Λ d ) = Λ d p and of item ( i ).To prove item ( ii ) we proceed by induction, showing in addition that for every E in Λ d p m we have C E = H ( E ). If m = 1 and Q p ( √ d ) is unramified over Q p , thenby item ( i ) and (4.5) in Lemma 4.7 every element E of Λ d is too supersingular andwe have t − ( E ) = T p ( { E } ). Using item ( i ) again, we obtain item ( ii ). That forevery E in Λ d p we have C E = H ( E ) was shown above. To complete the proofof the base step, assume m = 1 and that Q p ( √ d ) is ramified over Q p . Since forevery ˇ E in Λ d we have C ˇ E = H ( ˇ E ) and v p ( t ( ˇ E )) = by item ( i ), combining (4.5) inLemma 4.7 and Lemma 4.13( ii ) we obtain that ( t | A d p ) − (Λ d ) is contained in Λ d p .To prove the reverse inclusion, let E in Λ d p be given. Then E := E/C E is in Λ d byLemma 4.13( i ) and we have b v p ( E ) = by item ( i ). If we had b v p ( E ) = p , then byLemma 4.8 we would have b v p ( E ) = and E = t ( E ). By item ( i ) this would implythat E is in Λ d . This contradiction proves that b v p ( E ) = p . Using Lemma 4.8 INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 41 again we conclude that C E = H ( E ) and t ( E ) = E . This proves Λ d p ⊆ A d p and t (Λ d p ) ⊆ Λ d and completes the proof of the base step.To prove the induction step, let m ≥ ii ) holdsand such that for every E in Λ d p m we have C E = H ( E ). Combined with (4.5) inLemma 4.7 and Lemma 4.13( ii ), this last property implies that(4.19) t − (Λ d p m ) ⊆ Λ d p m +1) . To prove the reverse inclusion, let E in Λ d p m +1) be given. Then E := E/C E isin Λ d p m by Lemma 4.13( i ), so by the induction hypothesis we have b v p ( E ) = ( pp +1 · p − m if Q p ( √ d ) is unramified over Q p ; · p − m if Q p ( √ d ) is ramified over Q p . In particular, E is not too supersingular. Suppose that b v p ( E ) = p b v p ( E ). Thenby Lemma 4.8 we would have b v p ( E ) = p b v p ( E ) and t ( E ) = E . By the inductionhypothesis this would imply that E is in Λ d p m − , which is absurd. This contradic-tion proves that b v p ( E ) = p b v p ( E ). Using Lemma 4.8 again, we obtain C E = H ( E )and t ( E ) = E . This provesΛ d p m +1) ⊆ A d p m +1) and t (Λ d p m +1) ) ⊆ Λ d p m . Together with (4.19) this completes the proof of the induction step and of item ( ii ).The proof of the theorem is thus complete. (cid:3) Proof of Corollary 4.12.
To prove item ( i ), note that if E is a formal CM point,then E has supersingular reduction by Theorem 4.11. To prove the second assertion,assume E has supersingular reduction, let D be the discriminant of E and let D bethe unique p -adic discriminant containing D . Denote by d and f the fundamentaldiscriminant and conductor of D , respectively, so D = df and End( E ) is isomorphicto O d,f . Moreover, denote by d the fundamental p -adic discriminant and m ≥ D = d p m (Lemma A.1( i )). Then d is in d , m = ord p ( f )and End( E ) ⊗ Z p is a p -adic quadratic order isomorphic to Z p + p m O Q p ( √ d ) . Inparticular, the p -adic discriminant of End( E ) ⊗ Z p is equal to D by Lemma A.1( ii ).Consider the natural map End( E ) ⊗ Z p → End( F E ), induced by the ring homomor-phism End( E ) → End( F E ). Its image is a p -adic order of p -adic discriminant D .This implies that End( F E ) is a p -adic quadratic order and that there is an in-teger m ′ ≥ p -adic discriminant of End( F E ) is equal to d p m ′ .Combining Lemma 4.9 and Theorem 4.11 we obtain that m ′ = m and thereforethat the p -adic discriminant of End( F E ) is equal to D . Thus, E is in Λ D . Thiscompletes the proof of item ( i ).The first assertion of item ( ii ) is a direct consequence of item ( i ) and the factthat every discriminant in D is p -supersingular (Lemma 2.1). To prove the secondassertion, assume D is not in D . If D is not p -supersingular, then supp(Λ D ) is dis-joint from Y sups ( C p ) and therefore from Λ D by Theorem 4.11. Assume that D is p -supersingular and let D ′ be the unique p -adic discriminant containing D .Then supp(Λ D ) is contained in Λ D ′ by item ( i ) and it is therefore disjoint from Λ D .This completes the proof of item ( ii ) and of the corollary. (cid:3) Asymptotic distribution of CM points of fundamental discriminant
The goal of this section is to prove the following result, on the asymptotic distri-bution of CM points of fundamental discriminant. It is one of the main ingredients in the proof of Theorem A. Recall that for a p -adic discriminant D and everydiscriminant D in D , the set supp(Λ D ) is contained in Λ D (Corollary 4.12( ii )). Theorem 5.1.
For every fundamental p -adic discriminant d , the set Λ d is a com-pact subset of Y sups ( C p ) . Moreover, there is a Borel probability measure ν d whosesupport is equal to Λ d and such that for all ε > and δ > there is a con-stant C > , such that the following property holds. For every function F : Λ d → R that is constant on every ball of Λ d of radius δ and every fundamental discrimi-nant d in d , we have (5.1) (cid:12)(cid:12)(cid:12)(cid:12)Z F d δ d − Z F d ν d (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:18) sup Λ d | F | (cid:19) | d | − + ε . For each e in Y sups ( F p ), we introduce “zero-trace spheres” of L e, f and show thateach of these sets carries a natural homogeneous measure (Proposition 5.2 in Sec-tion 5.1). Zero-trace spheres parametrize formal CM points in D e with fundamental p -adic discriminant, via fixed points of the group action described in Section 2.6(Propositions 5.4 and 5.6 in Sections 5.2 and 5.3, respectively). After these consid-erations, we prove Theorem 5.1 in Section 5.5 using our results on the asymptoticdistribution of integer points on p -adic spheres in Section 3 and an equidistributionresult for CM points in supersingular residue discs (Theorem 5.7 in Section 5.4).5.1. Zero-trace spheres and their homogeneous measures.
Throughout thissection fix e in Y sups ( F p ). Let B e , R e and G e be as in Section 2.6 and let L ( e ), L e and L e, f be as in Section 4.1. Note that the degree function defines a ternaryquadratic form Q e on the lattice L ( e ), which is positive definite and defined over Z .Using the natural map End( e ) → R e defined by φ b φ as in Section 2.6, thequadratic form Q e extends to a quadratic form on L e taking values on Z p .For each ℓ in Z p such that − ℓ is in a fundamental p -adic discriminant, we call S ℓ ( e ) := { ϕ ∈ L e : nr( ϕ ) = ℓ } a zero-trace sphere , which we consider as a metric subspace of R e . The goal of thissection is to define a natural homogeneous measure on each zero-trace sphere.Given a fundamental p -adic discriminant d , put(5.2) L e, d := { ϕ ∈ L e : − nr( ϕ ) ∈ d } . Clearly, as d varies these sets form a partition of L e, f . Moreover, for each d we havethe partition(5.3) L e, d = G ∆ ∈ d S − ∆ ( e ) . The action of G e on B e by conjugation preserves the reduced trace and norm, soit restricts to a left action(5.4) G e × L e, d → L e, d ( g, ϕ ) gϕg − . Moreover, for every ∆ in d this action restricts to an action of G e on S − ∆ ( e ), whichis the restriction to G e of the action of the orthogonal group O Q e ( Z p ) on S − ∆ ( e ). Proposition 5.2.
For every e in Y sups ( F p ) and every fundamental p -adic discrim-inant d , the following properties hold. INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 43 ( i ) The set L e, d is nonempty and compact and for every ϕ in L e, d the subalge-bra Q p ( ϕ ) of B e is a field extension of Q p isomorphic to Q p ( √ d ) . ( ii ) For each ∆ in d , the set S − ∆ ( e ) is nonempty and compact and the group G e acts transitively and by isometries on S − ∆ ( e ) . In particular, the decompo-sition of L e, d into orbits with respect to the action (5.4) is given by (5.3) . ( iii ) For each ℓ in Z p such that − ℓ is in d , there exists a unique Borel proba-bility measure ν eℓ on S ℓ ( e ) that is invariant under the action of G e . Thismeasure is also uniquely determined as the unique Borel probability measureon S ℓ ( e ) that is invariant under the action of the orthogonal group O Q e ( Z p ) .Moreover, the support of ν eℓ is equal to all of S ℓ ( e ) . ( iv ) For every ∆ in d , e ′ in Y sups ( F p ) and g in G e,e ′ , the map R e → R e ′ givenby ϕ gϕg − maps S − ∆ ( e ) to S − ∆ ( e ′ ) and ν e − ∆ to ν e ′ − ∆ . The proof of this proposition is given after the following general lemma.
Lemma 5.3.
Let G be a group acting transitively and by isometries on a compactultrametric space X . Then there is a unique Borel probability measure on X thatis invariant by G . Moreover, the support of this measure is equal to all of X andthis measure is invariant under every isometry of X .Proof. Denote by dist X the distance on X . Given r >
0, let ∼ r be the equivalencerelation on X defined by dist X ( x, x ′ ) ≤ r , let X r be the finite set of equivalentclasses of ∼ r and let µ r be the probability measure on X r assigning the same mass toeach element of X r . For every r ′ in ]0 , r [ the partition of X into equivalence classesof ∼ r ′ is finer than that of ∼ r . The action of G on X descends to a transitive actionon X r ′ , so each equivalence class of ∼ r contains the same number of equivalenceclasses of ∼ r ′ . It follows that the natural projection X r ′ → X r maps µ r ′ to µ r .Since the collection of all equivalence classes of ∼ r , as r > X , by Carath´eodory’s theorem there is a unique Borel probabilitymeasure on X such that for every r > X r is µ r . By construction,the support of µ is equal to all of X .If g is an isometry of X , then for every r > g descends to a bijectionof X r and therefore leaves µ r invariant. From the definition of µ , we concludethat g ∗ µ = µ . In particular, µ is invariant by G . To prove uniqueness, let µ ′ bea Borel probability measure on X that is invariant under G . Then for each r > µ ′ projects to a measure µ ′ r on X r that is invariant under the inducedaction of G . Since this action is transitive, we have µ ′ r = µ r . Since this holds forevery r >
0, from the definition of µ we conclude that µ ′ = µ . This proves theuniqueness of µ and completes the proof of the lemma. (cid:3) Proof of Proposition 5.2.
To prove item ( i ), let ∆ in d be given. We use thatthere is an embedding of Q p ( √ ∆) into B e , see [Vig80, Chapitre
II,
Corollaire ϕ be the image of ∆+ √ ∆2 in B e . Then tr( ϕ ) = ∆ and nr( ϕ ) = ∆ − ∆4 bothbelong to Z p and therefore ϕ belongs to R e . On the other hand, δ := 2 ϕ − ∆satisfies tr( δ ) = 0 and nr( δ ) = − ∆ and it is therefore in S − ∆ ( e ). This provesthat S − ∆ ( e ) and therefore L e, d , are both nonempty. That L e, d is compact followsfrom the fact that the sets L e and d are both compact and the fact that the reducednorm is continuous. To prove the last assertion of item ( i ), note that ϕ = − nr( ϕ ),so ϕ belongs to d and therefore Q p ( ϕ ) is isomorphic to Q p ( √ d ). This completesthe proof of item ( i ). To prove item ( ii ), note that we already proved that S − ∆ ( e ) is nonempty.Since L e is compact and the reduced norm is continuous, S − ∆ ( e ) is compact. On theother hand, since the action of each element of G e on S − ∆ ( e ) is the composition ofa left and a right multiplication, it is an isometry. It remains to prove that G e actstransitively on S − ∆ ( e ). Let ϕ and ϕ ′ in S − ∆ ( e ) be given. Since ϕ and ϕ ′ both sat-isfy the equation X − ∆ = 0, there is an isomorphism of Q p -algebras between Q p ( ϕ )and Q p ( ϕ ′ ) mapping ϕ to ϕ ′ . By Skolem–Noether’s theorem this isomorphism ex-tends to an inner automorphism of B e , see [Vig80, Chapitre I, Th´eor`eme g in B × e such that g ϕg − = ϕ ′ . If we denote by ̟ a uni-formizer of B e , then g := g ̟ − ord B e ( g ) is in G e and satisfies gϕg − = ϕ ′ . Thiscompletes the proof of item ( ii ).Item ( iii ) is a direct consequence of item ( ii ) and Lemma 5.3.To prove item ( iv ), note that the map ϕ gϕg − is an isomorphism of Z p -algebrasthat extends by Q p -linearity to an isomorphism of Q p -algebras c : B e → B e ′ .Since the canonical involutions of B e and B e ′ are unique, for every ϕ in B e wehave c ( ϕ ) = c ( ϕ ). This implies that c preserves reduced traces and norms andthat it is an isometry. In particular, c maps S − ∆ ( e ) to S − ∆ ( e ′ ) isometrically. Byitem ( iii ) the image of ν e − ∆ by c is a Borel probability measure on S − ∆ ( e ′ ) that isinvariant under the action of G e ′ and therefore it is equal to ν e ′ − ∆ . This completesthe proof of item ( iv ) and of the lemma. (cid:3) Parametrizing fixed points.
The goal of this section is to prove the follow-ing proposition, giving a natural parametrization of the fixed points associated tothe elements of a given zero-trace sphere.
Proposition 5.4.
For every e in Y sups ( F p ) , every fundamental p -adic discrimi-nant d and every ∆ in d , the following properties hold. ( i ) If Q p ( √ d ) is unramified over Q p , then there is a continuous function x e, ∆ : S − ∆ ( e ) → b D e such that for every ϕ in S − ∆ ( e ) we have Fix e ( ϕ ) = { x e, ∆ ( ϕ ) } . ( ii ) If Q p ( √ d ) is ramified over Q p , then there are continuous functions x + e, ∆ , x − e, ∆ : S − ∆ ( e ) → b D e , such that for every ϕ in S − ∆ ( e ) we have x + e, ∆ ( ϕ ) = x − e, ∆ ( ϕ ) and Fix e ( ϕ ) = { x + e, ∆ ( ϕ ) , x − e, ∆ ( ϕ ) } . The proof of this proposition is given after the following lemma.
Lemma 5.5.
Fix an element e of Y sups ( F p ) , a fundamental p -adic discriminant d ,an element ∆ of d and a uniformizer ̟ of R e . Given ϕ in L e, d , put C ( ϕ ) := { ϕ ∈ S − ∆ ( e ) : ϕϕ − ∈ B e + ̟ R e } . Then there is a continuous function g : C ( ϕ ) → G e such that for every ϕ in C ( ϕ ) we have g ( ϕ ) ϕ g ( ϕ ) − = ϕ .Proof. For each ϕ in C ( ϕ ), we have that ς ( ϕ ) := ϕϕ − is in B e + ̟ R e . So, B e + ς ( ϕ ) is nonzero and g ( ϕ ) := 2( B e + ς ( ϕ )) − is in G e . The function g : C ( ϕ ) → G e INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 45 so defined is continuous. On the other hand, using ϕ ς ( ϕ ) = ς ( ϕ ) ϕ we obtain g ( ϕ ) ϕ g ( ϕ ) − = ( B e + ς ( ϕ )) − ϕ ( B e + ς ( ϕ ))= ( B e + ς ( ϕ )) − ( B e + ς ( ϕ )) ϕ = ϕ. (cid:3) Proof of Proposition 5.4.
