There are at most finitely many singular moduli that are S-units
aa r X i v : . [ m a t h . N T ] F e b THERE ARE AT MOST FINITELY MANY SINGULAR MODULITHAT ARE S -UNITS SEBASTI ´AN HERRERO, RICARDO MENARES, AND JUAN RIVERA-LETELIER
Abstract.
We show that for every finite set of prime numbers S , there are atmost finitely many singular moduli that are S -units. The key new ingredientis that for every prime number p , singular moduli are p -adically disperse. Introduction A singular modulus is the j -invariant of an elliptic curve with complex multipli-cation. These algebraic numbers lie at the heart of the theory of abelian extensionsof imaginary quadratic fields, as they generate the ring class fields of quadraticimaginary orders. This was predicted by Kronecker and referred to by himself ashis liebsten Jugendtraum .A result going back at least to Weber, ∗ states that every singular modulus isan algebraic integer. Thus, the absolute norm of a singular modulus is a rationalinteger, and the same holds for a difference of singular moduli. Gross and Zagiergave an explicit formula for the factorization of the absolute norms of differences ofsingular moduli [GZ85]. Roughly speaking, this formula shows that these absolutenorms are highly divisible numbers. In fact, Li showed recently that the absolutenorm of every difference of singular moduli is divisible by at least one prime num-ber [Li19]. Equivalently, that no difference of singular moduli is an algebraic unit.Li’s work extends previous results of Habegger [Hab15] and of Bilu, Habegger andK¨uhne [BHK20]. These results answered a question raised by Masser in 2011, whichwas motivated by results of Andr´e–Oort type.In view of these results, one is naturally led to look at differences of singularmoduli whose absolute norms are only divisible by a given set of prime numbers.To be precise, recall that for a set of prime numbers S , an algebraic integer is an S -unit if the only prime numbers dividing its absolute norm are in S . The followingis our main result. Theorem A.
Let S be a finite set of prime numbers and j a singular modulus.Then, there are at most finitely many singular moduli j such that j − j is an S -unit. The following corollary is a direct consequence of Theorem A with j = 0, whichis the j -invariant of every elliptic curve whose endomorphism ring is isomorphicto Z h √− i . Corollary 1.1.
For every finite set of prime numbers S , there are at most finitelymany singular moduli that are S -units. ∗ See [Web08,
Satz VI in § When restricted to S = ∅ , Theorem A is a particular instance of [Hab15, The-orem 2] and of [Li19, Corollary 1.3 with m = 1]. † This last result extends themain result of [BHK20], that in the case where S = ∅ the set of singular moduli inCorollary 1.1 is empty. In contrast to these results, the proof of Theorem A, whichfollows the strategy of proof of [Hab15, Theorem 2] in the case where S = ∅ , doesnot give an effectively computable upper bound.The number − is an example of a singular modulus that is a { } -unit. Infact, − is the j -invariant of every elliptic curve whose endomorphism ring isisomorphic to Z h √− i . Numerical computations suggest an affirmative answerto the following question. ‡ Question 1.2. Is − the unique singular modulus that is a { } -unit? For j = 0 or 1728 and for the infinite set of prime numbers S for which every el-liptic curve with j -invariant equal to j has potential ordinary reduction, Campagnashows the following in [Cam19]. If j is a singular modulus such that j − j is an S -unit, then j − j is in fact an algebraic unit. A combination of Theorem A andthe arguments of Campagna, shows that when j = 0 or 1728 the conclusion ofTheorem A holds for some infinite sets of prime numbers S , see Section 1.1.To prove Theorem A, we follow Habegger’s original strategy in the case where S = ∅ in [Hab15]. The main new ingredient is that for every prime number p ,singular moduli are p -adically disperse. To state this result precisely, let ( C p , | · | p )be a completion of an algebraic closure of the field of p -adic numbers Q p . For α in C p and r >
0, put D p ( α, r ) := { z ∈ C p : | z − α | p < r } . We identify the algebraicclosures of Q inside C and C p , and consider singular moduli as elements of C p . Inthis paper, a discriminant is the discriminant of an order in a quadratic imaginaryextension of Q . For a singular modulus j , the discriminant of the endomorphismring of an elliptic curve over C p whose j -invariant is equal to j only depends on j .Denote it by D j . For each discriminant D , the number of singular moduli j satisfying D j = D is equal to the class number h ( D ) of the order of discriminant D in Q ( √ D ). Theorem B (Singular moduli are p -adically disperse) . For all α in C p and ε > ,there is r > such that for every discriminant D such that | D | is sufficiently large,we have (1.1) { j ∈ D p ( α, r ) : j singular modulus with D j = D } ≤ εh ( D ) . In the complex setting, the analogous result is a direct consequence of the factthat the asymptotic distribution of singular moduli is given by a nonatomic mea-sure [Duk88, CU04]. This is used by Habegger in the proof of [Hab15, Theorem 2].In the p -adic setting there are infinitely many measures that describe the asymptoticdistribution of singular moduli. To prove Theorem B, we show that none of thesemeasures has an atom in C p (Theorem 2.1 in Section 2). We also prove an analogousresult for the Hecke orbit of every point in C p (Theorem 2.2 in Section 2.1). As aconsequence, we obtain that a Hecke orbit cannot have a significant proportion ofgood approximations of a given point in C p (Corollary 2.4 in Section 2.1), thus im-proving a result of Charles in [Cha18]. The proofs of these results are based on thedescription of all the measures describing the p -adic asymptotic distribution of sin-gular moduli and Hecke orbits, given in the companion papers [HMRL20, HMR21]. † See Section 1.1 for further comments on the latter. ‡ See, e.g. , Sutherland’s table https://math.mit.edu/~drew/NormsOfSingularModuli2000.pdf
HERE ARE AT MOST FINITELY MANY SINGULAR MODULI THAT ARE S -UNITS 3 Notes and further references.
