Exceptional splitting of reductions of abelian surfaces
aa r X i v : . [ m a t h . N T ] N ov EXCEPTIONAL SPLITTING OF REDUCTIONS OF ABELIAN SURFACES
ANANTH N. SHANKAR AND YUNQING TANG
Abstract.
Heuristics based on the Sato–Tate conjecture suggest that an abelian surface defined over anumber field has infinitely many places of split reduction. We prove this result for abelian surfaces withreal multiplication. As in [Cha14] and [Elk89], this shows that a density-zero set of primes pertaining tothe reduction of abelian varieties is infinite. The proof relies on the Arakelov intersection theory on Hilbertmodular surfaces. Introduction
Infinitely many nonsimple reductions of a given abelian surface.
Murty and Patankar con-jectured in [MP08] that an absolutely simple abelian variety over a number field has absolutely simplereduction for a density one set of primes (up to a finite extension) if and only if its endomorphism ring iscommutative. Chavdarov ([Cha97]) proved their conjecture in the case of abelian varieties of dimension 2or 6 whose geometric endomorphism ring is Z . Conditional upon the Mumford–Tate conjecture, Zywina([Zyw14]) established Murty and Patankar’s conjecture in full generality.It is natural to inquire whether the set of primes (conjecturally a density zero set!) at which a given abelianvariety does not have absolutely simple reduction is finite or infinite. Based on the Sato–Tate conjecture forabelian surfaces (see §1.2), it is expected that the (density zero) set of places of nonsimple reduction of asimple abelian surface is infinite. The main result of this paper is the following: Theorem 1.
Let A be an abelian surface over a number field K . Suppose that F ⊂ End( A ) ⊗ Q , where F is a real quadratic field. Then A modulo v is not absolutely simple for infinitely many primes v of K . A heuristic based on the Sato–Tate conjecture.
The classical Sato–Tate conjecture addressesthe distribution of Frobenius elements involved in the Galois representation on the étale cohomology of afixed elliptic curve defined over a number field. The work of Katz–Sarnak [KS99], Serre [Ser12], and Fité–Kedlaya–Rotger–Sutherland ([FKRS12]) generalizes this conjecture to higher dimensional abelian varieties.We focus on the case of abelian surfaces with real multiplication and offer a heuristic which indicates thatsuch surfaces have infinitely many places of nonsimple reduction.For simplicity, assume that A is an abelian surface defined over Q such that End( A ) ⊗ Q = End( A Q ) ⊗ Q = F , a real quadratic field. For each prime ℓ of good reduction for A , the characteristic polynomial of theFrobenius endomorphism of A modulo ℓ is of the form x + a x + a x + ℓa x + ℓ , with a , a ∈ Z . Theroots of this polynomial come in complex conjugate pairs λ , λ , λ , λ , and each λ i has absolute value ℓ / .Define s i,ℓ = λ i + λ i √ ℓ . The Sato–Tate conjecture for abelian surfaces with real multiplication predicts thatthe distribution of ( s ,ℓ , s ,ℓ ) ∈ [ − , × [ − , , as ℓ varies, converges to the measure on [ − , × [ − , defined by the function ( π ) p − s p − s (for instance, see [Ked15]). Assuming fast enough rates ofconvergence to this measure, the probability that | s ,ℓ − s ,ℓ | < √ ℓ is approximately the area of the region V ℓ = { ( s , s ) : | s − s | < √ ℓ } ⊂ [ − , × [ − , , which is approximately √ ℓ . If | s ,ℓ − s ,ℓ | < √ ℓ , it is easy to see that s ,ℓ and s ,ℓ must be equal. Then by Honda–Tate theory, A modulo ℓ is not simple. This gives a heuristic lower bound ( ≈ √ ℓ ) for the probability that A mod ℓ is More precisely, there exist absolute constants C , C > such that the probability that | s ,ℓ − s ,ℓ | < √ ℓ is greater than C / √ ℓ and less than C / √ ℓ . ot simple . On the other hand, X ℓ prime √ ℓ diverges, so A should have infinitely many primes of nonsimplereduction.1.3. Related results.