We use several times that for each ϕ in S − ∆ ( e ), theset C ( ϕ ) given by Lemma 5.5 is an open and closed subset of S − ∆ ( e ).Suppose Q p ( √ d ) is unramified over Q p and let x e, ∆ : S − ∆ ( e ) → b D e be the func-tion associating to each ϕ in S − ∆ ( e ) the unique element of Fix e ( ϕ ) (Lemma 4.5( ii )).Let ϕ in S − ∆ ( e ) be given, denote by x the unique element of Fix e ( ϕ ) andlet g : C ( ϕ ) → G e be the continuous function given by Lemma 5.5. Then foreach ϕ in C ( ϕ ) the point g ( ϕ ) · x is in Fix e ( ϕ ) and therefore g ( ϕ ) · x = x e, ∆ ( ϕ ).In particular, the restriction of x e, ∆ to C ( ϕ ) is continuous by Lemma 2.6( ii ).Suppose Q p ( √ d ) is ramified over Q p and denote by ι : S − ∆ ( e ) → S − ∆ ( e ) theinvolution given by ι ( ϕ ) = − ϕ . Note that for each ϕ in S − ∆ ( e ) the set C ( ϕ ) doesnot contain − ϕ . Moreover, for ϕ ′ in S − ∆ ( e ) the set C ( ϕ ′ ) is either disjoint fromor equal to C ( ϕ ). Since S − ∆ ( e ) is compact, it follows that there is a finite subset Φof S − ∆ ( e ) such that(5.5) {C ( ϕ ) , C ( − ϕ ) : ϕ ∈ Φ } is a partition of S − ∆ ( e ). For each element ϕ of Φ the set Fix e ( ϕ ) has precisely twoelements by Lemma 4.5( ii ). Denote them by x + ϕ and x − ϕ . Moreover, denote by g ϕ the continuous function given by Lemma 5.5. Using Lemma 2.6( ii ), that (5.5) isa partition of S − ∆ ( e ) and that for each ϕ in S − ∆ ( e ) we have ι ( C ( ϕ )) = C ( − ϕ ),we obtain that there are continuous functions x + and x − : S − ∆ ( e ) → b D e such thatfor each ϕ in Φ we have x ± | C ( ϕ ) ( ϕ ) = g ϕ ( ϕ ) · x ± ϕ and x ± | C ( − ϕ ) ( ϕ ) = g ϕ ( − ϕ ) · x ± ϕ . Since for each ϕ in S − ∆ ( e ) we have Fix e ( ϕ ) = Fix e ( − ϕ ) by Lemma 4.5( iv ), thepoints x + ( ϕ ) and x − ( ϕ ) belong to Fix e ( ϕ ). Thus, to prove item ( ii ) with x + e, ∆ = x + and x − e, ∆ = x − , it is enough to show that for every ϕ in Φ and every ϕ in C ( ϕ ) ∪ C ( − ϕ ) the points x + ( ϕ ) and x − ( ϕ ) are different. We have either x ± ( ϕ ) = g ϕ ( ϕ ) · x ± ϕ or x ± ( ϕ ) = g ϕ ( − ϕ ) · x ± ϕ . In both cases we conclude that x + ( ϕ ) and x − ( ϕ ) are different. This completes theproof of the proposition. (cid:3) From zero-trace spheres to CM points.
The goal of this section is toprove the following proposition. It relates zero-trace spheres to formal CM pointsand defines a natural measure on the set of formal CM points of a given fundamental p -adic discriminant and residue disc.Given e in Y sups ( F p ), for each subset S of L e, f putFix e ( S ) := [ g ∈ S Fix e ( g ) . The trace of a function b F : b D e → R , isTr e ( b F ) : L e, f → R g Tr e ( b F )( g ) := e ( g ) P x ∈ Fix e ( g ) b F ( x ) . Proposition 5.6.
For every e in Y sups ( F p ) and every fundamental p -adic discrim-inant d , the following properties hold. ( i ) For every ∆ in d we have (5.6) Π − e (Λ d ∩ D e ) = Fix e (cid:0) S − ∆ ( e ) (cid:1) and this set is compact. ( ii ) There is a Borel probability measure b ν e d on b D e that is uniquely determinedby the following property. For every ∆ in d and every continuous func-tion b F : b D e → R , we have (5.7) Z b F d b ν e d = Z Tr e ( b F ) d ν e − ∆ . Moreover, the support of b ν e d is equal to Π − e (Λ d ∩ D e ) .Proof. To prove item ( i ), note that S − ∆ ( e ) is compact by Proposition 5.2( ii ), soby Proposition 5.4 the setFix e ( S − ∆ ( e )) = ( x e, ∆ ( S − ∆ ( e )) if Q p ( √ d ) is unramified over Q p ; x + e, ∆ ( S − ∆ ( e )) ∪ x − e, ∆ ( S − ∆ ( e )) if Q p ( √ d ) is ramified over Q p , is also compact.To prove that the left-hand side of (5.6) is contained in the right-hand side, let x in Π − e (Λ d ∩ D e ) be given and put E := Π e ( x ). Then E is a formal CM point, soit is in Y sups ( Q p ) and x is in X e ( O Q p ). Let α : e F E → F e be an isomorphism suchthat ( F E , α ) represents x and consider the ring homomorphism(5.8) ι : End( F E ) → R e ϕ ι ( ϕ ) := α ◦ e ϕ ◦ α − . Since End( F E ) is isomorphic to O Q p ( √ d ) and O Q p ( √ d ) = Z p h ∆+ √ ∆2 i by (A.6) inLemma A.2( ii ), there is an element ϕ of Z p +2 End( F E ) satisfying the equation X − ∆ = 0. Then, ι ( ϕ ) is in Z p + 2 R e , satisfies the equation X − ∆ = 0 and therefore itbelongs to S − ∆ ( e ). Note also that the image of Aut( F E ) by ι equals O × Q p ( ι ( ϕ )) andby Lemma 4.1 this equals Z p [ U e ( ι ( ϕ ))] × and U e ( ι ( ϕ )) is in the image of Aut( F E )by ι . By Lemma 2.4 this implies that x is in Fix e ( ι ( ϕ )) and therefore in the right-hand side of (5.6).To prove the reverse inclusion, recall that S − ∆ ( e ) is nonempty by Proposi-tion 5.2( ii ) and let ϕ in S − ∆ ( e ) and x in Fix e ( ϕ ) be given. By Lemma 4.5( i )the point x is in X e ( O Q p ). Put E := Π e ( x ) and let α : e F E → F e be an isomorphismsuch that ( F E , α ) represents x . By Lemma 2.4 the unit U e ( ϕ ) is in the image ofthe map ι defined by (5.8). It follows that Z p [ U e ( ϕ )] and therefore ϕ , are all inthe image of ι . This implies that End( F E ) contains a solution of X − ∆ = 0 andtherefore that it is a p -adic quadratic order of p -adic discriminant d . This completesthe proof that the right-hand side of (5.6) is contained in the left-hand side and ofitem ( i ).To prove item ( ii ), fix ∆ in d and consider the Borel probability measure on b D e defined by b ν e d := ( ( x e, ∆ ) ∗ ν e − ∆ if Q p ( √ d ) is unramified over Q p ; (cid:16) ( x + e, ∆ ) ∗ ν e − ∆ + ( x − e, ∆ ) ∗ ν e − ∆ (cid:17) if Q p ( √ d ) is ramified over Q p . INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 47
Since the support of ν e − ∆ is equal to S − ∆ ( e ) by Proposition 5.2( iii ), item ( i ) impliesthat the support of b ν e d is equal to Π − e (Λ d ∩ D e ). To prove (5.7), let b F : b D e → R be a continuous function. By the change of variables formula we have Z b F d b ν e d = Z b F ◦ x e, ∆ d ν e − ∆ = Z Tr e ( b F ) d ν e − ∆ , if Q p ( √ d ) is unramified over Q p . If Q p ( √ d ) is ramified over Q p , then we have Z b F d b ν e d = Z
12 ( b F ◦ x + e, ∆ + b F ◦ x − e, ∆ ) d ν e − ∆ = Z Tr e ( b F ) d ν e − ∆ . This proves (5.7) for ∆ in d chosen above. To complete the proof of (5.7), it remainsto show that for every ∆ ′ in d the identity (5.7) holds with ∆ replaced by ∆ ′ . Let u in Z × p be such that ∆ ′ = u ∆. Then the left multiplication map ϕ uϕ inducesa bijective isometry S − ∆ ( e ) → S − ∆ ′ ( e ) and therefore it maps ν e − ∆ to ν e − ∆ ′ byProposition 5.2( iii ). Thus, by the change of variables formula and Lemma 4.5( iv ),for every continuous function b F : b D e → R we have Z Tr e ( b F )( ϕ ) d ν e − ∆ ′ ( ϕ ) = Z Tr e ( b F )( uϕ ) d ν e − ∆ ( ϕ ) = Z Tr e ( b F )( ϕ ) d ν e − ∆ ( ϕ ) . This proves the existence of b ν e d . Its uniqueness follows from the fact that (5.7) holdsfor every continuous function b F . This completes the proof of item ( ii ) and of theproposition. (cid:3) Equidistribution of CM points on supersingular residue discs.
Thepurpose of this section is to prove the following theorem.
Theorem 5.7.
For every ε > there is a constant C > such that the followingproperty holds. Let d be a p -supersingular fundamental discriminant and f ≥ aninteger. Then for every e in Y sups ( F p ) , we have (cid:12)(cid:12)(cid:12)(cid:12) deg(Λ df | D e )deg(Λ df ) − p − e ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | d | − + ε ( f | f | p ) − + ε . To state a corollary of this theorem, we introduce some notation. Consider thereal vector space R sups := n ( z e ) e ∈ Y sups ( F p ) : z e ∈ R o and let v sups be the vector in R sups defined by v sups e := p − e ) . The mass for-mula (2.5) implies that v sups is a probability vector. Given a divisor Λ on Y sups ( C p ),denote by v (Λ) the vector in R sups defined by v (Λ) e := deg(Λ | D e ).The following corollary is a direct consequence of Theorem 5.7. Corollary 5.8.
Let ( D n ) ∞ n =1 be a sequence of p -supersingular discriminants suchthat D n | D n | p → −∞ as n → ∞ . Then lim n →∞ v (Λ D n )deg(Λ D n ) = v sups . The hypothesis that D n | D n | p → −∞ as n → ∞ cannot be weakened to D n → −∞ as n → −∞ , see Remark 5.10 below. When restricted to discriminants for which p is inert in the corresponding qua-dratic imaginary extension of Q , Theorem 5.7 is a particular case of [JK11, Theo-rem 1.1] and of the “sparse equidistribution” result of Michel [Mic04, Theorem 3]in the case of fundamental discriminants.To prove Theorem 5.7 for fundamental discriminants, we construct an auxiliarymodular form of weight that is cuspidal and then derive the desired estimatesfrom Duke’s bounds of Fourier coefficients [Duk88]. The cuspidal modular form weuse in the proof of Theorem 5.7 also appears in the proof of [EOY05, Theorem 1.4].We also use Siegel’s classical estimate: For every ε > C > d we have(5.9) deg(Λ d ) ≥ C | d | − ε , see for example [Sie35] or [Gol74]. To pass from fundamental discriminants to thegeneral case, we use Zhang’s formula (Lemma 4.10) as in [CU04].The proof of Theorem 5.7 is at the end of this section, after some preparatorylemmas that are only needed in the case of discriminants that are not fundamental.In Lemma 5.9 we recall the description in [Gro87] of the action of Hecke correspon-dences on supersingular residue discs in terms of the Brandt matrices and we treatdiscriminants whose conductor is divisible by p . In Lemma 5.11 we apply Deligne’sbound to estimate the norm of eigenvalues of Brandt matrices.To state our first lemma, we introduce some notation. In the rest of this sectionwe consider vectors in R sups as column vectors. Given an integer m ≥ e and e ′ in Y sups ( F p ), denote by B ( m ) e,e ′ the number of subgroup schemes C oforder m of e such that e/C is isomorphic to e ′ . By [Gro87, Proposition 2.3], B ( m ) := ( B ( m ) e,e ′ ) e,e ′ ∈ Y sups ( F p ) is the Brandt matrix of degree m defined by (1.5) in loc. cit. Note that B (1) is the identity matrix. Recall that the Frobenius map Frobmaps Y sups ( F p ) onto itself and it induces an involution on this set, see Section 2.3. Itfollows that the induced linear map Frob ∗ : R sups → R sups defined by Frob ∗ ( v ) e := v Frob( e ) , is also an involution. Note also that Frob ∗ ( v sups ) = v sups , because forevery e in Y sups ( F p ) that does not have a representative elliptic curve defined over F p we have e ) = 2. Lemma 5.9. ( i ) For every integer m ≥ not divisible by p and every divisor Λ supportedon Y sups ( C p ) , we have v ( T m (Λ)) = B ( m ) ⊺ v (Λ) . ( ii ) We have
Frob ∗ = B ( p ) ⊺ as linear endomorphisms of R sups . Moreover, forevery integer r ≥ and every divisor Λ supported on Y sups ( C p ) , we have v ( T p r (Λ)) = σ ( p r ) · Frob r ∗ ( v (Λ)) . ( iii ) For every p -supersingular discriminant D and every integer r ≥ , we have (5.10) v (Λ Dp r )deg(Λ Dp r ) = v (Λ D )deg(Λ D ) . Proof.
By continuity, to prove item ( i ) we can assume that the divisor Λ is sup-ported on Y sups ( Q p ), see, e.g. , [HMRL20, Lemma 2.1]. In this case, the desiredassertion follows from the fact that for every E in Y sups ( Q p ) and every integer m ≥ p , the reduction map induces a bijection from the set INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 49 of subgroups of E of order m to the set of subgroup schemes of order m of e E , see forexample [Sil09, Chapter III, Corollary 6.4(b) and Chapter VII, Proposition 3.1(b)].The first assertion of item ( ii ) follows from the fact that each e in Y sups ( F p )has a unique subgroup scheme of order p and that this subgroup scheme is thekernel of the Frobenius map from e to Frob( e ). To prove (5.10) for r = 1,we use the fact that the reduction modulo p of the modular polynomial Φ p is e Φ p ( X, Y ) = ( X − Y p )( X p − Y ), see, e.g. , [Lan87, Chapter 5, Section 2, pp. 57-58].Together with (2.10) with n = p , (2.13) with q = p and the definition of Frob ∗ ,this implies (5.10) for r = 1. The case r ≥ Y sups ( F p ).To prove item ( iii ), denote by d and f the fundamental discriminant and theconductor of D , respectively, so that D = df . Put r := ord p ( f ) , f := p − r f and D := df . In view of item ( ii ), (4.9) and (4.10) in Lemma 4.10, to prove item ( iii ) it is sufficientto prove Frob ∗ ( v (Λ D )) = v (Λ D ). From (2.14) and items ( i ) and ( ii ), for eachinteger m ≥ p the maps Frob ∗ and B ( m ) ⊺ commute. Thus,in view of (4.8) in Lemma 4.10 with e f = 1, to prove Frob ∗ ( v (Λ D )) = v (Λ D ) it issufficient to prove Frob ∗ ( v (Λ d )) = v (Λ d ). To do this, note that, since for a given e in Y sups ( F p ) the endomorphism rings End( e ) and End(Frob( e )) are isomorphic,applying (4.4) and (4.4) again with e replaced by Frob( e ), we have v (Λ d ) e = deg(Λ d | D e ) = ǫ d h ( d, e ) = ǫ d h ( d, Frob( e )) = deg(Λ d | D Frob( e ) )= Frob ∗ ( v (Λ d )) e . This completes the proof of item ( iii ) and of the lemma. (cid:3)
Remark . For every p -supersingular discriminant D the sequence of vectors (cid:16) v (Λ Dp r )deg(Λ Dp r ) (cid:17) ∞ r =1 is constant by Lemma 5.9. Thus, unless we are in the unlikelysituation in which v (Λ D )deg(Λ D ) is exactly equal to v sups , this sequence cannot convergeto v sups . This proves that in Corollary 5.8 it is not sufficient so suppose that D n → −∞ as n → ∞ .To state the next lemma, we introduce some notation. Endow R sups with thescalar product h· , ·i sups and norm k · k sups , defined by(5.11) h v, v ′ i sups := X e ∈ Y sups ( F p ) v e v ′ e v sups e and k v k sups := q h v, v i sups . Lemma 5.11.