For distinct singular moduli j and j ′ , Ligives in [Li19] an explicit lower bound for the absolute norm of j − j ′ that impliesthat this algebraic integer is not an algebraic unit. When restricted to j ′ = 0,this is [BHK20, Theorem 1.1]. In fact, Li proves a stronger result for the valuesof modular polynomials at pairs of singular moduli. Li’s approach makes use of(extensions of) the work of Gross and Zagier in [GZ85], and it is different fromthose in [Hab15, BHK20]. Li does not treat the case of S -units in [Li19].In the case where j = 0 (resp. 1728), the conclusion of Theorem A holds forcertain classes of infinite sets of prime numbers S . In fact, if we put S := { q : prime number, q ≡ } (resp. { q : prime number, q ≡ } ),then the conclusion of Theorem A holds for every set of prime numbers S suchthat S r S is finite and does not contain { , , } (resp. { , , } ). This is a directconsequence of Theorem A and the proof of [Cam19, Theorems 1.2 and 6.1].The λ -invariants of CM points in the modular curve of level 2 share many arith-metic properties with singular moduli. However, there are infinitely many suchpoints whose λ -invariants are algebraic units, see [YYY18, Theorem 1.1].1.2. Organization.
In Section 2 we show that no measure describing the p -adicasymptotic distribution of singular moduli has an atom in C p (Theorem 2.1). To-gether with [HMR21, Theorems A and B], this implies Theorem B as a direct conse-quence. The proof of Theorem 2.1 is based on the description of all these measuresgiven in the companion papers [HMRL20, HMR21]. We also use an analogous de-scription for Hecke orbits given in loc. cit . We first establish a result analogous toTheorem 2.1 for Hecke orbits (Theorem 2.2) in Section 2.1, and in Section 2.2 wededuce Theorem 2.1 from this result. To prove Theorem 2.2, we first show that theimages of a point under Hecke correspondences of different prime indices are nearlydisjoint (Lemma 2.5). We use this to show that an atom in C p of an accumulationmeasure of a Hecke orbit would replicate indefinitely, thus creating infinite mass. § In Section 3 we estimate from below the p -adic distance from a fixed singularmodulus j to a varying singular modulus j , in terms of D j (Proposition 3.1). It isa p -adic counterpart of a result of Habegger in [Hab15]. After a brief review of thework of Gross and Hopkins on deformation spaces of formal modules in Section 3.1,in Section 3.2 we give the proof of Proposition 3.1. First, we use that singularmoduli are isolated in the ordinary reduction locus [HMRL20, Corollary B], torestrict to the case where j and j are both in the supersingular reduction locus. Inthe case where the conductors of D j and D j are both p -adic units, we use an idea inthe proof of [Cha18, Proposition 5.11]. To extend this estimate to the general case,we use a formula in [HMR21] that shows how the canonical branch of the Heckecorrespondence T p relates CM points whose conductors differ by a power of p .The proof of Theorem A is in Section 4. We follow Habegger’s original strategyin the case where S = ∅ in [Hab15]. In particular, we use Colmez’ bound [Col98, Th´eor`eme
1] in the form of [Hab15, Lemma 3] in a crucial way. The main newingredient to implement Habegger’s strategy is Theorem B. We also use the esti-mates from below of the archimedean and p -adic distances between singular moduli,mentioned above. § See Remark 2.6 for a different strategy of proof.
SEBASTI´AN HERRERO, RICARDO MENARES, AND JUAN RIVERA-LETELIER
Acknowledgments.
The authors would like to thank Shouwu Zhang, for pointingthat a statement like Theorem A might be obtained from knowledge about the p -adic asymptotic behavior of CM points. During the preparation of this work the firstnamed author was supported by ANID/CONICYT, FONDECYT PostdoctoradoNacional grant 3190086. The second named author was partially supported byFONDECYT grant 1171329. The third named author acknowledges partial supportfrom NSF grant DMS-1700291. The authors would like to thank the PontificiaUniversidad Cat´olica de Valpara´ıso, the University of Rochester and Universitat deBarcelona for hospitality during the preparation of this work.2. p -Adic limits of CM points The goal of this section is to prove Theorem 2.1 below. Together with [HMR21,Theorems A and B], which are summarized in Theorem 2.7 in Section 2.2, it impliesTheorem B as a direct consequence.Throughout this section, fix a prime number p and let ( C p , | · | p ) be as in theintroduction. Denote by Y ( C p ) the moduli space of elliptic curves over C p . Weconsider 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 . We endow the spaceof Borel measures on A with the weak topology with respect to the space ofbounded and continuous real functions. Denote by x can the “Gauss” or “canonical”point of A . For x in A denote by δ x the Dirac measure at x . An atom of aBorel measure ν on A is a point x in A such that ν ( { x } ) >
0. A measure is nonatomic if it has no atoms.The endomorphism ring of an elliptic curve over C p only depends on the class E in Y ( C p ) of the elliptic curve. It is isomorphic to Z or to an order in a quadraticimaginary extension of Q . In the latter case E is a CM point and its discriminant is the discriminant of its endomorphism ring. For every discriminant D , the set(2.1) Λ D := { E ∈ Y ( C p ) : CM point of discriminant D } is finite and nonempty. Denote by δ D,p the Borel probability measure on Y ( C p ),defined by(2.2) δ D,p := 1 D X E ∈ Λ D δ E . In contrast to the complex case, as the discriminant D tends to −∞ the mea-sure δ D,p does not converge in the weak topology. In fact, there are infinitely manydifferent accumulation measures [HMR21, Corollary 1.1].
Theorem 2.1.
Let p be a prime number. Then every accumulation measure of (2.3) (cid:8) δ D,p : D discriminant (cid:9) in the weak topology that is different from δ x can , is nonatomic. In particular, noaccumulation measure of (2.3) in the weak topology has an atom in Y ( C p ) . One of the main ingredients in the proof of this result is the description of allaccumulation measures of (2.3) given in the companion papers [HMRL20, HMR21].We also use an analogous description for Hecke orbits given in loc. cit . We firstestablish a result analogous to Theorem 2.1 for Hecke orbits (Theorem 2.2) inSection 2.1, and in Section 2.2 we deduce Theorem 2.1 from this result.