The Sato–Tate conjectures for elliptic curves and pairs of elliptic curves also suggeststhat the set of primes in either of the following two situations is infinite:(1) given an elliptic curve E over a number field, consider primes v such that E mod v is supersingular;(2) given a pair of non-isogenous elliptic curves E , E over a number field, consider primes v such thatthe reductions of E , E mod v become geometrically isogenous.As in our case, both sets of primes have density zero (after taking a finite extension). Indeed, Serreconjectured that up to taking a finite extension of the field of definition, a given abelian variety over anumber field has ordinary reduction at a density-one set of primes. Katz proved Serre’s conjecture in thecase of elliptic curves and abelian surfaces ([Ogu82, pages 370–372]). Sawin (in [Saw16]) made explicit thesmallest field extension that is required for abelian surfaces. The second set also has density zero (aftertaking a finite extension) by Faltings’ isogeny theorem ([Fal86]).Elkies proved (1) in [Elk87, Elk89] when the elliptic curve is defined over a number field with at least onereal embedding and Charles proved (2) in [Cha14]. Theorem 1 is an analogue of these two results. Indeed,all three results establish that certain thin sets of primes related to the reduction of abelian varieties areinfinite.1.4. The strategy of the proof.
The proof of Theorem 1 builds on the idea of the proof of the maintheorem in [Cha14], where Charles uses Arakelov intersection theory on the modular curve X (1) to provehis result. In our case, we use Arakelov intersection theory on the Hilbert modular surface H (see §2.1 forthe precise definition).Let [ A ] ∈ H ( K ) denote the point determined by A . Loosely speaking, the modular curve embeds canon-ically into H (we label its image ∆ ) and parameterizes the locus of split abelian surfaces (along with theproduct polarization). A natural strategy is to consider the Arakelov intersection of ∆ with Hecke orbits of [ A ] . The local contribution at a finite prime v is positive precisely when the Hecke orbit of [ A ] intersects ∆ modulo v . The reduction of A modulo v would be geometrically nonsimple for such v .In our proof, we replace ∆ with a compact Hirzebruch–Zagier divisor T (see §2.1 and §5.1 for definitions).Hirzebruch–Zagier divisors of H have the feature that the rank of the Néron–Severi group of an abeliansurface B increases if [ B ] lies on these divisors. This has the consequence that over a finite field, an abeliansurface is not absolutely simple if it lies on a special divisor. There are two advantages of using a compact Hirzebruch–Zagier divisor T : the first is that we do not have to deal with places of bad reduction for A . Thesecond is that we are able to avoid all the cusps of H in the archimedean contribution to the global Arakelovintersection.In order to prove Theorem 1, it would suffice to prove that the set of primes which contribute to theintersection is infinite as we vary over infinitely many well-chosen Hecke orbits of [ A ] . There are two stepsinvolved in proving this: • A local step, where we bound the local contribution of the intersection at every place. • A global step, where we compute the growth of the Arakelov intersection of Hecke orbits of [ A ] with T . The growth is expressed in terms of the degree of the Hecke operators, and is seen to growasymptotically faster than the local contributions at each place.Consequently, it follows that more and more primes contribute to this intersection as we vary the Heckeorbit of A .The abelian surfaces parametrized by Hirzebruch–Zagier divisors have extra special endomorphisms (re-called in §2.1). In the non-archimedean case, our methods are very different from the ones used in [Cha14].For a finite place v , we use Grothendieck–Messing theory prove statements about the rate of decay of specialendomorphisms of A [ ℓ ∞ ] modulo higher and higher powers of v . This method avoids the use of CM liftsand can be used in other Arakelov-theoretic situations. We use these results and Geometry-of-numbers ar-guments to bound the number of special endomorphisms of A modulo powers of v . This allows us to prove This does not take into account those primes modulo which A is simple, but not absolutely simple. Here we refer to the -cycle, given by taking the Zariski closure of the Hecke orbit of [ A ] in H over Spec Z . hat the v -adic contribution grows asymptotically slower than the global intersection for most of the Heckeorbits that we consider. Indeed, if there were too many Hecke orbits T p ([ A ]) having large v -adic intersectionwith T , then A modulo v n would have too many special endomorphisms as n → ∞ .The arguments used to bound the archimedean contribution are very different from the ones used to boundthe finite contributions. A key step in bounding the archimedean contribution is the following statement:for a fixed infinite place, if [ A ] is close to two Hirzebruch–Zagier divisors, then [ A ] must be close to theirintersection, which is a CM abelian surface.In order to prove the global part of our result, it is necessary to relate the global Arakelov intersectionof T and certain Hecke orbits T p ([ A ]) (see §2.2.1 for the precise definition) to the intersection of [ A ] and T .We accomplish this in two steps: • We use Borcherds’ theory (briefly recalled after Lemma 5.1.1) to construct a compact special divisor,whose class in the Picard group of a toroidal compactification of H equals, up to multiplying by aconstant in Z > , the class of the Hodge bundle. Consequently, the global Arakelov intersection of [ A ] with T (endowed with a suitable Hermitian metric), up to multiplying by a suitable constant,equals the Faltings height of A . • We relate the Faltings height of T p ([ A ]) to the Faltings height of A when A has potentially goodreduction at p in Proposition 5.1.6. This extends a result of Autissier ([Aut05, Theorem 5.1]), whichonly applies to A with potentially ordinary reduction at p .It follows that the global intersection number grows faster than any local contribution. Hence, infinitelymany primes occur in the intersection of T p ([ A ]) and T as p → ∞ .1.5. Organization of the paper.