There is an orthonormal basis B of R sups containing the vector v sups ,such that for every m ≥ each vector in B is an eigenvector of B ( m ) ⊺ . Further-more, for v in B let λ v : N → C be defined by B ( m ) v = λ v ( m ) v . Then the followingproperties hold. ( i ) For every integer m ≥ that is not divisible by p and every integer r ≥ ,we have λ v sups ( p r m ) = σ ( m ) . ( ii ) For every ε > there is a constant C > such that for every v in B different from v sups and every integer m ≥ , we have | λ v ( m ) | ≤ C m + ε . Proof.
We first recall some facts about the space M (Γ ( p )) of holomorphic modularforms of weight 2 for Γ ( p ). This space contains the Eisenstein series F p ( τ ) := p −
124 + ∞ X r =0 X m ≥ ,p ∤ m σ ( m ) exp(2 πimp r τ ) , see [Gro87, (5.7)]. The subspace of cuspidal modular forms S (Γ ( p )) has codi-mension one in M (Γ ( p )), so M (Γ ( p )) = C F p ⊕ S (Γ ( p )), see, e.g. , [Miy89,Theorems 2.5.2 and 4.2.7]. Since the constant coefficient of F p is nonzero, it followsthat every modular form in M (Γ ( p )) whose constant coefficient is zero is cuspidal.To prove the first assertion and item ( i ), note that k v sups k sups = 1 and thatfor every integer m ≥ p and every integer r ≥
0, wehave B ( p r m ) ⊺ v sups = σ ( m ) · v sups , see [Gro86, Proposition 2.7(1, 6)]. Moreover,for every m in N the matrix B ( m ) ⊺ is self-adjoint with respect to the inner prod-uct (5.11) and for every m ′ in N the matrices B ( m ) ⊺ and B ( m ′ ) ⊺ commute, see[Gro86, Proposition 2.7(5, 6)]. It follows that there is an orthonormal basis B of R sups containing v sups and such that for every positive integer m , each vectorin B is an eigenvector of B ( m ) ⊺ . This proves the first assertion and item ( i ).To prove ( ii ) note that by [Gro87, Propositions 4.4 and 5.6] for all v and v ′ in R sups , the following series in τ in H belongs to M (Γ ( p )) φ ( v, v ′ )( τ ) := p − h v, v sups i sups h v ′ , v sups i sups + ∞ X m =1 h B ( m ) ⊺ v, v ′ i sups exp(2 πimτ ) . In particular, for each v in B different from v sups the modular form f v := φ ( v, v )has Fourier expansion f v ( τ ) = ∞ X m =1 λ v ( m ) exp(2 πimτ ) . Since the constant term of f v is zero, f v is cuspidal and item ( ii ) follows from (2.2)and Deligne’s bound [Del74, Th´eor`eme (cid:3)
Note that for every fundamental discriminant d and every integer f ≥
2, we have(5.12) deg(Λ df ) = deg(Λ d ) w d, (cid:0) R − d ∗ σ (cid:1) ( f ) , by (2.10) and (4.8) in Lemma 4.10 with e f = 1. Lemma 5.12.
For every ε > there is a constant C > , such that for every m in N and every fundamental discriminant d we have (cid:12)(cid:12) R − d ( m ) (cid:12)(cid:12) ≤ Cm ε and (cid:0) R − d ∗ σ (cid:1) ( m ) ≥ C − m − ε . Proof.
Recall that ψ d : N → {− , , } is the arithmetic function given by the Kro-necker symbol (cid:0) d · (cid:1) . Denote by µ the M¨obius function and note that R − d = µ ∗ ( µ · ψ d ).Thus, for every prime number q we have R − d ( q s ) = s = 0; − − ψ d ( q ) if s = 1; ψ d ( q ) if s = 2;0 if s ≥ . INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 51
This implies that for every m in N we have | R − d ( m ) | ≤ d ( m ), so the first inequalityfollows from (2.2).To prove the second inequality, let N in N be such that for every q ≥ N wehave q − q ≥ q − ε and let C ′ in ]0 ,
1[ be such that for every q in { , . . . , N } wehave q − q ≥ C ′ q − ε . Noting that for every integer s ≥ R − d ∗ σ )( q s ) = q s − ψ d ( q ) q s − ≥ q s − ( q − , we conclude that for every m in N we have( R − d ∗ σ )( m ) m ≥ Y q | m, prime q − q ≥ ( C ′ ) N m − ε . This completes the proof of the lemma. (cid:3)
Proof of Theorem 5.7.
Fix ε > ε ′ := ε . Let C (resp. C , C , C ) be theconstant given by Siegel’s estimate (5.9) (resp. (2.2), Lemma 5.11( ii ), Lemma 5.12)with ε replaced by ε ′ . Given a fundamental discriminant b d and an integer b f ≥ w b d, b f be as in Section 4.1 and put u ( b d b f ) := w b d, b f .Assume first f = 1, so D = d is a fundamental discriminant. For each integer m ≥ H p ( m ) the modified Hurwitz numbers defined by Gross [Gro87,(1.8)] and for each e ∈ Y sups ( F p ) put a e ( m ) := e )2 X D ′ discriminant D ′ | m h ( D ′ , e ) u ( D ′ ) , if − m is a discriminant and a e ( m ) := 0 otherwise. Then the following series in τ in H are modular forms of weight for Γ (4 p ): θ e ( τ ) := 1 + ∞ X m =1 a e ( m ) exp(2 πimτ ) and E p ( τ ) := p −
112 + 2 ∞ X m =1 H p ( m ) exp(2 πimτ ) , see [Gro87, (12.8), Proposition 12.9 and (12.11)]. Moreover, the modular form(5.13) θ e ( τ ) − p − E p ( τ ) = ∞ X m =1 (cid:18) a e ( m ) − p − H p ( m ) (cid:19) exp(2 πimτ )is cuspidal [EOY05, (3.6), (3.13) and (3.14)]. Then by (4.4) we have a e ( | d | ) = e )2 u ( d ) h ( d, e ) = e )2 ǫ d u ( d ) deg(Λ d | D e )and by [Gro87, (1.7) and (1.8)] we have H p ( | d | ) = h ( d )2 ǫ d u ( d ) = deg(Λ d )2 ǫ d u ( d ) . We thus have (cid:12)(cid:12)(cid:12)(cid:12) a e ( | d | ) − p − H p ( | d | ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:18) e )2 ǫ d u ( d ) (cid:19) deg(Λ d ) (cid:12)(cid:12)(cid:12)(cid:12) deg(Λ d | D e )deg(Λ d ) − p − e ) (cid:12)(cid:12)(cid:12)(cid:12) . Combined with Siegel’s bound (5.9) and Duke’s bound [Duk88, Theorem 5] forthe | d | -th coefficient of the cuspidal modular form (5.13), we obtain the desiredestimate in the case where f = 1. To prove the theorem in the case where f ≥
2, note that by Lemma 5.9( iii ) wecan suppose that f is not divisible by p . Then w d,f = 1 and by (4.8) in Lemma 4.10with e f = 1 and Lemma 5.9( i ), we have v (Λ D ) = 1 w d, X f ∈ N ,f | f R − d (cid:18) ff (cid:19) B ( f ) ⊺ v (Λ d ) . Writing v (Λ d ) as a linear combination of the elements in the base B , we obtain v (Λ D )deg(Λ D ) = deg(Λ d )deg(Λ D ) w d, X v ∈B (cid:0) R − d ∗ λ v (cid:1) ( f ) (cid:28) v (Λ d )deg(Λ d ) , v (cid:29) sups v. Noting that h v (Λ d ) , v sups i sups = X e ∈ Y sups ( F p ) deg(Λ d | D e ) = deg(Λ d ) , by Lemma 5.11( i ) and (5.12) we obtain(5.14) v (Λ D )deg(Λ D ) − v sups = X v ∈B ,v = v sups (cid:0) R − d ∗ λ v (cid:1) ( f ) (cid:0) R − d ∗ σ (cid:1) ( f ) (cid:28) v (Λ d )deg(Λ d ) − v sups , v (cid:29) sups v. By our choice of C , C and C , for every v in B different from v sups we have (cid:12)(cid:12)(cid:0) R − d ∗ λ v (cid:1) ( f ) (cid:12)(cid:12) ≤ C C X f ∈ N ,f | f (cid:18) ff (cid:19) ε ′ f + ε ′ ≤ C C C f +2 ε ′ . Combined with (5.14) and our choice of C , this implies (cid:13)(cid:13)(cid:13)(cid:13) v (Λ D )deg(Λ D ) − v sups (cid:13)(cid:13)(cid:13)(cid:13) sups ≤ C C C f − + ε (cid:13)(cid:13)(cid:13)(cid:13) v (Λ d )deg(Λ d ) − v sups (cid:13)(cid:13)(cid:13)(cid:13) sups . So the desired estimate follows from the definition of k · k sups and the case f = 1,established above. This completes the proof of the theorem. (cid:3) Proof of Theorem 5.1.
That Λ d is contained in Y sups ( C p ) is given by The-orem 4.11. That Λ d is compact then follows from Proposition 5.6( i ), the fact thatthe set Y sups ( F p ) is finite and the fact that Π e is continuous.Given e in Y sups ( F p ), let b ν e d be as in Proposition 5.6( ii ). Since Π e is continuous, ν e d := (Π e ) ∗ b ν e d is a Borel probability measure on Y ( C p ) whose support is Λ d ∩ D e . Then, the Borelmeasure on Y ( C p ), ν d := 24 p − X e ∈ Y sups ( F p ) e ) ν e d is a probability measure by the mass formula (2.5) and its support is Λ d .To complete the proof of Theorem 5.1, it remains to prove (5.1). We use thefollowing consequence of Theorem 5.7. Lemma 5.13.
Let e in Y sups ( F p ) and ε > be given. Then, for every p -supersingularfundamental discriminant d such that | d | is sufficiently large we have V | d | ( e ) ≥ | d | − ε . INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 53
Proof.
Theorem 4.2 and Lemma 4.5( ii ) imply that for every p -supersingular fun-damental discriminant d , we have V | d | ( e ) ≥
13 deg(Λ d | D e ) . Together with Theorem 5.7 and Siegel’s estimate (5.9), this implies the desiredassertion. (cid:3)
The estimate (5.1) is a direct consequence of Theorem 5.7, the lemma above andthe following proposition.
Proposition 5.14.
Let e be in Y sups ( F p ) and let d be a fundamental p -adic dis-criminant. Then for all ε > and δ > there is a constant C > such that thefollowing property holds. For every function F : Λ d ∩ D e → R that is constant onevery ball of Λ d ∩ D e of radius δ and every fundamental discriminant d in d forwhich V | d | ( e ) is nonempty, we have (cid:12)(cid:12)(cid:12)(cid:12)Z F d δ Λ d | D e − Z F d ν e d (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:18) sup Λ d ∩ D e | F | (cid:19) | d | + ε V | d | ( e ) . Proof.
Given a function F : Λ d ∩ D e → R , put ˘ F := Tr e ( F ◦ Π e ) and note that byTheorem 4.2, the definition of Tr e , Proposition 5.6( ii ) and the change of variablesformula, for every fundamental discriminant d in d for which V | d | ( e ) is nonemptywe have(5.15) Z F d δ Λ d | D e = 1 V | d | ( e ) X φ ∈ V | d | ( e ) ˘ F ( b φ ) and Z F d ν e d = Z ˘ F d ν e | d | . Since L e, d is compact and Π e is continuous, by Proposition 5.4 there is ˘ δ > F is constant on every ball of Λ d ∩ D e of radius δ , then ˘ F is constant onevery ball of L e, d of radius ˘ δ .Given ε >
0, let
C > n = 3, δ replaced by ˘ δ , Q = Q e and S = 4. Moreover, let d be a fundamental discriminantin d for which V | d | ( e ) is nonempty and let F : Λ d ∩ D e → R be constant on everyball of radius δ . Then, by Proposition 5.2( ii ) the hypotheses of Corollary 3.2 aresatisfied with ℓ = | d | , m = | d | and with F replaced by ˘ F . The desired estimate isthen a direct consequence of Corollary 3.2 and (5.15). (cid:3) Equidistribution of partial Hecke orbits
The goal of this section is to prove the following quantitative version of Theo-rem C in Section 1.2.
Theorem C’.
For every E in Y sups ( C p ) and every coset N in Q × p / Nr E containedin Z p , the closure Orb N ( E ) in Y sups ( C p ) of the partial Hecke orbit Orb N ( E ) iscompact. Moreover, there is a Borel probability measure µ E N on Y ( C p ) whose supportis equal to Orb N ( E ) and such that the following property holds. For every ε > and every locally constant function F : Y sups ( C p ) → R , there is a constant C > such that for every E ′ in Orb Nr E ( E ) and every n in N ∩ N we have (6.1) (cid:12)(cid:12)(cid:12)(cid:12)Z F d δ T n ( E ′ ) − Z F d µ E N (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cn − + ε . To prove Theorem C’, for all e and e ′ in Y sups ( F p ) we introduce “supersingularspheres” of the p -adic space Hom F p ( F e , F e ′ ) and show that each of these sets carriesa natural homogeneous measure (Proposition 6.2 in Section 6.1). We show that eachclosure of a partial Hecke orbit restricted to a residue disc is the projection of asupersingular sphere by an evaluation map (Proposition 6.4 in Section 6.2). Thenwe prove Theorem C’ in Section 6.3, using our results on the asymptotic distributionof integer points on p -adic spheres in Section 3. We also show that distinct partialHecke orbits have different limit measures (Proposition 6.9 in Section 6.4).The following corollary of Theorem C’ is used in Section 7.2. To state it, we usethe action of Hecke correspondences on sets and measures, see Section 2.8. Given E in Y sups ( C p ), denote by · the multiplication in the quotient group Q × p / Nr E . Corollary 6.1.
Let E be in Y sups ( C p ) and let N and N ′ be cosets in Q × p / Nr E contained in Z p . Then, for every E ′ in Orb Nr E ( E ) and every n in N ∩ N we have T n (cid:16) Orb N ′ ( E ′ ) (cid:17) = Orb N · N ′ ( E ) and σ ( n ) ( T n ) ∗ µ E ′ N ′ = µ E N · N ′ . Proof.