HERE ARE AT MOST FINITELY MANY SINGULAR MODULI THAT ARE S -UNITS 5 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 the formalgroup of E and by End( F E ) the ring of endomorphisms of F E that are definedover 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 issaid to have formal complex multiplication or to be a formal CM point .An elliptic curve class E in Y ( C p ) has supersingular reduction , if there is arepresentative elliptic curve over O p whose reduction is smooth and supersingular.Denote by Y sups ( C p ) the set of all elliptic curve classes in Y ( C p ) with supersingularreduction.2.1. On the limit measures of Hecke orbits.
The goal of this section is to proveTheorem 2.2 below, which is the main ingredient in the proof of Theorem 2.1. Tostate it, we introduce some notation.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 aredefined bydeg( 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 ,p := 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, where the sum runs over all subgroups C of E of order n . For background on Heckecorrespondences, see [Shi71, Sections 7.2 and 7.3] for the general theory, or thesurvey [DI95, Part II]. Theorem 2.2.
For each E in Y ( C p ) , every accumulation measure of ( δ T n ( E ) ,p ) ∞ n =1 in the weak topology that is different from δ x can , is nonatomic. In particular, noaccumulation measure of ( δ T n ( E ) ,p ) ∞ n =1 in the weak topology has an atom in Y ( C p ) . To prove Theorem 2.2, we first recall some results in [HMR21]. For E in Y sups ( C p ),define a subgroup Nr E of Z × p as follows. 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 SEBASTI´AN HERRERO, RICARDO MENARES, AND JUAN RIVERA-LETELIER 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.For a coset N in Q × p / Nr E contained in Z p , the partial Hecke orbit of E along N isOrb N ( E ) := [ n ∈ N ∩ N supp( T n ( E )) . In the following theorem we use the action of Hecke correspondences on com-pactly supported measures, see, e.g. , [HMR21, Section 2.8].
Theorem 2.3 ([HMR21, Theorem C and Corollary 6.1]) . Let N and N ′ be cosetsin Q × p / Nr E contained in Z p . ( i ) The closure
Orb N ( E ) of Orb N ( E ) in Y sups ( C p ) is compact. Moreover,there is a Borel probability measure µ E N on Y ( C p ) whose support is equalto Orb N ( E ) , and such that for every sequence ( n j ) ∞ j =1 in N ∩ N tendingto ∞ , we have the weak convergence of measures δ T nj ( E ) ,p → µ E N as j → ∞ . ( ii ) 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 ′ . The following corollary is an immediate consequence of Theorems 2.2 and 2.3and [HMRL20, Theorem C].
Corollary 2.4.
For every E in Y ( C p ) , α in C p , and ε > , there exists r > suchthat the following set is finite: (cid:8) n ∈ N : deg( T n ( E ) | D p ( α,r ) ) ≥ εσ ( n ) (cid:9) . Previously, Charles showed that the set above with N replaced by N r p N haszero density [Cha18, Proposition 3.2].The proof of Theorem 2.2 is given after the following lemma. Lemma 2.5.
Let E be in Y ( C p ) . If for distinct prime numbers q and q ′ we put I := supp( T q ( E )) ∩ supp( T q ′ ( E )) , then we have deg( T q ( E ) | I ) ≤ .Proof. For each E in I choose an isogeny φ E in Hom q ′ ( E, E ). Let φ be inHom q ( E , E ) and set ψ := φ E ◦ φ . The isogeny ψ determines both E in I and φ .Indeed, suppose there are E ′ in I and φ ′ in Hom q ( E , E ′ ) with φ E ′ ◦ φ ′ = ψ . Thegroup Ker( ψ ) has qq ′ elements, so it has a unique subgroup of order q . Since Ker( φ )and Ker( φ ′ ) are two such subgroups, we have Ker( φ ) = Ker( φ ′ ). Then E = E ′ by HERE ARE AT MOST FINITELY MANY SINGULAR MODULI THAT ARE S -UNITS 7 [Sil09, Chapter III, Proposition 4.12], and from the equality φ E ◦ φ = φ E ◦ φ ′ wededuce φ = φ ′ . We thus have(2.4) deg( T q ( E ) | I ) = X E ∈ I q ( E , E ) / E ) ≤ X E ∈ I q ( E , E ) ≤ X E ∈ I { φ E ◦ φ : φ ∈ Hom q ( E , E ) } ≤ { ψ ∈ End( E ) : deg( ψ ) = qq ′ } . If E is not a CM point, then this last number is equal to zero and the lemma followsin this case. Suppose E is a CM point, so the field of fractions K of End( E ) is aquadratic imaginary extension of Q . Denote by O K the ring of integers of K . Sinceeach of the ideals q O K and q ′ O K is either prime or a product of two conjugateprime ideals, there are at most four ideals of O K of norm qq ′ . We thus have { ψ ∈ End( E ) : deg( ψ ) = qq ′ } ≤ { x ∈ O K : xx = qq ′ } ≤ O × K ≤ . Together with (2.4) this completes the proof of the lemma. (cid:3)
Proof of Theorem 2.2.
By [HMRL20, Theorem C], it is sufficient to assume that E is in Y sups ( C p ). Moreover, using Theorem 2.3( i ) and [HMRL20, Theorem C] again,it is sufficient to prove that for every coset N in Q × p / Nr E contained in Z p themeasure µ E N has no atom in Orb N ( E ).Fix E in Orb N ( E ) and let N ≥ P of 2 N primenumbers that are contained in ( Z × p ) and that are larger than 100 N . Moreover, forall distinct q and q ′ in P denote by I ( q, q ′ ) the set I in Lemma 2.5 and put S q := supp( T q ( E )) r [ q ′ ∈ P,q ′ = q I ( q, q ′ ) . Then by the inequality q > N and Lemma 2.5, we have(2.5) deg( T q ( E ) | S q ) ≥ deg( T q ( E )) − X q ′ ∈ P,q ′ = q deg( T q ( E ) | I ( q,q ′ ) ) ≥ q + 1 − P − ≥ q + 12 . On the other hand, note that the signed measure µ E N − µ E N ( { E } ) δ E is nonnegative,thus the same holds for ( T q ) ∗ ( µ E N − µ E N ( { E } ) δ E ). Combined with Theorem 2.3( ii ),this implies that for every E ′ in Y ( C p ) we have µ E N ( { E ′ } ) = (cid:18) q + 1 ( T q ) ∗ µ E N (cid:19) ( { E ′ } ) ≥ µ E N ( { E } ) q + 1 (( T q ) ∗ δ E ) ( { E ′ } )= µ E N ( { E } ) q + 1 deg( T q ( E ) | { E ′ } ) . Together with (2.5) this implies1 = µ E N (Orb N ( E )) ≥ X q ∈ P µ E N ( S q ) ≥ X q ∈ P µ E N ( { E } ) q + 1 deg( T q ( E ) | S q ) ≥ N µ E N ( { E } ) . Since N is arbitrary, this implies that E is not an atom of µ E N and completes theproof of Theorem 2.2. (cid:3) SEBASTI´AN HERRERO, RICARDO MENARES, AND JUAN RIVERA-LETELIER
Remark . A different strategy to prove Theorem 2.2 is to use that for every E in Y sups ( C p ) and every coset N in Q × p / Nr E contained in Z p , the measure µ E N is theprojection of a certain homogeneous measure under an analytic map of finite degree.Theorem 2.2 then follows from the fact that the partial Hecke orbit Orb N ( E ) isinfinite.2.2. On the limit measures of CM points.
The goal of this section is to proveTheorem 2.1. The proof is based on Theorem 2.2 and on the description of allaccumulation measures of (2.3) given in the companion papers [HMRL20, HMR21].We start recalling some results in the latter.A fundamental discriminant is the discriminant of the ring of integers of a qua-dratic imaginary extension of Q . For each discriminant D , there is a unique funda-mental discriminant d and a unique integer f ≥ D = df . In this case, d and f are the fundamental discriminant and conductor of D , respectively. Adiscriminant is prime , if it is fundamental and divisible by only one prime number.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 .The p -adic discriminant of a formal CM point E , is the p -adic discriminant ofthe p -adic quadratic order End( F E ). Given a p -adic discriminant D , putΛ D := { E ∈ Y ( C p ) : formal CM point of p -adic discriminant D } . Theorem 2.7 ([HMR21, Theorems A and B]) . For every p -adic discriminant D ,the following properties hold. ( i ) The set Λ D is a compact subset of Y ( C p ) , and there is a Borel probabilitymeasure ν D on Y ( C p ) whose support is equal to Λ D and such that thefollowing equidistribution property holds. Let ( D n ) ∞ n =1 be a sequence ofdiscriminants in D tending to −∞ , such that for every n the fundamentaldiscriminant of D n is either not divisible by p , or not a prime discriminant.Then we have the weak convergence of measures δ D n ,p → ν D as n → ∞ . ( ii ) Suppose that there is a prime discriminant d and an integer m ≥ such that D := dp m is in D . Then there are Borel probability measures ν + D and ν − D on Y ( C p ) such that the following equidistribution property holds. For everysequence ( f n ) ∞ n =0 in N tending to ∞ such that for every n we have (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 ) ,p → ν + D (resp. δ D ( f n ) ,p → ν − D ) as n → ∞ . The proof of Theorem 2.1 is given after the following proposition, in which wegather further properties of the limit measures in Theorem 2.7. To state it, weintroduce some notation.A p -adic discriminant is fundamental , if it is the p -adic discriminant of the ring ofintegers of a quadratic extension of Q p . Let d be a fundamental p -adic discriminant. HERE ARE AT MOST FINITELY MANY SINGULAR MODULI THAT ARE S -UNITS 9 For ∆ in d , the field Q p ( √ ∆) depends only on d , but not on ∆. Denote it by Q p ( √ d ).Choose a formal CM point E d such that End( F E ) is isomorphic to the ring ofintegers of Q p ( √ d ), as follows. If d does not contain a prime discriminant thatis divisible by p , then choose an arbitrary formal CM point E d as above. In thecase where d contains a prime discriminant d that is divisible by p , then d is theunique fundamental discriminant in d with this property and we choose E d in Λ d .Note that if Q p ( √ d ) is unramified over Q p then Nr E d = Z × p , and that if Q p ( √ d ) isramified over Q p then Nr E d is a subgroup of Z × p of index two, see, e.g. , [HMR21,Lemma 2.3].Denote by v p Katz’ valuation 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 the Hecke correspondence T p is the map t : N p → Y sups ( C p )defined by t ( E ) := E/H ( E ). The map t is analytic in the sense that it is givenby a finite sum of Laurent series, each of which converges on all of N p , see, e.g. ,[HMRL20, Theorem B.1].Given a fundamental p -adic discriminant d and an integer m ≥
0, define theaffinoid(2.6) 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 ≥ . In the following proposition we summarize some of the results in [HMR21, Propo-sition 7.1, (7.13) and Sections 7.2 and 7.3].
Proposition 2.8.
For every fundamental p -adic discriminant d we have (2.7) ν d = µ E d Z × p if Q p ( √ d ) is unramified over Q p ; (cid:16) µ E d Nr d + µ E d Z × p r Nr d (cid:17) if Q p ( √ d ) is ramified over Q p , and for every integer m ≥ we have (2.8) ν d p m = p m − ( p +1) ( t m (cid:12)(cid:12) A d p m ) ∗ ν d if Q p ( √ d ) is unramified over Q p ; p m ( t m (cid:12)(cid:12) A d p m ) ∗ ν d if Q p ( √ d ) is ramified over Q p . If in addition d contains a prime discriminant, then we also have (2.9) ν + d = µ E d Nr d and ν − d = µ E d Z × p r Nr d , and (2.8) holds for ν + d p m (resp. ν − d p m ), with ν d replaced by ν + d (resp. ν − d ).Proof of Theorem 2.1. Let ( D n ) ∞ n =1 be a sequence of discriminants tending to −∞ such that the sequence of measures ( δ D n ,p ) ∞ n =1 converges weakly to a measure dif-ferent from δ x can . By [HMRL20, Theorem A], there is a constant c > n we have | D n | p > c and Λ D n ⊆ Y sups ( C p ). This implies that ( D n ) ∞ n =1 iscontained in a finite union of p -adic discriminants, see, e.g. , [HMRL20, Lemmas 2.1and A.1]. Taking a subsequence if necessary, assume that ( D n ) ∞ n =1 is contained ina p -adic discriminant D . Let d be the fundamental p -adic discriminant and m ≥ D = d p m , see, e.g. , [HMRL20, Lemma A.1( i )]. Passing to a subsequence if necessary, there are two cases.