Let ( n j ) ∞ j =1 be a sequence in N ′ ∩ N tending to ∞ , such that for every j theinteger n j is coprime to n . On one hand, the sequence ( δ T n · nj ( E ′ ) ) ∞ j =1 convergesto µ E N · N ′ as j → ∞ by Theorem C’. Noting that by (2.14) for every j we have δ T n · nj ( E ′ ) = 1 σ ( n ) ( T n ) ∗ δ T nj ( E ′ ) , on the other hand ( δ T n · nj ( E ′ ) ) ∞ j =1 converges to σ ( n ) ( T n ) ∗ µ E ′ N ′ as j → ∞ by Theo-rem C’ with E = E ′ . This proves the equality of measures. The equality of setsfollows by comparing the supports of these measures using Theorem C’ again. (cid:3) Supersingular spheres and their homogeneous measures.
Throughoutthis section we fix e and e ′ in Y sups ( F p ).The group Hom( e, e ′ ) is a free Z -module of rank 4. Given an isogeny φ in Hom( e, e ′ ),denote by φ its dual isogeny in Hom( e ′ , e ). The ring End( e ) is a maximal order inthe quaternion algebra End( e ) ⊗ Q over Q and the map End( e ) → End( e ) givenby φ φ extends by Q -linearity to the canonical involution in End( e ) ⊗ Q . Thering End( e ) has characteristic zero and the subring generated by the identity map e on e is equal to the subset of endomorphisms φ satisfying φ = φ . We identify thissubring with Z . Then for every φ in Hom( e, e ′ ) we have φφ = deg( φ ).The Z -bilinear map h , i : Hom( e, e ′ ) × Hom( e, e ′ ) → End( e )( φ , φ )
7→ h φ , φ i := φ φ + φ φ . takes values in Z and induces the quadratic form Q e,e ′ : Hom( e, e ′ ) → Z φ Q e,e ′ ( φ ) := h φ, φ i . This quadratic form is positive definite and defined over Z . Furthermore, for every φ in Hom( e, e ′ ) we have Q e,e ′ ( φ ) = Q e ′ ,e ( φ ) = φφ = deg( φ )and for every e ′′ in Y sups ( C p ) and every ψ in Hom( e ′ , e ′′ ) we have(6.2) Q e,e ′′ ( ψφ ) = Q e ′ ,e ′′ ( ψ ) Q e,e ′ ( φ ) . INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 55
Define R e,e ′ := Hom F p ( F e , F e ′ ) and G e,e ′ := Iso F p ( F e , F e ′ )and note that in the case where e ′ = e we have R e,e = R e and G e,e = G e .Endow R e,e ′ with the unique distance such that for every ϕ in G e,e ′ the map R e → R e,e ′ defined by ψ ϕ ◦ ψ is an isometry. The natural map Hom( e, e ′ ) → R e,e ′ ,denoted by φ b φ as in Section 2.6, extends to an isomorphism of Z p -modulesHom( e, e ′ ) ⊗ Z p ∼ −→ R e,e ′ . We extend accordingly the map Hom( e, e ′ ) → Hom( e ′ , e ), φ φ to a Z p -linearmap R e,e ′ → R e ′ ,e , h , i to a Z p -bilinear map R e,e ′ × R e,e ′ → Z p and Q e,e ′ to aquadratic form on R e,e ′ taking values in Z p . Note that the identity (6.2) extendsby continuity to every φ in R e,e ′ and every ψ in R e ′ ,e ′′ . In particular, for every ϕ in R e,e ′ and ℓ in Z p , we have(6.3) Q e,e ( ϕ ) = nr( ϕ ) and Q e,e ′ ( ℓϕ ) = ℓ Q e,e ′ ( ϕ ) . For each nonzero ℓ in Z p we call S ℓ ( e, e ′ ) := { ϕ ∈ R e,e ′ : Q e,e ′ ( ϕ ) = ℓ } , a supersingular sphere , which we consider as a metric subspace of R e,e ′ . On theother hand, by (6.2) with e = e ′ = e ′′ , the set S ( e, e ) is a subgroup of G e and forevery ϕ in S ( e, e ) we have ϕ − = ϕ .Note that S ( e, e ) is a subgroup of G e and that G e acts on R e ′ ,e by(6.4) G e × R e ′ ,e → R e ′ ,e ( g, ϕ ) gϕ. For each nonzero ℓ in Z p this action restricts to an action of S ( e, e ) on S ℓ ( e ′ , e ).This action is also the restriction of the natural action of the orthogonal group O Q e,e ( Z p )on S ℓ ( e ′ , e ), to its subgroup S ( e, e ). Proposition 6.2.
For all e and e ′ in Y sups ( F p ) and every nonzero ℓ in Z p , thefollowing properties hold. ( i ) The supersingular sphere S ℓ ( e, e ′ ) is nonempty and compact. If in addition ℓ belongs to Z × p , then it is contained in G e,e ′ . ( ii ) The action of S ( e, e ) on S ℓ ( e ′ , e ) induced by (6.4) is faithful, transitive andby isometries. ( iii ) There exists a unique Borel probability measure µ e ′ ,eℓ on S ℓ ( e ′ , e ) that isinvariant under the action of S ( e, e ) . This measure is also uniquely deter-mined as the unique Borel probability measure on S ℓ ( e ′ , e ) that is invariantunder the action of the orthogonal group O Q e ′ ,e ( Z p ) . Moreover, the supportof µ e ′ ,eℓ is equal to all of S ℓ ( e ′ , e ) . ( iv ) For every e ′′ in Y sups ( F p ) and g in G e,e ′ , the map ϕ ϕg maps S ℓ ( e ′ , e ′′ ) to S ℓ nr( g ) ( e, e ′′ ) and µ e ′ ,e ′′ ℓ to µ e,e ′′ ℓ nr( g ) . We call µ e ′ ,eℓ the homogeneous measure of S ℓ ( e ′ , e ).The proof of Proposition 6.2 is given after the following lemma. Lemma 6.3.
For all e and e ′ in Y sups ( F p ) , we have G e,e ′ = { ϕ ∈ R e,e ′ : Q e,e ′ ( ϕ ) ∈ Z × p } . Proof.
For each ϕ in G e,e ′ we have by (6.2) with e ′′ = eQ e,e ′ ( ϕ − ) Q e ′ ,e ( ϕ ) = Q e,e ( ϕ − ϕ ) = Q e,e ( c e ) = 1 . This implies that Q e,e ′ ( ϕ ) belongs to Z × p .Let ϕ be an element of R e,e ′ such that ℓ := Q e,e ′ ( ϕ ) belongs to Z × p . Then ℓ − ϕ belongs to R e ′ ,e and we have( ℓ − ϕ ) ϕ = ℓ − ( ϕϕ ) = ℓ − Q e,e ′ ( ϕ ) = 1and ϕ ( ℓ − ϕ ) = ℓ − ( ϕϕ ) = ℓ − Q e ′ ,e ( ϕ ) = ℓ − Q e,e ′ ( ϕ ) = 1 . This proves that ℓ − ϕ is the inverse of ϕ and hence that ϕ belongs to G e,e ′ . Theproof of the lemma is thus complete. (cid:3) Proof of Proposition 6.2.
The last assertion of item ( i ) is a direct consequence ofLemma 6.3. To prove that S ℓ ( e, e ′ ) is nonempty, note that by (6.2) with e ′′ = e it is sufficient to prove that Q e,e is surjective. Let n ≥ u in Z × p . By [Vig80, Chapitre
II,
Corollaire B e contains anelement θ satisfying θ = − p and a subalgebra K isomorphic to Q p . By Lemma 2.3there is v in O K satisfying nr( v ) = u . Then v and v ′ := θ n v are both in R e and Q e,e ( v ′ ) = nr( v ′ ) = p n u . This proves that Q e,e is surjective and completes theproof that for every ℓ in Z p the set S ℓ ( e, e ′ ) is nonempty. That S ℓ ( e, e ′ ) is compactfollows from the fact that R e,e ′ is compact and Q e,e ′ is continuous. The proof ofitem ( i ) is thus complete.To prove item ( ii ), let ϕ and ϕ ′ be given elements of S ℓ ( e ′ , e ). Then the ele-ment g := ℓ − ϕ ′ ϕ of B e belongs to R e , satisfies gϕ = ℓ − ϕ ′ ℓ = ϕ ′ and by (6.2) wehave Q e,e ( g ) = ℓ − Q e ′ ,e ( ϕ ′ ) Q e ′ ,e ( ϕ ) = 1 . This proves that g belongs to S ( e, e ) and that the action of S ( e, e ) on S ℓ ( e ′ , e ) istransitive. To prove that this action is faithful, note that for g in S ( e, e ) and ϕ in S ℓ ( e ′ , e ) satisfying gϕ = ϕ , we have g = ℓ − g ( ϕϕ ) = ℓ − ( gϕ ) ϕ = ℓ − ϕϕ = ℓ − ℓ = 1 . Finally, since for each g in G e the left multiplication map is an isometry, it followsthat the action of S ( e, e ) on S ℓ ( e ′ , e ) is by isometries. This completes the proof ofitem ( ii ).Item ( iii ) is a direct consequence of item ( ii ) and Lemma 5.3.The first part of item ( iv ) follows from (6.2) and (6.3), while the second partfollows from item ( iii ) and the fact that the left (resp. right) multiplication mapby g is an isometry. This completes the proof of the proposition. (cid:3) From supersingular spheres to Hecke orbits.
In this section we provethe following proposition. It relates supersingular spheres to partial Hecke orbitsand defines a natural measure on the closure of a partial Hecke orbit inside a residuedisc.For e and e ′ in Y sups ( F p ) and x in b D e , define the evaluation mapEv x,e ′ : G e,e ′ → b D e ′ g Ev x,e ′ ( g ) := g · x. INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 57
Proposition 6.4.
For all e and e ′ in Y sups ( F p ) , every E in D e and every coset N in Z × p / Nr E , the following properties hold for every x in Π − e ( E ) and ℓ in N . ( i ) We have (6.5) Π − e ′ (cid:16) Orb N ( E ) ∩ D e ′ (cid:17) = Ev x,e ′ ( S ℓ ( e, e ′ )) and this set is compact. ( ii ) The measure on b D e ′ defined by (6.6) b µ E,e ′ N := (Ev x,e ′ ) ∗ ( µ e,e ′ ℓ ) depends only on E and N and not on x or ℓ . Moreover, the support of b µ E,e ′ N is equal to Π − e ′ (cid:16) Orb N ( E ) ∩ D e ′ (cid:17) . The proof of this proposition is given at the end of this section. For e in Y sups ( F p )and x in b D e , denote the stabilizer of x in G e by G e,x := { g ∈ G e : g · x = x } . Lemma 6.5.
Let e be in Y sups ( F p ) , E in D e and x in Π − e ( E ) . Then (6.7) Nr E = { nr( g ) : g ∈ G e,x } , and this set contains ( Z × p ) . In particular, Nr E is an open subgroup of Z × p whoseindex is at most two if p is odd and at most four if p = 2 .Proof. If E is not in Y sups ( Q p ), then Nr E = ( Z × p ) by definition. On the otherhand, x is not in X e ( O Q p ) and therefore G e,x = Z × p by Lemma 4.5( i ). This impliesthat the right-hand side of (6.7) is equal to ( Z × p ) and proves the lemma in the casewhere E is not in Y sups ( Q p ).Assume E is in Y sups ( Q p ) and let α : e F E → F e be an isomorphism of for-mal Z p -modules such that ( F E , α ) represents x . We use the ring homomorphismEnd( F E ) → End F p ( F e ), given by ϕ α ◦ e ϕ ◦ α − . Assume E is a formal CM point,let K be the field of fractions of End( F E ) and nr : K → Q p its norm map. Then wehave nr( ϕ ) = nr( α ◦ e ϕ ◦ α − ) and therefore(6.8) Nr E = (cid:8) nr( α ◦ e ϕ ◦ α − ) : ϕ ∈ Aut( F E ) (cid:9) . If E is not a formal CM point, then the group Aut( F E ) is isomorphic to Z × p andthe equality above also holds. Then (6.7) is a direct consequence of (6.8) andLemma 2.4. That Nr E contains ( Z × p ) follows from (6.8) and from the fact that F E is a formal Z p -module. (cid:3) Item ( ii ) of the following lemma is a reformulation of [Men12, Theorem 1.2]. Lemma 6.6.
For all e and e ′ in Y sups ( F p ) , the following properties hold. ( i ) For every integer n ≥ that is not divisible by p , we have n ( e, e ′ ) = e ′ ) · deg( T n ( e ) | { e ′ } ) . ( ii ) For every ε > there is a constant C > , such that for every integer n ≥ that is not divisible by p we have (cid:12)(cid:12)(cid:12)(cid:12) n ( e, e ′ ) σ ( n ) − p − (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cn − + ε . Proof.
Item ( ii ) is a direct consequence of item ( i ), [Men12, Theorem 1.2] and thefact that e ′ ) ≤ i ), note that for each isogeny φ in Hom n ( e, e ′ ) the endomor-phism φφ is equal to the morphism of multiplication by n on e , which is separable,see, e.g. , [Sil09, Chapter III, Corollary 5.4]. This proves that φ is separable andtherefore that its kernel Ker( φ ) is in C := (cid:8) C ≤ e ( F p ) : C = n, e/C = e ′ (cid:9) . Thus, φ Ker( φ ) defines a surjective map K : Hom n ( e, e ′ ) → C , see, e.g. , [Sil09,Chapter III, Proposition 4.12]. The desired identity follows from C = deg( T n ( e ) | { e ′ } )and from the fact that for every C in C we have K − ( C ) = e ′ ), see, e.g. ,[Sil09, Chapter III, Corollary 4.11]. This proves item ( i ) and completes the proofof the lemma. (cid:3) Proof of Proposition 6.4.