Case 1.
For every n the fundamental discriminant of D n is either not divisible by p ,or not a prime discriminant. In this case the sequence ( δ D n ,p ) ∞ n =1 converges to ν D by Theorem 2.7( i ). Then (2.7) in Proposition 2.8 and Theorem 2.2 imply that ν d is nonatomic. This is the desired assertion in the case where m = 0. If m ≥ ν D is nonatomic follows from the fact that ν d is nonatomic, from (2.8) inProposition 2.8 and from the fact that the canonical branch t of T p is analytic. Case 2.
There is a prime discriminant d that is divisible by p and a sequence ( f n ) ∞ n =1 in N such that for every n we have D n = df n and (cid:16) df n (cid:17) = 1 (resp. (cid:16) df n (cid:17) = − δ D n ,p ) ∞ n =1 converges weakly to ν + D (resp. ν − D ) by Theo-rem 2.7( ii ). Then (2.9) in Proposition 2.8 and Theorem 2.2 imply that ν + d and ν − d are both nonatomic. This is the desired assertion in the case where m = 0. If m ≥ ν + D and ν − D are both nonatomic follows from the fact that ν + d and ν − d areboth nonatomic, from the last assertion of Proposition 2.8 and from the fact thatthe canonical branch t of T p is analytic. (cid:3) p -Adic distance between singular moduli The goal of this section is to prove the following estimate. Throughout thissection we fix a prime number p and use the notation in Section 1. Proposition 3.1.
Let j be a singular modulus. Then, there exist constants A ≥ and B such that for every singular modulus j different from j we have − log | j − j | p ≤ A log | D j | + B. The archimedean counterpart of this estimate was shown by Habegger in [Hab15].When restricted to the one-dimensional setting, Habegger’s question in [Hab14,p. 3306] is whether the estimate in Proposition 3.1 holds in the more general casein which j is an arbitrary algebraic integer.After some preliminaries in Section 3.1, the proof of Proposition 3.1 is given inSection 3.2. To prove Proposition 3.1, we use that CM points outside Y sups ( C p )are isolated [HMRL20, Corollary B] to restrict to the case where the CM pointscorresponding to j and j are both in Y sups ( C p ). In the case where the conduc-tors of D j and D j are both p -adic units, we use an idea in the proof of [Cha18,Proposition 5.11]. To deduce the general case from this particular case, we use aformula in [HMR21] showing how the canonical branch of T p relates CM pointswhose conductors differ by a power of p (Theorem 3.5).Denote by Y sups ( F p ) the (finite) set of isomorphism classes of supersingular el-liptic curves over F 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 class is e . The set D e is a residuedisc in Y ( C p ).3.1. Formal Z p -modules and elliptic curves. In this section, we briefly recallthe work of Gross and Hopkins in [HG94], on deformation spaces of formal modules.See also [HMR21, Sections 2.4 to 2.7] for a more detailed account of the resultsneeded here. For every e in Y sups ( F p ), we describe an action of (End( e ) ⊗ Z p ) × ona certain ramified covering of D e . In the proof of Proposition 3.1 we use a relationbetween the metric on D e and the natural metric of the covering, which is statedas Theorem 3.2 below. HERE ARE AT MOST FINITELY MANY SINGULAR MODULI THAT ARE S -UNITS 11 Fix e in Y sups ( F p ) and a representative elliptic curve defined over F p that wealso denote by e . Denote by F e the formal group of e endowed with its naturalstructure of formal Z p -module and set B e := End F p ( F e ) ⊗ Q p , R e := End F p ( F e ) and G e := Aut F p ( F e ) . Then, B e is a division quaternion algebra over Q p and the sets R e and G e embedin B e as the maximal order and its group of units, respectively. Denote by g g the involution of B e , and for g in B e denote by nr( g ) := gg in Q p its reducednorm . On the other hand, the function ord B e : B e → Z ∪ {∞} defined for g in B e by ord B e ( g ) := ord p (nr( g )), is the unique valuation extending the valuation 2 ord p on Q p . Identifying R e and G e with their images in B e , we have R e = { g ∈ B e : ord B e ( g ) ≥ } and G e = { g ∈ B e : ord B e ( g ) = 0 } . The function dist B e : B e × B e → R defined for g and g ′ in B e bydist B e ( g, g ′ ) := p − ord B e ( g − g ′ ) , defines an ultrametric distance on B e that makes B e into a topological algebraover Q p .Identify the residue field of C p with an algebraic closure F p of F p and denote by π : O p → F p the reduction map. Moreover, denote by Q p the unique unramifiedquadratic extension of Q p inside C p , and by Z p its ring of integers.Let R be a complete, local, Noetherian Z p -algebra with maximal ideal M andresidue field isomorphic to a subfield k of F p that contains F p . Fix a reductionmap R → k . We are mainly interested in the special case where R the ring ofintegers of a finite extension of Q p contained in C p together with the restrictionof π , or a quotient of such ring of integers together with the morphism induced bythe restriction of π . We stick to the general case for convenience.A deformation of F e over R is a pair ( F , α ), where F is a formal Z p -moduleover R and α : e F → F e is an isomorphism of formal Z p -modules defined over k .Here, e F is the formal group over k obtained as the base change of F under thereduction map R → k . Two such deformations ( F , α ) and ( F ′ , α ′ ) are isomor-phic , if there exists an isomorphism ϕ in Iso R ( F , F ′ ) with reduction e ϕ such that α ′ ◦ e ϕ = α . Denote by X e ( R ) the set of isomorphism classes of deformations of F e over R .For the rest of this section, we further assume that our choice of the represen-tative elliptic curve e is such that F e is isomorphic over F p to the specializationof a universal formal Z p -module of height two, see [HMR21, Lemma 2.5]. Then,a consequence of the work of Gross and Hopkins is that there exists a bijection Aconsequence of the work of Gross and Hopkins is that there exists a bijection(3.1) M → X e ( R )that is functorial in R , see [HG94, Section 12] and [HMR21, Section 2.5] for details.Moreover, we have the actionAut k ( F e ) × X e ( R ) → X e ( R )( β, ( F , α )) β · ( F , α ) := ( F , β ◦ α ) . Let K be a finite extension of Q p inside C p , with ring of integers O K andresidue field k . Consider the reduction map O K → k obtained as the restrictionof π to O K . Denote by Y ( e, O K ) the space of isomorphism classes of pairs ( E, α ) formed by an elliptic curve E given by a Weierstrass equation with coefficientsin O K and having smooth reduction, and an isomorphism α : e E → e defined over k .Here, 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 . Consider the natural map(3.2) Y ( e, O K ) → X e ( O K )that associates to a class in Y ( e, O K ) represented by a pair ( E, α ), the classin X e ( O K ) represented by the deformation ( F E , b α ). Here, b α : e F E → F e is the iso-morphism induced by α . This map is known to be a bijection thanks to the so-calledWoods-Hole Theory, see [LST64, Section 6] or [MC10, Theorem 4.1]. We obtain amap(3.3) Π e, K : X e ( O K ) → Y sups ( Q p ) ∩ D e , by composing the inverse of (3.2) with the natural map from Y ( e, O K ) to Y sups ( Q p ) ∩ D e .Consider K := { finite extensions of Q p inside C p } as a directed set with respect to the inclusion. For each K in K , consider theparametrization (3.1) with R = O K . Taking a direct limit over K and then acompletion, we obtain a set b D e that is parametrized by the maximal ideal of O p .The action of G e on the system { X e ( O K ) : K ∈ K } extends to a continuous map G e × b D e → b D e that is analytic in the second variable, see [HMR21, Section 2.6]for details.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,Proposition 1.2(c)].
Theorem 3.2 ([HMR21, Theorem 2.7]) . Fix e in Y sups ( F p ) . Then, the system { Π e, K : K ∈ K } given by (3.3) defines a ramified covering map Π e : b D e → D e , such that for every x in b D e and every E in D e we have (3.4) 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 ) } . Proof of Proposition 3.1.
The proof of Proposition 3.1 is at the end of thissection.Let e be in Y sups ( F p ). The set L ( e ) := { φ ∈ Z + 2 End( e ) : tr( φ ) = 0 } , is a Z -lattice of dimension three inside End( e ). Define for each integer m ≥ V m ( e ) := { φ ∈ L ( e ) : nr( φ ) = m } . For each fundamental p -adic discriminant d and every discriminant D in d , theimage of the set V | D | ( e ) by the natural map End( e ) → End F p ( F e ), denoted by φ b φ , is contained in L e, d := { ϕ ∈ Z p + 2 R e : tr( ϕ ) = 0 , − nr( ϕ ) ∈ d } , HERE ARE AT MOST FINITELY MANY SINGULAR MODULI THAT ARE S -UNITS 13 see [HMR21, Lemma 2.1]. Let U e, d : L e, d → G e be the function defined by U e, d ( ϕ ) := ( ϕ + ϕ if ϕ + ϕ belongs to G e ;1 + ϕ + ϕ otherwise , and for each ϕ in L e, d defineFix e ( ϕ ) := n x ∈ b D e : U e, d ( ϕ ) · x = x o . Given a fundamental p -adic discriminant d , denote by Q p ( √ d ) the compositumof Q p and Q p ( √ d ). Proposition 3.3 ([HMR21, Lemmas 4.5( iv ) and 4.15, and Proposition 5.6( i )]) . Fix e in Y sups ( F p ) and a fundamental p -adic discriminant d . ( i ) For ϕ and ϕ ′ in L e, d 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 ) . ( ii ) We have Π − e (Λ d ∩ D e ) ⊆ X e ( O Q p ( √ d ) ) , and for every ∆ in d we have Π − e (Λ d ∩ D e ) = Fix e ( { ϕ ∈ L e, d : nr( ϕ ) = − ∆ } ) . For a fundamental p -adic discriminant d , put ε d := if Q p ( √ d ) is ramifiedover Q p and ε d := 1 if Q p ( √ d ) is unramified over Q p . Lemma 3.4.
For every prime number p , every e in Y sups ( F p ) and every funda-mental p -adic discriminant d , the following property holds. For all ϕ and ˇ ϕ in L e, d and all x in Fix e ( ϕ ) and ˇ x in Fix e ( ˇ ϕ ) , we have | x − ˇ x | p ≥ p − ε d dist B e ( ϕ ˇ ϕ, ˇ ϕϕ ) . Proof.
Let ̟ and ̟ be uniformizers of O Q p ( √ d ) and O Q p ( ϕ ) , respectively, and notethat(3.5) ord p ( ̟ ) = ε d = 12 ord B e ( ̟ ) . If x = ˇ x , then ˇ ϕ is in Q p ( ϕ ) by Proposition 3.3( i ) and therefore ϕ ˇ ϕ = ˇ ϕϕ . So thedesired property holds in this case. Assume x = ˇ x . By Proposition 3.3( ii ), x and ˇ x are both in Π − e (Λ d ∩ D e ) and therefore in X e ( O Q p ( √ d ) ). In particular, there isan integer N ≥ | x − ˇ x | p = | ̟ | Np . Let ( F , α ) and ( ˇ F , ˇ α ) represent x and ˇ x , respectively, and denote by F N and ˇ F N the base change of F and ˇ F under theprojection map O Q p ( √ d ) → R := O Q p ( √ d ) /π N O Q p ( √ d ) . Since (3.1) is a bijection,there is an isomorphism ψ : F N → ˇ F N defined over R such that ˇ α = α ◦ e ψ . Thisimplies that the mapsEnd R ( F N ) → R e and End R ( ˇ F N ) → R e , given by φ α ◦ e φ ◦ α − and φ ˇ α ◦ e φ ◦ ˇ α − , respectively, have the same image. Thus, by [Gro86, Proposition 3.3] we have O Q p ( ϕ ) + ̟ N − R e = O Q p ( ˇ ϕ ) + ̟ N − R e . It follows that ϕ ˇ ϕ − ˇ ϕϕ is in ̟ N − R e . Together with (3.5), this impliesdist B e ( ϕ ˇ ϕ, ˇ ϕϕ ) ≤ | ̟ | N − p = p ε d | x − ˇ x | p . (cid:3) In the following theorem we use the canonical branch t of T p , recalled in Sec-tion 2.2. Theorem 3.5 ([HMR21, Theorem 4.6]) . Let d be a fundamental discriminant suchthat Λ d ⊆ Y sups ( C p ) . Then for every integer r ≥ and every integer f ≥ that isnot divisible by p , we have Λ d ( fp r ) = t − (cid:0) Λ df (cid:1) ∩ 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:0) Λ df (cid:1) if r ≥ and p is inert in Q ( √ d ) . The following lemma is [HMRL20, Lemma 4.9], see also [CM06, Lemma 4.8] and[Gro86, Proposition 5.3].