To prove item ( i ), note first that for each ℓ in N theset Ev x,e ′ ( S ℓ ( e, e ′ )) is compact, because Ev x,e ′ is continuous and S ℓ ( e, e ′ ) is compactby Proposition 6.2( i ).We prove (6.5) first in the case where E is in Y sups ( Q p ). We start showing thatthe left-hand side is contained in the right-hand side. Since the right-hand side iscompact and Π e is continuous, it is sufficient to show that every element x ′ ofΠ − e ′ (Orb N ( E ) ∩ D e ′ )belongs to the right-hand side of (6.5). Put E ′ := Π e ′ ( x ′ ) and let α ′ : f E ′ → e ′ be anisomorphism such that the induced isomorphism of formal Z p -modules α ′ : e F E ′ → F e ′ is such that the deformation ( e F E ′ , α ′ ) of F e ′ represents x ′ . On the other hand,let α : e E → e be an isomorphism of elliptic curves, so that the induced isomor-phism α : e F E → F e is such that ( F E , α ) represents x . By definition of Orb N ( E ),there is n in N ∩ N such that E ′ is in the support of T n ( E ). That is, there is anisogeny φ : E → E ′ such that deg( φ ) belongs to N . Denote by ϕ : F E → F E ′ theinduced isomorphism and note that the element g := α ′ ◦ e ϕ ◦ α − of G e,e ′ satis-fies Q e,e ′ ( g ) = deg( φ ). On the other hand, the deformation g · ( F E , α ) = ( F E , α ′ ◦ e ϕ )is isomorphic to ( F E ′ , α ′ ) via the isomorphism ϕ , so g · x = x ′ . Since Q e,e ′ ( g ) and ℓ are both in N , by Lemma 6.5 there is g in G e such that g · x = x and nr( g ) = ℓQ e,e ′ ( g ) − . We thus have Q e,e ′ ( gg ) = ℓ and ( gg ) · x = g · x = x ′ . This proves that x ′ belongs to the right-hand side of (6.5) and completes theproof that the left-hand side of (6.5) is contained in the right-hand side when E isin Y sups ( Q p ).To prove the reverse inclusion, recall that S ℓ ( e, e ′ ) is nonempty by Proposi-tion 6.2( i ) and let g be a given element of this set. Let ( m j ) ∞ j =1 be a sequence in N tending to ∞ that is contained in the coset ℓ ( Z × p ) of Z × p / ( Z × p ) and that convergesto ℓ in Z p . For each j let u j in Z × p be such that m j = ℓu j and such that u j → Z × p as j → ∞ . By (2.1), (2.10), Proposition 6.2 and Lemma 6.6( ii ), for every c in ] ,
1[ the hypotheses of Theorem D are satisfied for n = 4 and Q = Q e,e ′ . Ap-plying this theorem and using that the support of the limit measure µ e,e ′ ℓ is equal INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 59 to S ℓ ( e, e ′ ) and therefore contains g , we obtain that for each j we can choose φ j in Hom m j ( e, e ′ ) in such a way that M − u j ( b φ j ) → g in S ℓ ( e, e ′ ) as j → ∞ . Since u j → Z p as j → ∞ , we conclude that b φ j → g in G e,e ′ as j → ∞ andtherefore that b φ j · x → g · x in b D e ′ as j → ∞ . Noting that the sequence ( m j ) ∞ j =1 iscontained in the coset N , for each j we haveΠ e ′ ( b φ j · x ) ∈ supp (cid:0) T m j ( E ) (cid:1) ∩ D e ′ ⊆ Orb N ( E ) ∩ D e ′ . This proves that g · x belongs to the closure of Π − e ′ (Orb N ( E ) ∩ D e ′ ) and completesthe proof of (6.5) in the case where E is in Y sups ( Q p ). In particular, this provesthat Π − e ′ (cid:16) Orb N ( E ) ∩ D e ′ (cid:17) is compact for E in Y sups ( Q p ).It remains to prove (6.5) in the case where E is not in Y sups ( Q p ). We use the factthat G e acts by isometries on b D e (Lemma 2.6( i )). In view of (2.16), this impliesthat for every x in b D e the Hausdorff distance betweenΠ − e ′ (cid:16) Orb N ( E ) ∩ D e ′ (cid:17) and Π − e ′ (cid:16) Orb N (Π e ( x )) ∩ D e ′ (cid:17) , and between Ev x,e ′ ( S ℓ ( e, e ′ )) and Ev x ,e ′ ( S ℓ ( e, e ′ )) , are both bounded from above by the distance between x and x . Since Y sups ( Q p )is dense in Y sups ( C p ) and for every x in Π − e ( Y sups ( Q p ) ∩ D e ) the equality (6.5)holds with x replaced by x , we conclude that the Hausdorff distance between theleft-hand side and the right-hand side of (6.5) is equal to zero. Since both of thesesets are closed, it follows that they are equal. This completes the proof of item ( i ).To prove item ( ii ), for each g in G e denote by T g : G e,e ′ → G e,e ′ the rightmultiplication map h hg . We use that for every x in b D e , we have(6.9) Ev x,e ′ ◦ T g = Ev g · x,e ′ . Let x and x ′ in Π − e ( E ) and ℓ and ℓ ′ in N be given. Then there exists an auto-morphism φ in Aut( e ) such that b φ · x ′ = x and by Lemma 6.5 there is g in G e such that g · x ′ = x ′ and nr( g ) ℓ = ℓ ′ . Then Q e,e ( b φ ) = deg( φ ) = 1, so b φ belongsto S ( Q e,e ) and therefore ( T b φg ) ∗ µ e,e ′ ℓ = µ e,e ′ ℓ ′ by Proposition 6.2( iv ). Combinedwith (6.9), this implies(Ev x ′ ,e ′ ) ∗ µ e,e ′ ℓ ′ = (Ev x ′ ,e ′ ) ∗ (( T b φg ) ∗ µ e,e ′ ℓ ) = (Ev ( b φg ) · x ′ ,e ′ ) ∗ µ e,e ′ ℓ = (Ev x,e ′ ) ∗ µ e,e ′ ℓ . This proves the first assertion of item ( ii ).To prove the remaining assertions of item ( ii ), fix x in Π − e ( E ) and ℓ in N . Thenthe support of µ e,e ′ ℓ is equal to S ℓ ( e, e ′ ) by Proposition 6.2( iii ) and therefore thesupport of (Ev x,e ′ ) ∗ µ e,e ′ ℓ is equal to Ev x,e ′ ( S ℓ ( e, e ′ )). Then the desired assertionfollows from item ( i ). This completes the proof of item ( ii ) and of the proposition. (cid:3) Proof of Theorem C’.
In this section we prove Theorem C’. The mainingredient is the following proposition, whose proof is based on our results on theasymptotic distribution of integer points on p -adic spheres in Section 3.Let E in Y sups ( C p ) and N in Z × p / Nr E be given. That Orb N ( E ) is compactfollows from Proposition 6.4( i ), the fact that the set Y sups ( F p ) is finite and the factthat Π e is continuous. For e in Y sups ( F p ), let b µ E,e N be the measure on b D e givenby (6.6) in Proposition 6.4( ii ). Since Π e is continuous, µ E,e N := (Π e ) ∗ b µ E,e N is a Borel probability measure on Y ( C p ) whose support is Orb N ( E ) ∩ D e . Then,the Borel measure on Y ( C p ), µ E N := 24 p − X e ∈ Y sups ( F p ) e ) µ E,e N is a probability measure by the mass formula (2.5) and its support is Orb N ( E ). Proposition 6.7.
For every E in Y sups ( C p ) , e in Y sups ( F p ) and every locally con-stant function F : D e → R , the following property holds for every ε > and everycoset N in Z × p / Nr E . There is a constant C > such that for every e ′ in Y sups ( F p ) ,every E ′ in Orb Nr E ( E ) ∩ D e ′ and every n in N ∩ N for which Hom n ( e ′ , e ) isnonempty, we have (cid:12)(cid:12)(cid:12)(cid:12)Z F d δ T n ( E ′ ) | D e − Z F d µ E,e N (cid:12)(cid:12)(cid:12)(cid:12) ≤ C n + ε n ( e ′ , e ) . The proof of this proposition is given after the following lemma.
Lemma 6.8.
Let E be in Y sups ( C p ) and let N and N ′ be cosets in Z × p / Nr E . Thenfor every E ′ in Orb N ( E ) , we have Nr E ′ = Nr E , Orb N ′ ( E ′ ) = Orb N · N ′ ( E ) and µ E ′ N ′ = µ E N · N ′ . Proof.
Let e and e ′ in Y sups ( F p ) be such that E and E ′ are in D e and D e ′ , respec-tively. Moreover, fix ℓ in N , ℓ ′ in N ′ , x in Π − e ( E ) and x ′ in Π − e ′ ( E ′ ). By Propo-sition 6.4( i ) there is g in S ℓ ( e, e ′ ) such that g · x = x ′ . Then G e ′ ,x ′ = g G e,x g − and therefore Nr E ′ = Nr E by Lemma 6.5.To prove the second and third equalities, let e ′′ in Y sups ( F p ) be given andlet T : G e ′ ,e ′′ → G e,e ′′ be the right multiplication map h hg . By Proposi-tion 6.2( iv ), we have T ( S ℓ ′ ( e ′ , e ′′ )) = S ℓℓ ′ ( e, e ′′ ) and T ∗ ( µ e ′ ,e ′′ ℓ ′ ) = µ e,e ′′ ℓℓ ′ . Hence,Ev x ′ ,e ′′ ( S ℓ ′ ( e ′ , e ′′ )) = { g · ( g · x ) : g ∈ S ℓ ′ ( e ′ , e ′′ ) } = { ( gg ) · x : g ∈ S ℓ ′ ( e ′ , e ′′ ) } = { ˘ g · x : ˘ g ∈ S ℓℓ ′ ( e, e ′′ ) } = Ev x,e ′′ ( S ℓℓ ′ ( e, e ′′ ))and Ev x,e ′′ ∗ ( µ e,e ′′ ℓℓ ′ ) = (cid:16) Ev x,e ′′ ◦ T (cid:17) ∗ ( µ e ′ ,e ′′ ℓ ′ ) = Ev x ′ ,e ′′ ∗ ( µ e ′ ,e ′′ ℓ ′ ) . Together with Proposition 6.4 and the definition of the measures µ E ′ N ′ and µ E N · N ′ , thisimplies the second and third equalities and completes the proof of the lemma. (cid:3) INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 61
Proof of Proposition 6.7.
Let e in Y sups ( F p ) be such that E is in D e and let x bein Π − e ( E ). Since Π e and Ev x ,e are both continuous, the function ˘ F := F ◦ Π e ◦ Ev x ,e is locally constant. Let δ > F is constant on every ball of G e ,e of ra-dius δ . Fix ε > C be given by Corollary 3.2 with n = 4. Let e ′ in Y sups ( F p )and E ′ in Orb Nr E ( E ) ∩ D e ′ be given. Then by Proposition 6.4( i ) with N replacedby Nr E , there is g in S ( e , e ′ ) such that x ′ := g · x belongs to Π − e ′ ( E ′ ). Denoteby T : G e ′ ,e → G e ,e the right multiplication map given by g ′ g ′ g and by S E the union of the compact sets Orb N ′ ( E ) with N ′ running over Z × p / Nr E . Then S E is compact by Lemma 6.5 and ˘ F ′ := ˘ F ◦ T satisfies(6.10) ˘ F ′ = F ◦ Π e ◦ Ev x ′ ,e and sup G e ′ ,e | ˘ F ′ | = sup S E ∩ D e | F | . On the other hand, since T maps G e ′ ,e to G e ,e isometrically, the function ˘ F ′ isconstant on every ball of G e ′ ,e of radius δ .Let n in N ∩ N be such that Hom n ( e ′ , e ) is nonempty. Then by (2.16) and (6.10),we have(6.11) Z F d δ T n ( E ′ ) | D e = 1 n ( e ′ , e ) X φ ∈ Hom n ( e ′ ,e ) ˘ F ′ ( b φ ) . On the other hand, by Lemma 6.8, the definition of µ E ′ ,e N and the change of variablesformula, we have Z F d µ E,e N = Z F d µ E ′ ,e N = Z ˘ F ′ d µ e ′ ,en . Together with Proposition 6.2, (6.11) and Corollary 3.2 with ℓ and m equal to n ,this implies (cid:12)(cid:12)(cid:12)(cid:12)Z F d δ T n ( E ′ ) | D e − Z F d µ E,e N (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n ( e ′ , e ) X φ ∈ Hom n ( e ′ ,e ) ˘ F ′ ( b φ ) − Z ˘ F ′ d µ e ′ ,en (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:18) sup S E ∩ D e | F | (cid:19) n + ε n ( e ′ , e ) . (cid:3) Proof of Theorem C’.
In view of the considerations at the beginning of this section,in the case where N is contained in Z × p it only remains to prove the estimate (6.1).In that case, (6.1) is a direct consequence of Proposition 6.7 combined with (2.1),(2.10), the definition of µ E N and [Men12, Theorem 1.2] or Lemma 6.6( ii ).Assume N is not contained in Z × p and let N be the coset in Z × p / Nr E and k ≥ N = p k N . Then, for every E ′ in Y sups ( C p ) we haveOrb N ( E ′ ) = T p k (Orb N ( E ′ )) and Orb N ( E ′ ) = T p k (cid:16) Orb N ( E ′ ) (cid:17) and this last set is compact. Moreover, the support of the measure µ E N := 1 σ ( p k ) ( T p k ) ∗ µ E N . is equal to Orb N ( E ) and the estimate (6.1) is a direct consequence of the same for-mula with N replaced by N , using the change of variables formula. This completesthe proof of the theorem. (cid:3) On partial Hecke orbits and their limit measures.
This section is de-voted to prove the following proposition, which is used in Section 7.1.
Proposition 6.9.
For every E in Y sups ( C p ) , the following properties hold. ( i ) For distinct cosets N and N ′ in Z × p / Nr E , the partial Hecke orbits Orb N ( E ) and Orb N ′ ( E ) are disjoint. ( ii ) For distinct cosets N and N ′ in Q × p / Nr E contained in Z p , the measures µ E N and µ E N ′ are different. The proof of this proposition is at the end of this section. As in Section 4.2, we de-note Katz’ valuation by v p and by b v p : Y sups ( C p ) → h , pp +1 i , the map b v p = min n v p , pp +1 o . Lemma 6.10.
For every x in i , pp +1 i , every divisor D supported on b v − p ( x ) andevery integer n ≥ , the divisor T p n D is supported on b v − p (cid:16)h p − n x, pp +1 i(cid:17) and wehave (6.12) ( T p n D ) | b v − p ( p − n x ) = ( t n | b v − p ( p − n x ) ) ∗ D . The proof of this lemma is based on the following lemma.
Lemma 6.11 ([HMRL20, Proposition 4.5]) . Denote by τ the identity on Div (cid:16)h , pp +1 i(cid:17) ,let τ be the piecewise-affine correspondence on h , pp +1 i defined by τ ( x ) := [ px ] + p [ xp ] if x ∈ h , p +1 i ;[1 − x ] + p [ xp ] if x ∈ i p +1 , pp +1 i , and for each integer m ≥ define the correspondence τ m on h , pp +1 i recursively,by τ m := τ ◦ τ m − − p τ m − . Then for every integer m ≥ and every integer n ≥ not divisible by p , we have (6.13) ( b v p ) ∗ ◦ T p m n | Y sups ( C p ) = σ ( n ) · τ m ◦ ( b v p ) ∗ . Proof of Lemma 6.10.
By Lemma 6.11, for every x ′ in i , pp +1 i and every divisor D supported on b v − p (cid:16)h x ′ , pp +1 i(cid:17) , the divisor T p D is supported on b v − p (cid:16)h p − x ′ , pp +1 i(cid:17) .Together with (2.15) and an induction argument, this implies the first assertion.To prove the second assertion, we proceed by induction on n . The case n = 1is a direct consequence of (4.5) in Lemma 4.7 and Lemma 4.8. Let n ≥ T p n +1 D ) | b v − p ( p − ( n +1) x ) = ( T p ( T p n D )) | b v − p ( p − ( n +1) x ) . On the other hand, by (4.5) in Lemma 4.7, Lemma 4.8 and the induction hypothesis,we have ( T p ( T p n D )) | b v − p ( p − ( n +1) x ) = ( t | b v − p ( p − ( n +1) x ) ) ∗ (( T p n D ) | b v − p ( p − n x ) ))= ( t | b v − p ( p − ( n +1) x ) ) ∗ (( t n | b v − p ( p − n x ) ) ∗ D )= ( t n +1 | b v − p ( p − ( n +1) x ) ) ∗ D . INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 63
Together with (6.14) this completes the proof of the induction step and of thelemma. (cid:3)
Proof of Proposition 6.9.