Lemma 3.6.
Denote Katz’ valuation by v p , as in Section 2.2. Let D be a discrim-inant such that Λ D ⊆ Y sups ( C p ) and let m ≥ be the largest integer such that p m divides the conductor of D . Then for every E in supp(Λ D ) we have min n v p ( E ) , pp +1 o = ( · p − m if p ramifies in Q ( √ D ); pp +1 · p − m if p is inert in Q ( √ D ) .Proof of Proposition 3.1. Let E be the CM point such that j ( E ) = j . SinceCM points outside Y sups ( C p ) are isolated [HMRL20, Corollary B], we can assumethat E is in Y sups ( C p ). Let e be the element of Y sups ( F p ) such that E is in D e .Let d be the fundamental p -adic discriminant and m ≥ E is in Λ d p m . Let j be a singular modulus different from j and let E be the CM pointsatisfying j ( E ) = j . Without loss of generality, assume that D j = D j and that E is in D e . In view of Lemma 3.6, we can also assume that there is a fundamental p -adic discriminant d ′ such that D j is in p m d ′ , see, e.g. , [HMR21, Lemma 2.1].Since Λ d p m and Λ d ′ p m are both compact by Theorem 2.7( i ) and they are disjointif d ′ = d , we can also assume that d ′ = d . On the other hand, by Theorem 3.5 andthe fact that the canonical branch t of T p is analytic, it is sufficient to prove thelemma in the case where m = 0, so E and E are both in Λ d ∩ D e .Using δ e ≤
12 and Theorem 3.2, we can find x in Π − e ( E ) and x in Π − e ( E )such that(3.6) | j − j | p ≥ | x − x | p . On the other hand, by Proposition 3.3( ii ) there is φ in End( e ) such that b φ sat-isfies the equation X − D j = 0, is in L e, d and is such that x is in Fix e ( b φ ).Similarly, we can find φ in End( e ) such that b φ satisfies the equation X − D j = 0,is in L e, d and is such that x is in Fix e ( b φ ). Note that Proposition 3.3( i ) and ourassumption D j = D j , imply that φ φ − φφ is nonzero. Combined with the factthat deg is a positive definite quadratic form on End( e ) and [Sil09, Chapter V,Lemma 1.2], this impliesord B e ( b φ b φ − b φ b φ ) = ord p (nr( b φ b φ − b φ b φ )) = ord p (deg( φ φ − φφ )) ≤ log p (deg( φ φ − φφ )) ≤ log p (4 deg( φ ) deg( φ )) = log p (4 deg( φ ) | D j | ) . Together with (3.6) and Lemma 3.4 this implies the desired estimate. (cid:3)
HERE ARE AT MOST FINITELY MANY SINGULAR MODULI THAT ARE S -UNITS 15 Proof of Theorem A
We follow Habegger’s original strategy in the case where S = ∅ in [Hab15],the main new ingredient being Theorem B. We also use Proposition 3.1 and itsarchimedean counterpart shown by Habegger in [Hab15].Denote by M Q the set of all prime numbers together with ∞ and by Q thealgebraic closure of Q inside C . Put C ∞ := C and denote by | · | ∞ the usualabsolute value on C . Moreover, for each prime number p let ( C p , | · | p ) be as inthe introduction and identify the algebraic closure of Q inside C p with Q . We thusconsider singular moduli as elements of C p . Moreover, for all v in M Q , α in C v and r >
0, put D v ( α, r ) := { z ∈ C v : | z − α | v < r } . Let log + : [0 , ∞ [ → R and log − : [0 , ∞ [ → R ∪ {−∞} be the functions defined bylog + ( x ) := log max { , x } and log − ( x ) := log min { , x } . For α in Q , denote by O( α ) its orbit by the absolute Galois group Gal( Q | Q ). The logarithmic Weil height of α is given byh W ( α ) := 1 α ) X v ∈ M Q X α ′ ∈ O( α ) log + | α ′ | v . If in addition α = 0, then by the product formula we have(4.1) h W ( α ) = − α ) X v ∈ M Q X α ′ ∈ O( α ) log − | α ′ | v . Note that for all α and α in Q , we have(4.2) h W ( α − α ) ≥ h W ( α ) − h W ( α ) − log 2 . As in Section 1, for every discriminant D denote by h ( D ) the class number of theorder of discriminant D in Q ( √ D ) and for each singular modulus j denote by D j the discriminant of the corresponding CM point. Note thatO( j ) = { singular moduli j ′ with D j ′ = D j } and j ) = h ( D j ) , see, e.g. , [Lan87, Chapter 10, Theorem 5].Fix a singular modulus j . Combining [Hab15, Lemma 3], which is based onColmez’ lower bound [Col98, Th´eor`eme 1 ], with (4.2), we obtain that there areconstants
A > B such that for every singular modulus j we have(4.3) h W ( j − j ) ≥ A log | D j | + B. For each v in M Q there are constants A v and B v such that for every singularmodulus j different from j we have(4.4) − log | j − j | v ≤ A v log | D j | + B v , see [Hab15, Lemmas 5 and 8 and formula (11), or the proof of Lemma 6] in thecase where v = ∞ and Proposition 3.1 in the case where v is finite. Moreover, wehave(4.5) j ) · j − j ) = [ Q ( j ) : Q ] · [ Q ( j − j ) : Q ] ≥ [ Q ( j , j ) : Q ] ≥ [ Q ( j ) : Q ] = j ) = h ( D j ) . Let S be a finite set of prime numbers and suppose that there are infinitelymany singular moduli j such that j − j is an S -unit. Then we can find a sequence of discriminants ( D n ) ∞ n =1 tending to −∞ , such that for each n there is a CMpoint E n of discriminant D n such that j ( E n ) − j is an S -unit. Put j n := j ( E n ).We use Theorem B with p replaced by ∞ , which is a direct consequence of [CU04, Th´eor`eme p in S , this implies thatthere is r in ]0 ,
1[ such that for every v in S ∪{∞} and every sufficiently large n ≥ X j ∈ O( j ) j n ) ∩ D v ( j , r )) ≤ A A v ( S + 1) j ) h ( D n ) . Together with (4.5), this implies that for every sufficiently large n we have j n − j ) ∩ D v (0 , r )) j n − j ) ≤ j n − j ) { ( j , j ′ ) ∈ O( j n ) × O( j ) : | j − j ′ | v < r }≤ j ) h ( D n ) X j ′ ∈ O( j ) j n ) ∩ D v ( j ′ , r )) ≤ A A v ( S + 1) . Combining (4.1) with (4.4) and with our assumption that j n − j is an S -unit, thisimplies that for some constant B ′ independent of n we haveh W ( j n − j ) = − j n − j ) X v ∈ S ∪{∞} X α ∈ O( j n − j ) | α | v Int.Math. Res. Not. IMRN , (24):10005–10041, 2020.[Cam19] Francesco Campagna. On singular moduli that are S-units. Manuscripta Math., toappear, arXiv e-prints , page arXiv:1904.08958, April 2019.[Cha18] Fran¸cois Charles. Exceptional isogenies between reductions of pairs of elliptic curves. Duke Math. J. , 167(11):2039–2072, 2018.[CM06] Robert Coleman and Ken McMurdy. Fake CM and the stable model of X ( Np ). Doc.Math. , (Extra Vol.):261–300, 2006.[Col98] Pierre Colmez. Sur la hauteur de Faltings des vari´et´es ab´eliennes `a multiplicationcomplexe. Compositio Math. , 111(3):359–368, 1998.[CU04] Laurent Clozel and Emmanuel Ullmo. ´Equidistribution des points de Hecke. In Con-tributions to automorphic forms, geometry, and number theory , pages 193–254. JohnsHopkins Univ. Press, Baltimore, MD, 2004.[DI95] Fred Diamond and John Im. Modular forms and modular curves. In Seminar on Fer-mat’s Last Theorem (Toronto, ON, 1993–1994) , volume 17 of CMS Conf. Proc. , pages39–133. Amer. Math. Soc., Providence, RI, 1995.[Duk88] W. Duke. Hyperbolic distribution problems and half-integral weight Maass forms. Invent. Math. , 92(1):73–90, 1988. HERE ARE AT MOST FINITELY MANY SINGULAR MODULI THAT ARE S -UNITS 17 [Fr¨o68] A. Fr¨ohlich. Formal groups . Lecture Notes in Mathematics, No. 74. Springer-Verlag,Berlin-New York, 1968.[Gro86] Benedict H. Gross. On canonical and quasicanonical liftings. Invent. Math. , 84(2):321–326, 1986.[GZ85] Benedict H. Gross and Don B. Zagier. On singular moduli. J. Reine Angew. Math. ,355:191–220, 1985.[Hab14] Philipp Habegger. The Tate-Voloch conjecture in a power of a modular curve. Int.Math. Res. Not. IMRN , (12):3303–3339, 2014.[Hab15] Philipp Habegger. Singular moduli that are algebraic units. Algebra Number Theory ,9(7):1515–1524, 2015.[HG94] M. J. Hopkins and B. H. Gross. Equivariant vector bundles on the Lubin-Tate modulispace. In Topology and representation theory (Evanston, IL, 1992) , volume 158 of Contemp. Math. , pages 23–88. Amer. Math. Soc., Providence, RI, 1994.[HMR21] S. Herrero, R. Menares, and J. Rivera-Letelier. p -Adic distribution of CM points andHecke orbits. II: Linnik equidistribution on the supersingular locus. Preprint, 2021.[HMRL20] Sebasti´an Herrero, Ricardo Menares, and Juan Rivera-Letelier. p -adic distribution ofCM points and Hecke orbits I: Convergence towards the Gauss point. Algebra NumberTheory , 14(5):1239–1290, 2020.[Kat73] Nicholas M. Katz. p -adic properties of modular schemes and modular forms. pages69–190. Lecture Notes in Mathematics, Vol. 350, 1973.[Lan87] Serge Lang. Elliptic functions , volume 112 of Graduate Texts in Mathematics .Springer-Verlag, New York, second edition, 1987. With an appendix by J. Tate.[Li19] Yingkun Li. Singular Units and Isogenies Between CM Elliptic Curves. arXiv e-prints ,page arXiv:1810.13214v3, September 2019.[LST64] J. Lubin, J.-P. Serre, and J. Tate. Elliptic curves and formal groups. In Seminar atWoods Hole Institute on algebraic geometry, 1964.[MC10] Ken McMurdy and Robert Coleman. Stable reduction of X ( p ). Algebra NumberTheory , 4(4):357–431, 2010. With an appendix by Everett W. Howe.[Shi71] Goro Shimura. Introduction to the arithmetic theory of automorphic functions . Pub-lications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers,Tokyo; Princeton University Press, Princeton, N.J., 1971. Kanˆo Memorial Lectures,No. 1.[Sil09] Joseph H. Silverman. The arithmetic of elliptic curves , volume 106 of Graduate Textsin Mathematics . Springer, Dordrecht, second edition, 2009.[Web08] Heinrich Weber. Lehrbuch der Algebra , volume III. Braunschweig, 2nd edition, 1908.[YYY18] Tonghai Yang, Hongbo Yin, and Peng Yu. The lambda invariants at CM points. Int.Math. Res. Not., to appear, arXiv e-prints , page arXiv:1810.07381, October 2018. 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. Email address : [email protected] URL ::