To prove item ( i ), suppose that Orb N ( E ) and Orb N ′ ( E )intersect and let E ′ be a common element. Let e and e ′ in Y sups ( F p ) be such that E and E ′ belong to D e and D e ′ , respectively and fix x in Π − e ( E ) and x ′ in Π − e ′ ( E ′ ).By Proposition 6.4( i ) we can find g and g ′ in G e,e ′ so that Q e,e ′ ( g ) and Q e,e ′ ( g ′ ) arein N and N ′ , respectively and such that g · x = x ′ = g ′ · x . This implies that g − g ′ is in G e,x and therefore that Q e,e ′ ( g ) − Q e,e ′ ( g ′ ) = nr( g − g ′ ) ∈ Nr E . This implies that N = N ′ and completes the proof of item ( i ).To prove item ( ii ), assume that µ E N = µ E N ′ . In particular, Orb N ( E ) = Orb N ′ ( E )by Theorem C. Let n ≥ n ′ ≥ N and N ′ the cosetsin Z × p / Nr E such that N = p n N and N ′ = p n ′ N ′ . Then we have(6.15) Orb N ( E ) = T p n (cid:16) Orb N ( E ) (cid:17) , Orb N ′ ( E ) = T p n ′ (cid:16) Orb N ′ ( E ) (cid:17) , and(6.16) µ E N = 1 σ ( p n ) ( T p n ) ∗ µ E N and µ E N ′ = 1 σ ( p n ′ ) ( T p n ′ ) ∗ µ E N ′ by Corollary 6.1. Put x E := b v p ( E ) and note that by (6.13) in Lemma 6.11with m = 0, the sets Orb N ( E ) and Orb N ′ ( E ) are both contained in b v − p ( x E ).Then (6.15) and Lemmas 6.10 and 6.11 imply that Orb N ( E ) is contained in b v − p (cid:16)h p − n x E , pp +1 i(cid:17) and intersects b v − p ( p − n x E ) and that Orb N ′ ( E ) is contained in b v − p (cid:16)h p − n ′ x E , pp +1 i(cid:17) and intersects b v − p ( p − n ′ x E ). We conclude that n = n ′ . In the case where n = 0 thedesired assertion follows from item ( i ). Assume n ≥ δ of t n | b v − p ( p − n x E ) is equal to p n if x E < pp +1 and to ( p + 1) p n − if x E = pp +1 . In all the cases, ( t n ) ∗ ( t n | b v − p ( p − n x E ) ) ∗ is equal to δ times the identity on b v − p ( x E ). We thus have by (6.16) and Lemma 6.10, δµ E N = ( t n ) ∗ (cid:16) ( t n | b v − p ( p − n x E ) ) ∗ µ E N (cid:17) = ( t n ) ∗ (cid:16) σ ( p n ) µ E N | b v − p ( p − n x E ) (cid:17) = ( t n ) ∗ (cid:16) σ ( p n ) µ E N ′ | b v − p ( p − n x E ) (cid:17) = ( t n ) ∗ (cid:16) ( t n | b v − p ( p − n x E ) ) ∗ µ E N ′ (cid:17) = δµ E N ′ . In particular, Orb N ( E ) = Orb N ′ ( E ) by Theorem C and therefore N = N ′ byitem ( i ). Since n = n ′ , this implies N = N ′ . This completes the proof of item ( ii )and of the proposition. (cid:3) Equidistribution of CM points along a p -adic discriminant In this section we prove Theorems A and B. For a fundamental p -adic discrimi-nant d , we start showing how Λ d is decomposed into closures of partial Hecke orbits(Proposition 7.1 in Section 7.1). The set Λ d coincides with a partial Hecke orbitif Q p ( √ d ) is unramified over Q p . If Q p ( √ d ) is ramified over Q p , then Λ d is parti-tioned into precisely two closures of partial Hecke orbits. In this case we use genus theory to determine for each discriminant D in d , how supp(Λ D ) is distributed be-tween these closures of partial Hecke orbits (Proposition 7.4 in Section 7.2). Oncethese results are established, in Section 7.3 we deduce Theorems A and B in thecase of fundamental p -adic discriminants from Theorems 5.1 and C’. We deduce thegeneral case from that of fundamental p -adic discriminants using the (formal) CMpoints formulae in Sections 4.2 and 4.3.7.1. Hecke orbits of formal CM points.
Given a fundamental p -adic discrimi-nant d , the goal of this section is to prove the following proposition describing Λ d in terms of closures of partial Hecke orbits. Put Nr d := n nr( g ) : g ∈ O × Q p ( √ d ) o . Proposition 7.1.
Let d be a fundamental p -adic discriminant. Then, for every E in Λ d we have Nr E = Nr d and the following properties hold. ( i ) If Q p ( √ d ) is unramified over Q p , then Nr d = Z × p , (7.1) Λ d = Orb Nr d ( E ) and ν d = µ E Nr d . ( ii ) If Q p ( √ d ) is ramified over Q p , then Nr d has index two in Z × p , (7.2) Λ d = Orb Nr d ( E ) ⊔ Orb Z × p r Nr d ( E ) and ν d = 12 (cid:16) µ E Nr d + µ E Z × p r Nr d (cid:17) . In particular, ν d (cid:16) Orb Nr d ( E ) (cid:17) = ν d (cid:16) Orb Z × p r Nr d ( E ) (cid:17) = 12 ,µ E Nr d = 2 ν d | Orb Nr d ( E ) and µ E Z × p r Nr d = 2 ν d | Orb Z × p r Nr d ( E ) . The proof of this proposition is given after a couple of lemmas.
Lemma 7.2.
Fix a fundamental p -adic discriminant d and ∆ in d . Then, forall e and e ′ in Y sups ( F p ) and all ϕ in S − ∆ ( e ) and ϕ ′ in S − ∆ ( e ′ ) , there is g in S ( e, e ′ ) ∪ S − ( e, e ′ ) such that gϕg − = ϕ ′ or gϕg − = − ϕ ′ . Proof.
Fix g in G e,e ′ and note that g − ϕ ′ g belongs to S − ∆ ( e ) by Proposition 5.2( iv ).By Proposition 5.2( ii ) there is ρ in G e such that ρ − g − ϕ ′ g ρ = ϕ . Supposethat Q e,e ′ ( g ρ ) (resp. − Q e,e ′ ( g ρ )) belongs to Nr d and let ψ in Q p ( ϕ ) be suchthat nr( ψ ) = Q e,e ′ ( g ρ ) (resp. nr( ψ ) = − Q e,e ′ ( g ρ )). Then g := g ρψ − belongsto S ( e, e ′ ) (resp. S − ( e, e ′ )) and we have gϕg − = ( g ρ ) ψ − ϕψ ( g ρ ) − = ( g ρ ) ϕ ( g ρ ) − = ϕ ′ . It remains to consider the case where neither Q e,e ′ ( g ρ ) nor − Q e,e ′ ( g ρ ) is in Nr d .In this case there is γ in G e such that γϕγ − = ϕ and γ = − Q e,e ′ ( g ρ ) − , see Lemma 2.2. Then g := g ργ belongs to S − ( e, e ′ ) and we have gϕg − = ( g ρ ) γϕγ − ( g ρ ) − = ( g ρ ) ϕ ( g ρ ) − = − ( g ρ ) ϕ ( g ρ ) − = − ϕ ′ . This completes the proof of the lemma. (cid:3)
INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 65
Lemma 7.3.
Let d be a fundamental p -adic discriminant such that Q p ( √ d ) isramified over Q p and let e be in Y sups ( F p ) . Moreover, let ϕ be in L e, d and recallthat Fix e ( ϕ ) has precisely two elements. Then there is g in G e mapping Fix e ( ϕ ) toitself, interchanging its elements. Moreover, for every such g the reduced norm nr( g ) is in Nr d if and only if − is not in Nr d .Proof. Recall that Nr d has index two in Z × p by Lemma 2.3, so there is γ in G e such that γϕγ − = ϕ and γ ∈ Z × p r Nr d , see Lemma 2.2. Thus, by Lemma 4.5( iv ) we have γ · Fix e ( ϕ ) = Fix e ( ϕ ) = Fix e ( ϕ ) . On the other hand, γ is not in Q p ( ϕ ), so γ cannot have a fixed point in Fix e ( ϕ )by Lemma 4.5( iv ). Since Fix e ( ϕ ) has only two elements, γ must interchange them.This completes the proof of the first assertion of the lemma.To prove the second assertion, let g in G e be such that g · Fix e ( ϕ ) = Fix e ( ϕ )and such that g interchanges the elements of Fix e ( ϕ ). Then γg fixes each elementof Fix e ( ϕ ), so it belongs to Q p ( ϕ ) by Lemma 4.5( iv ). In particular,nr( γg ) = nr( γ ) nr( g ) = − γ nr( g ) ∈ Nr d . We conclude that nr( g ) is in Nr d if and only if − Nr d . This completesthe proof of the lemma. (cid:3) Proof of Proposition 7.1.
Let e in Y sups ( F p ) be such that E belongs to D e and fix x in Π − e ( E ) and ∆ in d . By Proposition 5.6( i ) there is ϕ in S − ∆ ( e ) such that x belongs to Fix e ( ϕ ).To prove the first assertion, note that by the definition of Λ d and the fact thatthe p -adic discriminant is a complete isomorphism invariant for p -adic quadraticorders (Lemma A.1( ii )), the p -adic quadratic orders End( F E ) and O Q p ( √ d ) areisomorphic. Thus, Nr E = Nr d .To prove items ( i ) and ( ii ), let e ′ be in Y sups ( F p ) and let b F : b D e ′ → R be acontinuous function. Note that for every u in Z × p and every g in G e , we have byProposition 6.2( iv ) and the change of variables formula Z Tr e ′ ( b F )( ρ ( gϕg − ) ρ − ) d µ e,e ′ u ( ρ ) = Z Tr e ′ ( b F )( ρgϕ ( ρg ) − ) d µ e,e ′ u ( ρ )= Z Tr e ′ ( b F )( b ρϕ b ρ − ) d µ e,e ′ u nr( g ) ( b ρ ) . (7.3)Together with Lemmas 4.5( iv ) and 7.2, this implies for every ϕ ′ in S − ∆ ( e ′ ) we have Z Tr e ′ ( b F )( ρϕρ − ) d (cid:16) µ e,e ′ + µ e,e ′ − (cid:17) ( ρ ) = Z Tr e ′ ( b F )( ρϕ ′ ρ − ) d (cid:16) µ e,e ′ + µ e,e ′ − (cid:17) ( ρ ) . Together with Propositions 5.2( iv ) and 5.6( ii ) and the change of variables formula,this implies(7.4) Z Tr e ′ ( b F )( ρϕρ − ) d (cid:16) µ e,e ′ + µ e,e ′ − (cid:17) ( ρ )= Z Z Tr e ′ ( b F )( ρϕ ′ ρ − ) d (cid:16) µ e,e ′ + µ e,e ′ − (cid:17) ( ρ ) d ν e − ∆ ( ϕ ′ )= Z Z Tr e ′ ( b F )( ρϕ ′ ρ − ) d ν e − ∆ ( ϕ ′ ) d (cid:16) µ e,e ′ + µ e,e ′ − (cid:17) ( ρ )= Z Z Tr e ′ ( b F )( ˘ ϕ ) d ν e ′ − ∆ ( ˘ ϕ ) d (cid:16) µ e,e ′ + µ e,e ′ − (cid:17) ( ρ )= 2 Z Tr e ′ ( b F ) d ν e ′ − ∆ = 2 Z b F d b ν e ′ d . If Q p ( √ d ) is unramified over Q p , then Nr d = Z × p by Lemma 2.3 and by Propo-sitions 5.4( i ) and 6.4( ii ) and the change of variables formula, we have for each u in { , − } Z Tr e ′ ( b F )( ρϕρ − ) d µ e,e ′ u ( ρ ) = Z b F ( x e, ∆ ( ρϕρ − )) d µ e,e ′ u ( ρ )= Z b F (cid:16) Ev x,e ′ ( ρ ) (cid:17) d µ e,e ′ u ( ρ )= Z b F d b µ E,e ′ Nr d . Together with (7.4), this implies b ν e ′ d = b µ E,e ′ Nr d . Since this holds for every e ′ in Y sups ( F p ),we obtain ν d = µ E Nr d . The equality of sets in (7.1) follows from a comparison of thesupports of these measures, using Theorems 5.1 and C. This completes the proofof item ( i ).Suppose Q p ( √ d ) is ramified over Q p , so Nr d has index two in Z × p by Lemma 2.3and Fix e ( ϕ ) has precisely two elements by Lemma 4.5( ii ). Denote by ˘ x the ele-ment of Fix e ( ϕ ) that is different from x and put ˘ E := Π e (˘ x ). Then, by Proposi-tions 5.4( ii ) and 6.4( ii ) and the change of variables formula, we have Z Tr e ′ ( b F )( ρϕρ − ) d (cid:16) µ e,e ′ + µ e,e ′ − (cid:17) ( ρ )= 12 Z b F ( x + e, ∆ ( ρϕρ − )) + b F ( x − e, ∆ ( ρϕρ − )) d (cid:16) µ e,e ′ + µ e,e ′ − (cid:17) ( ρ )= 12 Z b F (cid:16) Ev x,e ′ ( ρ ) (cid:17) + b F (cid:16) Ev ˘ x,e ′ ( ρ ) (cid:17) d (cid:16) µ e,e ′ + µ e,e ′ − (cid:17) ( ρ )= 12 Z b F d (cid:16)b µ E,e ′ Nr d + b µ ˘ E,e ′ Nr d + b µ E,e ′ − Nr d + b µ ˘ E,e ′ − Nr d (cid:17) . Since (7.4) and the previous formula hold for every e ′ in Y sups ( F p ), we obtain(7.5) ν d = 14 (cid:16) µ E Nr d + µ ˘ E Nr d + µ E − Nr d + µ ˘ E − Nr d (cid:17) . INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 67
On the other hand, Proposition 6.4( i ) and Lemma 7.3 imply˘ E ∈ ( Orb Z × p r Nr d ( E ) if − Nr d ;Orb Nr d ( E ) if − Nr d . So, by Lemma 6.8 we have µ E − Nr d = µ E Nr d and µ ˘ E − Nr d = µ ˘ E Nr d = µ E Z × p r Nr d , if − Nr d . If − Nr d , then we have µ ˘ E Nr d = µ E Nr d and µ E − Nr d = µ ˘ E − Nr d = µ E Z × p r Nr d . Thus, in the all the cases (7.5) yields the equality of measures in (7.2). That theclosures of the partial orbits in the first equality of (7.2) are disjoint is given byProposition 6.9( i ). Then the equality of sets in (7.2) and the remaining assertionsof item ( ii ) follow from a comparison of the supports of the measures ν d , µ E Nr d and µ E Z × p r Nr d , using Theorems 5.1 and C. This completes the proof of item ( ii ) andof the proposition. (cid:3) Symmetry breaking.
Fix a fundamental p -adic discriminant d for which Q p ( √ d )is ramified over Q p and recall that Nr d has index two in Z × p (Lemma 2.3). Wechoose a point E d in Λ d , as follows. Suppose d contains a prime discriminant d that is divisible by p . Then d is the unique fundamental discriminant in d with thisproperty and we choose an arbitrary E d in supp(Λ d ). If d does not contain a primediscriminant divisible by p , then we choose an arbitrary E d in Λ d . With this choiceof E d , put Λ + d := Orb Nr d ( E d ) and Λ − d := Orb Z × p r Nr d ( E d )and note that by Proposition 7.1( ii ) we have the partition(7.6) Λ d = Λ + d ⊔ Λ − d . The goal of this section is to prove the following proposition, describing for eachdiscriminant D in d how Λ D is distributed between Λ + d and Λ − d . To state it, definethe divisors Λ + D := Λ D | Λ + d and Λ − D := Λ D | Λ − d , and note that Λ D = Λ + D + Λ − D . Recall that (cid:0) ·· (cid:1) denotes the Kronecker symbol. Proposition 7.4.
Let d be a fundamental discriminant divisible by p . Then, forevery integer f ≥ that is not divisible by p the following properties hold. ( i ) If d is not a prime discriminant, then deg(Λ + df ) = deg(Λ − df ) . ( ii ) If d is a prime discriminant, then Λ ± df = Λ df if (cid:16) df (cid:17) = ± if (cid:16) df (cid:17) = ∓ . The proof of this proposition is given at the end of this section. The followingcorollary is a direct consequence of Corollary 6.1 and Proposition 7.1. To state it,define ν + d := µ E d Nr d and ν − d := µ E d Z × p r Nr d . Note that the support of ν + d (resp. ν − d ) is equal to Λ + d (resp. Λ − d ) by Theorem Cand that by Proposition 7.1( ii ) we have ν d = 12 (cid:0) ν + d + ν − d (cid:1) , ν d (Λ + d ) = ν d (Λ − d ) = 12 , ν + d = 2 ν d | Λ + d and ν − d = 2 ν d | Λ − d . Corollary 7.5.
For every fundamental p -adic discriminant d and every integer n ≥ that is not divisible by p , we have T n (Λ d ) = Λ d and σ ( n ) ( T n ) ∗ ( ν d ) = ν d . If in addition Q p ( √ d ) is ramified over Q p , then we also have T n (Λ ± d ) = Λ ± d and σ ( n ) ( T n ) ∗ ( ν ± d ) = ν ± d if n is in Nr d and if n is not in Nr d then we have T n (Λ ± d ) = Λ ∓ d and σ ( n ) ( T n ) ∗ ( ν ± d ) = ν ∓ d . A quadratic fundamental discriminant is the discriminant of the ring of integersof a quadratic (real or imaginary) extension of Q . So, a quadratic fundamentaldiscriminant is a fundamental discriminant if and only if it is negative. A quadraticdiscriminant is prime , if it is fundamental and divisible by only one prime number.Every quadratic fundamental discriminant can be written uniquely up to order asa product of prime quadratic discriminants that are mutually coprime, see, e.g. ,[Lem00, Proposition 2.2]. Note that a quadratic fundamental discriminant d divisi-ble by p is prime if and only if p is odd and d = ( − p − p , or if p = 2 and d = − , − m and n , denote by ( n, m ) p the Hilbert symbol over Q p ,see, e.g. , [Ser73, Chapter III] or [Lem00, Section 2.5]. Lemma 7.6.
Let d be a fundamental discriminant divisible by p and let p ∗ be theunique prime quadratic discriminant divisible by p in the factorization of d intoprime quadratic discriminants. Then the following properties hold. ( i ) For every n in N coprime to d , we have ( n, d ) p = (cid:16) p ∗ n (cid:17) ; ( ii ) If d = p ∗ , then there is a prime number q such that ( q, d ) p = − and (cid:18) dq (cid:19) = 1 . In the proofs of Lemma 7.6 and of Proposition 7.4 given below, we use severalproperties of the Hilbert symbol that can be found, e.g. , in [Ser73, Theorems 1and 2, Chapter III]. We also use the following notation. Given a quadratic exten-sion K of Q denote by Cl( K ) the ideal class group of K , and for a fractional ideal a of K denote by [ a ] its class in Cl( K ) and by Nr ( a ) its norm. Proof of Lemma 7.6.
Note that d ′ := dp ∗ is a quadratic fundamental discriminant.Since ( · , d ) p and (cid:16) p ∗ · (cid:17) are both completely multiplicative, it is sufficient to prove INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 69 item ( i ) in the case where n is a prime number q not dividing d . We have( q, d ) p = ( q, d ′ ) p ( q, p ∗ ) p = ( q, p ∗ ) p = (cid:16) qp (cid:17) if p is odd;( − q − if p = 2 and p ∗ = − − q − + q − if p = 2 and p ∗ = − − q − if p = 2 and p ∗ = 8 . In all the cases the number above is equal to (cid:16) p ∗ q (cid:17) by the quadratic reciprocity lawand its complementary laws, see, e.g. , [Ser73, Theorems 5 and 6, Chapter I]. Thiscompletes the proof of item ( i ).Put K := Q ( √ d ) and let χ : Cl( K ) → { , − } be the unique quadratic charactersuch that for every prime ideal p of O K we have χ ([ p ]) = (cid:16) p ∗ Nr ( p ) (cid:17) if gcd( Nr ( p ) , p ∗ ) = 1; (cid:16) d ′ Nr ( p ) (cid:17) if gcd( Nr ( p ) , d ′ ) = 1 , see, e.g. , [Lem00, Section 2.3]. It follows from genus theory that there exists anideal class [ a ] in Cl( K ) such that χ ([ a ]) = −
1, see, e.g. , [Lem00, Theorem 2.17].Let b be an ideal of O K in [ a ] whose norm is coprime to d . By decomposing b intoprimes ideals we find a prime ideal q of O K such that χ ([ q ]) = −
1. Then Nr ( q ) iscoprime to d and by item ( i ) we have( Nr ( q ) , d ) p = (cid:18) p ∗ Nr ( q ) (cid:19) = χ ([ q ]) = − . This implies that q := Nr ( q ) is a prime number. Since q does not divide d , wehave (cid:16) dq (cid:17) = 1. This proves item ( ii ). (cid:3) Proof of Proposition 7.4.
Let d be the fundamental p -adic discriminant contain-ing d . We use several times that, if n in N is coprime to d , then we have ( n, d ) p = 1if and only if n is in Nr d , see, e.g. , [Ser73, Proposition 1, Chapter III]. Put K := Q ( √ d ), and recall that R d is the function ∗ ψ d . Fix a field isomorphismbetween C p and C , and for each E in Y ( C p ) denote by E ⊗ C the element of Y ( C )obtained from E by base change to C . Moreover, denote by E : Cl( K ) → supp(Λ d )the bijection so that for each fractional ideal a of K , the quotient C / a is isomorphicto ( E ([ a ]) ⊗ C )( C ), see, e.g. , [Sil94, Chapter II, Section 1].We first prove the proposition in the case where f = 1. To prove item ( i )when f = 1, let q be a prime number such that ( q, d ) p = − (cid:16) dq (cid:17) = 1(Lemma 7.6(ii)). In particular, q is different from p , it is split in K , and it isnot in Nr d . If follows that there is an ideal q of O K of norm q such that qq = q O K .Thus, the map a aq induces a bijection ι of supp(Λ d ) given by E ([ a ])
7→ E ([ aq ]),whose inverse is given by E ([ a ])
7→ E ([ aq ]). Since for every fractional ideal a of K each of the natural maps C / aq → C / a and C / aq → C / a is an isogeny of degree q , byCorollary 7.5 the involution ι interchanges supp(Λ + d ) and supp(Λ − d ). In particular,deg(Λ + d ) = deg(Λ − d ).To prove item ( ii ) when f = 1, note that the point E d used to define Λ + d and Λ − d at the beginning of the section, is in supp(Λ + d ) by definition. Let E be agiven element of supp(Λ d ), and let φ : E d → E be an isogeny whose degree is not divisible by p [HMRL20, Lemma 4.8]. Let a , a be ideals of O K such that E ([ a ]) = E , E ([ a a ]) = E d , and such that the natural map C / a a → C / a corresponds to theisogeny φ . Consider the prime factorization a = q α · · · q α n n . Then for each j in { , . . . , n } the norm of q j is either a prime number q j and then (cid:16) dq j (cid:17) = 1, orthe square of a prime number q ′ j and then (cid:16) dq ′ j (cid:17) = −
1. In all the cases we get (cid:16) d Nr ( a ) (cid:17) = 1. Thus, by Lemma 7.6( i ) we have(deg( φ ) , d ) p = ( Nr ( a ) , d ) p = (cid:18) d Nr ( a ) (cid:19) = 1 , hence deg( φ ) is in Nr d and therefore E is in Λ + d by Corollary 7.5. This provesitem ( ii ) when f = 1.It remains to consider the case where f ≥
2. In this case, we have (5.12).Moreover, by Corollary 7.5 we also have(7.7) deg(Λ ± df ) = deg(Λ ± d ) w d, X f ∈ N ,f | ff ∈ Nr d R − d (cid:18) ff (cid:19) σ ( f )+ deg(Λ ∓ d ) w d, X f ∈ N ,f | ff Nr d R − d (cid:18) ff (cid:19) σ ( f ) . Combined with item ( i ) with f = 1, this implies item ( i ) for every f ≥
2. In viewof Lemma 7.6( i ) and (7.7), to deduce item ( ii ) for f ≥ f = 1it is sufficient to show the following: For every r in N that is in Z × p r Nr d , wehave R − d ( r ) = 0. Since the function R − d is multiplicative and ( · , d ) p is completelymultiplicative, it is sufficient to show that for every prime number q different from p such that ( q , d ) p = − s ≥
1, we have R − d ( q s ) = 0. Notingthat ψ d ( q ) = (cid:16) dq (cid:17) = − i ) and denoting the M¨obius function by µ ,this follows from a direct computation using the formula R − d = µ ∗ ( µ · ψ d ). (cid:3) Proof of Theorems A and B.
The proof of Theorems A and B is given atthe end of this section.For a fundamental p -adic discriminant d for which Q p ( √ d ) is ramified over Q p ,let Λ + d , Λ − d , ν + d and ν − d be as in Section 7.2. Proposition 7.7.
For every ε > and every locally constant function F : Y sups ( C p ) → R ,there is a constant C > such that the following property holds. Let d be a fun-damental p -adic discriminant, d be a fundamental discriminant in d and f ≥ aninteger that is not divisible by p . Then (7.8) (cid:12)(cid:12)(cid:12)(cid:12)Z F d δ df − Z F d ν d (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cf − + ε , if Q p ( √ d ) is unramified over Q p , and if Q p ( √ d ) is ramified over Q p then (7.9) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z F d δ df − deg(Λ + df )deg(Λ df ) Z F d ν + d − deg(Λ − df )deg(Λ df ) Z F d ν − d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cf − + ε . INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 71
Proof.
Put ε ′ := ε and let C and C be the constants given by (2.2) and Lemma 5.12,respectively, with ε replaced by ε ′ .Assume first that Q p ( √ d ) is unramified over Q p , so Nr d = Z × p . Fix E in Λ d and let C be the constant given by Theorem C’ in Section 6 with ε replacedby ε ′ and N = Nr d . Then we have µ E Nr d = ν d by Proposition 7.1( i ). Thus,applying (2.10), (4.8) in Lemma 4.10 with e f = 1, Theorem C’ to each element E ′ of supp(Λ d ) and each divisor f ≥ f and (5.12), we obtain(7.10) (cid:12)(cid:12)(cid:12)(cid:12)Z F d δ df − Z F d ν d (cid:12)(cid:12)(cid:12)(cid:12) = w d,f w d, deg(Λ df ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X f ∈ N ,f | f R − d (cid:18) ff (cid:19) σ ( f ) X E ′ ∈ supp(Λ d ) (cid:18)Z F d δ T f ( E ′ ) − Z F d µ E Nr d (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C w d,f deg(Λ d ) w d, deg(Λ df ) X f ∈ N ,f | f (cid:12)(cid:12)(cid:12)(cid:12) R − d (cid:18) ff (cid:19) σ ( f ) (cid:12)(cid:12)(cid:12)(cid:12) f − + ε ′ . Using (5.12) again, we have by our choice of C and C (7.11) w d,f deg(Λ d ) w d, deg(Λ df ) X f ∈ N ,f | f (cid:12)(cid:12)(cid:12)(cid:12) R − d (cid:18) ff (cid:19) σ ( f ) (cid:12)(cid:12)(cid:12)(cid:12) f − + ε ′ ≤ C C f − ε ′ X f ∈ N ,f | f (cid:18) ff (cid:19) ε ′ f +2 ε ′ ≤ C C f − +4 ε ′ . Together with (7.10), this gives (7.8) with C = C C C and completes the proofof the proposition in the case where Q p ( √ d ) is unramified over Q p .Assume that Q p ( √ d ) is ramified over Q p and recall that Nr d has index two in Z × p .Fix E + in Λ + d and E − in Λ − d and let C ′ be the maximum value of the constantgiven by Theorem C’ with ε replaced by ε ′ and with E = E + or E − and N = Nr d or Z × p r Nr d . Applying (4.8) in Lemma 4.10 with e f = 1 and Corollary 7.5, weobtainΛ + df w d,f = X f ∈ N ,f | ff ∈ Nr d R − d (cid:18) ff (cid:19) T f (cid:18) Λ + d w d, (cid:19) + X f ∈ N ,f | ff Nr d R − d (cid:18) ff (cid:19) T f (cid:18) Λ − d w d, (cid:19) . On the other hand, by Lemma 6.8 we have µ E ± Nr d = ν ± d (resp. µ E ± Z × p r Nr d = ν ∓ d ) . Thus, applying (2.10), Theorem C’ to each element E ′ of supp(Λ d ) and each divi-sor f ≥ f , (7.7) and (7.11), we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Λ + d F d δ df − deg(Λ + df )deg(Λ df ) Z F d ν + d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = w d,f w d, deg(Λ df ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X f ∈ N ,f | ff ∈ Nr d R − d (cid:18) ff (cid:19) σ ( f ) X E ′ ∈ supp(Λ + d ) (cid:18)Z F d δ T f ( E ′ ) − Z F d µ E + Nr d (cid:19) + X f ∈ N ,f | ff Nr d R − d (cid:18) ff (cid:19) σ ( f ) X E ′ ∈ supp(Λ − d ) (cid:18)Z F d δ T f ( E ′ ) − Z F d µ E − Z × p r Nr d (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ′ w d,f deg(Λ d ) w d, deg(Λ df ) X f ∈ N ,f | f (cid:12)(cid:12)(cid:12)(cid:12) R − d (cid:18) ff (cid:19) σ ( f ) (cid:12)(cid:12)(cid:12)(cid:12) f − + ε ′ ≤ C C C ′ f − +4 ε ′ . A similar argument shows that the same estimate holds with Λ + d , Λ + d and ν + d replaced by Λ − d , Λ − d and ν − d , respectively. Combined, these estimates yield (7.9)with C = 2 C C C ′ and complete the proof of the proposition. (cid:3) Proposition 7.8.
Let d be a fundamental p -adic discriminant. For all ε > and δ > there is a constant C ′ > such that the following property holds. Forevery function F : Λ d → R that is constant on every ball of Λ d of radius δ , everyfundamental discriminant d in d and every integer f ≥ that is not divisible by p ,we have (cid:12)(cid:12)(cid:12)(cid:12)Z F d δ df − Z F d ν d (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ′ (cid:18) sup Λ d | F | (cid:19) | d | − + ε f ε . The proof of this proposition is given after the following lemma.
Lemma 7.9.
For every δ in ]0 , and every function F : Y sups ( C p ) → R that isconstant on every ball of radius δ , the following property holds. For every inte-ger n ≥ that is not divisible by p , the function T n F is constant on every ball ofradius δ .Proof. Let e in Y sups ( F p ) be given and recall that δ e = e ) / ≤
12. By (2.16),for each x in b D e we have T n F ◦ Π e ( x ) = X e ′ ∈ Y sups ( F p ) e ′ ) X φ ∈ Hom n ( e,e ′ ) F ◦ Π e ′ ( b φ · x ) . Since for each e ′ in Y sups ( F p ) the action of G e ′ on b D e ′ is by isometries (Lemma 2.6( i )),by (2.9) in Theorem 2.7 the function T n F ◦ Π e is constant on every ball of b D e ofradius δ . Using δ e ≤
12 and (2.9) in Theorem 2.7 again, we conclude that thefunction T n F is constant on every ball of D e of radius δ . Since e in Y sups ( F p ) isarbitrary, this implies the lemma. (cid:3) INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 73
Proof of Proposition 7.8.
Put ε ′ := ε and let C (resp. C ) be the constant givenby (2.2) (resp. Lemma 5.12) with ε replaced by ε ′ . Moreover, let C > δ replaced by δ ′ := min { , δ } .Let d be a fundamental discriminant in d and let f ≥ p . By Lemma 7.9, the function G := 1 (cid:0) R − d ∗ σ (cid:1) ( f ) X f ∈ N ,f | f R − d (cid:18) ff (cid:19) T f F is constant on every ball of radius δ ′ . On the other hand, by (4.8) in Lemma 4.10with e f = 1, (2.10), Theorem 5.1, Corollary 7.5 and the change of variables formula,we have(7.12) (cid:12)(cid:12)(cid:12)(cid:12)Z F d δ df − Z F d ν d (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z G d δ d − Z G d ν d (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:18) sup Λ d | G | (cid:19) | d | − + ε . On the other hand, for every E in Λ d we have by Corollary 7.5 and our choice of C and C | G ( E ) | ≤ (cid:0) R − d ∗ σ (cid:1) ( f ) X f ∈ N ,f | f (cid:12)(cid:12)(cid:12)(cid:12) R − d (cid:18) ff (cid:19) σ ( f ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) sup Λ d | F | (cid:19) ≤ C C (cid:18) sup Λ d | F | (cid:19) f − ε ′ X f ∈ N ,f | f (cid:18) ff (cid:19) ε ′ f ε ′ ≤ C C (cid:18) sup Λ d | F | (cid:19) f ε ′ . Together with (7.12) this implies the proposition with C ′ equal to CC C . (cid:3) Let d be a fundamental p -adic discriminant and m ≥ p -adicdiscriminant D := d p m , define the Borel measure ν D on Y ( C p ), by(7.13) ν D := ( p m ( t m (cid:12)(cid:12) A D ) ∗ ν d if Q p ( √ d ) is ramified over Q p ; p m − ( p +1) ( t m (cid:12)(cid:12) A D ) ∗ ν d if Q p ( √ d ) is unramified over Q p . It is a probability measure by the fact that ν d is a probability measure and thefact that for every integer m ≥ t | A d p m is of degree p , unless m = 1and Q p ( √ d ) is unramified over Q p in which case the degree is p + 1, see Lemma 4.8. Proof of Theorems A and B.
Denote by d the fundamental p -adic discriminant andby m ≥ D = d p m (Lemma A.1( i )).We first consider the case where m = 0, so D = d . The first assertion ofTheorem A is given by Theorem 5.1. The second assertion of Theorem A andTheorem B are a direct consequence of Propositions 7.4, 7.7 and 7.8.Assume m ≥
1. The first assertion of Theorem A follows from the fact that Λ d is compact, from Theorem 4.11( ii ), from the fact that A D is an affinoid and fromthe fact that t is analytic by Lemma 4.7. Using Theorem 4.11( ii ) again and thatthe support of ν d is equal to Λ d , we obtain that the support of ν D is equal to Λ D .The equidistribution statement in Theorem A for D follows from that for d , usingTheorem 4.6, the change of variables formula and the fact that the fundamentaldiscriminant of every discriminant in D is in d (Lemma 2.1). This completes theproof of Theorem A. To complete the proof of Theorem B, note that the compactsets Λ + D := ( t m (cid:12)(cid:12) A D ) − (Λ + d ) and Λ − D := ( t m (cid:12)(cid:12) A D ) − (Λ − d ) form a partition of Λ D . Define the Borel probability measure ν + D (resp. ν − D ) by (7.13)with ν d replaced by ν + d (resp. ν − d ). Then the remaining assertions of Theorem Bfor D follow from those for d , using Theorem 4.6 and the change of variables formula. (cid:3) Appendix A. Quadratic extensions of Q p and p -adic discriminants For the reader’s convenience, in this appendix we give a proof of Lemma 2.1and gather other basic facts about quadratic field extensions of Q p and p -adicdiscriminants. The proof of Lemma 2.1 is given at the end of this appendix. Weuse the notation and terminology in Section 2.1. Lemma A.1. (1) For every p -adic discriminant D there is a unique fundamental p -adic dis-criminant d , and a unique integer m ≥ , such that D = d p m . Conversely,every set of this form is a p -adic discriminant. ( ii ) For each fundamental p -adic discriminant d , and each integer m ≥ , ev-ery p -adic quadratic order of p -adic discriminant d p m is isomorphic to the Z p -order Z p + p m O Q p ( √ d ) in Q p ( √ d ) . In particular, the p -adic discrimi-nant is a complete isomorphism invariant of p -adic quadratic orders. ( iii ) The set of all fundamental p -adic discriminants is given by (A.1) (cid:8) Z × p r Z p , p ( Z × p ) , p ( Z × p r Z p ) (cid:9) if p is odd, and if p = 2 by (A.2) {− Z , − Z ,
12 + 32 Z , Z , − Z ,
24 + 64 Z , −
24 + 64 Z } . The proof of this lemma is given after the following lemma. Denote by Q p theset of all quadratic extensions of Q p inside C p . Recall that Q p denotes the uniqueunramified extension of Q p in Q p , and that for each ∆ in Q p we denote by Q p ( √ ∆)the unique element of Q p containing a root of X − ∆. For a quadratic extension K of Q p and x in K , consider tr( x ) , nr( x ) and ∆( x ) as defined in Section 2. Lemma A.2. If p is odd, then let A be an integer that is not a square modulo p . ( i ) Every quadratic field extension of Q p is isomorphic to a unique elementof Q p . Moreover, we have Q p = ( Q p ( √ A ) if p is odd ; Q ( √− if p = 2 , and Q p = (n Q p ( √ A ) , Q p ( √ p ) , Q p ( √ Ap ) o if p is odd ; (cid:8) Q ( √ d ) : d ∈ {− , − , − , − , − , − , − } (cid:9) if p = 2 . ( ii ) Let d be in { A, p, Ap } if p is odd and in {− , − , − , − , − , − , − } if p = 2 . Then, for Q := Q p ( √ d ) we have (A.3) O Q = ( Z h √− i if p = 2 and d = − Z p (cid:2) √ d (cid:3) otherwise , INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 75 the p -adic discriminant of O Q is equal to (A.4) ( d ( Z × p ) if p is odd, or p = 2 and d = − d ( Z × p ) if p = 2 and d = − , and the subgroup nr( O ×Q ) of Z × p is equal to (A.5) nr( O ×Q ) = Z × p if Q = Q p ;( Z × p ) if p is odd and d = p or Ap ;1 + 4 Z if p = 2 and d = − or − Z ) ∪ (3 + 8 Z ) if p = 2 and d = − or − Z ) ∪ ( − Z ) if p = 2 and d = − or − . In particular, for every ∆ in the p -adic discriminant of O Q we have (A.6) Q = Q p ( √ ∆) and O Q = Z p h ∆+ √ ∆2 i , and the index of nr( O ×Q ) in Z × p is one if Q = Q p and two if Q is ramifiedover Q p .Proof. Since C p contains an algebraic closure of Q p , every quadratic extension of Q p is isomorphic to one in Q p . Two distinct elements of Q p can not be isomorphicsince every quadratic extension of fields is normal. This proves that every quadraticextension of Q p is isomorphic to a unique element of Q p . The explicit descriptionof Q p given in items ( i ) and ( ii ) can be verified from straightforward computa-tions using explicit representatives of cosets in Q × p / ( Q × p ) as found, e.g. , in [Ser73,Section 3.3, Chapter II]. To prove the assertions about unramified extensions, notethat in the case where p is odd (resp. p = 2), Q := Q p ( √ A ) (resp Q ( √− X − A (resp. X + X + 1) over Q p . Since the reduction of thispolynomial is irreducible over F p , it follows Q is an unramified extension of Q p .This completes the proof of item ( i ).To prove (A.3) in item ( ii ), assume first p = 2 and d = − u := √− .Then, tr( u ) = nr( u ) = 1 and therefore u is in O Q . This proves that Z [ u ] iscontained in O Q . To prove the reverse inclusion, let α and β in Q be such that h := α + βu belongs to O Q . Then ∆( h ) = − β belongs to Z , and therefore β and α = h − βu are both in Z . This proves that O Q = Z [ u ]. Assume p is odd,or that p = 2 and d = −
3, and let α and β in Q p be such that h := α + β √ d belongs to O Q . Then tr( h ) = 2 α and ∆( h ) = d (2 β ) are both in Z p . Since d is an integer that is not divisible by p , this implies that 2 β belongs to Z p . If p is odd, then this implies that α and β are both in Z p , and weobtain O Q = Z p (cid:2) √ d (cid:3) . If p = 2 and d = −
3, then we also have − d ≡ , α ) − d (2 β ) = 4 nr( h ) ≡ , and we conclude that α and β are both in Z . This proves that O Q = Z (cid:2) √ d (cid:3) ,and completes the proof of (A.3).To prove (A.4), note that by (A.3) the p -adic discriminant of O Q is equal to ( d ( Z × p ) if p = 2 and d = − d ( Z × p ) otherwise . This is (A.4) in the case where p = 2. In the case where p is odd, the desiredassertion follows from the fact that 4 belongs to ( Z × p ) .To prove (A.5), assume first p is odd and d = A . Since Z × p r ( Z × p ) = A ( Z × p ) ,and the norm map from the residue field of Q to F p is surjective, we have nr( O ×Q ) = Z × p .Suppose p is odd and d = A or that p = 2. Then O Q = Z p (cid:2) √ d (cid:3) by (A.3). If p divides d , then O ×Q = Z × p + √ d Z p , andnr( O ×Q ) = (cid:8) − d ℓ : ℓ ∈ Z p (cid:9) ( Z × p ) = ( ( Z × p ) if p is odd;(1 + 8 Z ) ∪ (1 − d + 8 Z ) if p = 2 . It remains to consider the case where p = 2 and d = − − −
5. Since( Z × ) = 1 + 8 Z , in the case where d = − √− , √− , and 13 = nr(1 + 2 √− . This implies nr( O ×Q ) = Z × . If p = 2 and d = − −
5, then O ×Q = Z × + (1 + √ d ) Z and nr( O ×Q ) = (cid:8) ℓ + (1 − d ) ℓ : ℓ ∈ Z (cid:9) ( Z × ) = 1 + 4 Z . This completes the proof (A.5), of item ( ii ) and of the lemma. (cid:3) Proof of Lemma A.1.
To prove items ( i ) and ( ii ), note first that for every funda-mental p -adic discriminant d ′ and every integer m ≥
0, the p -adic discriminant ofthe Z p -order Z p + p m O Q p ( √ d ′ ) in Q p ( √ d ′ ) is equal to d ′ p m . Let O be a p -adic qua-dratic order, and let D be its p -adic discriminant. Then the field of fractions of O has the same discriminant as Q p ( √ D ), and it is therefore isomorphic to it. So, thereis an integer m ≥ O is isomorphic to the Z p -order Z p + p m O Q p ( √ D ) in Q p ( √ D ). Thus, if we denote by d the p -adic discriminant of O Q p ( √ D ) , then d is afundamental p -adic discriminant and D = d p m . This implies Q p ( √ d ) = Q p ( √ D ),and completes the proof of item ( ii ). To complete the proof of item ( i ), it re-mains to prove the uniqueness statement. To do this, let d and d ′ be fundamental p -adic discriminants, and m ≥ m ′ ≥ d p m = d ′ p m ′ .Then, Q p ( √ d ) = Q p ( √ d ′ ), and d and d ′ are both equal to the p -adic discriminantof O Q p ( √ d ) . It follows that m = m ′ . This completes the proof of item ( i ).To prove item ( iii ), note that (A.1) is a direct consequence of (A.4) in Lemma A.2,and the fact that for every integer A that is not a square modulo p we have A ( Z × p ) = Z × p r Z p . The identity (A.2) is a direct consequence of (A.4) in Lemma A.2,and the fact that ( Z × ) = 1 + 8 Z . This completes the proof of item ( iii ) and ofthe lemma. (cid:3) Proof of Lemma 2.1.
In view of Lemma A.1( i ), to prove the first assertion it issufficient to show that a fundamental discriminant d belongs to a fundamental p -adic discriminant if and only if it is p -supersingular. If p is odd, then by (A.1)the union of all fundamental p -adic discriminants is equal to Z p r (cid:0) p Z p ∪ Z p (cid:1) , so d belongs to a fundamental p -adic discriminant if and only if (cid:16) dp (cid:17) = 1. As remarkedabove, this last condition holds precisely when d is p -supersingular. If p = 2, thenby (2.3) we have d ≡ d ≡ − , . INNIK EQUIDISTRIBUTION ON THE SUPERSINGULAR LOCUS 77
Together with (A.2), we obtain that d belongs to a fundamental 2-adic discriminantif and only if d d is 2-supersingular. This completes the proof the first assertion.In view of Lemma A.1( i ), to prove the second assertion we can restrict to thecase where the p -adic discriminant d := D is fundamental. To do this, let ∆ in d be given, and fix an integer r ≥
6. Note that by (A.1) and (A.2), every ∆ ′ in Z p satisfying ord p (∆ − ∆ ′ ) ≥ r belongs to d . Assume p is odd, and note that (A.1)implies that ∆ is either in Z × p or in p Z × p . By Dirichlet’s theorem on prime numbersin arithmetic progressions there is a prime number p ′ such that p ′ ≡ − p (∆ + p ′ ) ≥ r in the former case, and p ′ ≡ − p mod 4 and ord p (∆ /p + p ′ ) ≥ r in the latter case. Putting d := − p ′ in the former case and d := − pp ′ in thelatter, we have that d is a fundamental discriminant, and that ord p (∆ − d ) ≥ r . Inparticular, d belongs to d . This completes the proof of the lemma when p is odd.Assume p = 2, and note that (A.2) implies that ∆ is either in − Z , − Z , or 8 + 16 Z . Let p ′ be a prime number satisfyingord (∆ + p ′ ) ≥ r, ord (∆ / p ′ ) ≥ r, or ord (∆ / p ′ ) ≥ r, and put d := − p ′ , − p ′ , or − p ′ , respectively. Then d is a fundamental discriminantthat satisfies ord (∆ − d ) ≥ r , and is therefore in d . This completes the proof ofthe second assertion, and of the lemma. (cid:3) References [BGJGP05] Matthew H. Baker, Enrique Gonz´alez-Jim´enez, Josep Gonz´alez, and Bjorn Poonen.Finiteness results for modular curves of genus at least 2.
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Instituto de Matem´aticas, Pontificia Universidad Cat´olica de Valpara´ıso, BlancoViel 596, Cerro Bar´on, Valpara´ıso, Chile.
Email address : [email protected] Facultad de Matem´aticas, Pontificia Universidad Cat´olica de Chile, Vicu˜na Mackenna4860, Santiago, Chile.
Email address : [email protected] Department of Mathematics, University of Rochester. Hylan Building, Rochester,NY 14627, U.S.A.
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