In §2, we recall the definitions of the Hilbert modular surface and theHirzebruch–Zagier divisors. In §3, we bound the archimedean contribution. We spend §4 counting specialendomorphisms and bounding the non-archimedean contribution. We use Borcherds’ theory in §5.1 to choosea compact Hirzebruch–Zagier divisor and relate the Arakelov intersection number to Faltings height. Wealso extend Autissier’s result to the setting of Hilbert modular surfaces. Finally, we assemble all these resultstogether in §5.2 to prove Theorem 1.1.6.
Notation and conventions.
We use K to denote a number field and let O K be its ring of integers.As in Theorem 1, we use F to denote a fixed real quadratic field with discriminant D ; its ring of integers isdenoted by O F and d F is its different ideal. For a ∈ F , we use Nm a to denote its F/ Q -norm.The statement of Theorem 1 is invariant under isogeny. Therefore, we will always assume that A has realmultiplication by the maximal order O F , and is equipped with an a -polarization for some (integral) ideal a of O F ; see [Pap95, § 2.1 item 2 before Def. 2.1.1] for the definition of an a -polarization. We may also assumethat A has semistable reduction over K . For any abelian varieties B, B ′ , we use End( B ) and Hom(
B, B ′ ) todenote the endomorphism ring of B and the Z -module given by homomorphisms from B to B ′ .Let H be the Hilbert modular surface over Z associated to F which is the moduli stack of abelian surfaces B with real multiplication by O F and an a -polarization. This is a Deligne–Mumford stack and we use [ B ] to denote the point of H determined by B . Sometimes, we may denote a point of H by [ B ′ ] ; this means that B ′ is the abelian surface determines this given point.We always use mathcal letters to mean the natural extension over certain ring of integers. For example,we use A to denote the everywhere semistable semi-abelian scheme over O K such that A K = A K .Throughout the text, v means a place of K , either archimedean or finite. If v is finite, we use F v to denoteits residue field and e v to denote the degree of ramification of K at v . If A has good reduction at v , weuse A v,n to denote A modulo v n . We always use p to denote a prime number which is totally split in thenarrow Hilbert class field of F and denote by p , p ′ the two primes ideals of F above p . We use A [ p ] and A [ p ] to denote the p -torsion and p -torsion subgroups of A . Acknowledgements.
We thank George Boxer, Francesc Castella, Kęstutis Česnavičius, François Charles,William Chen, Noam Elkies, Ziyang Gao, Chi-Yun Hsu, Nicholas Katz, Ilya Khayutin, Djordjo Milovic,Lucia Mocz, Peter Sarnak, William Sawin, Arul Shankar, Jacob Tsimerman, Tonghai Yang, and Shou-WuZhang for useful comments and/or discussions. We are also grateful to Kęstutis Česnavičius, Chao Li, Mark Technically speaking, since we will not make [ A ] into an arithmetic cycle, here by Arakelov intersection, we mean the heightof the -cycle [ A ] with respect to the Arakelov divisor T , which is endowed with a Hermitian metric by Borcherds’ theory. isin, and Lucia Mocz for very useful comments on previous versions of this paper. We would like to thankDavesh Maulik for pointing a gap in Theorem 4.1.1 in an earlier version of this paper. The second author,during her stay at the Institute for Advanced Study, was supported by the NSF grant DMS-1128115 to IAS.2. Hirzebruch–Zagier divisors and Hecke orbits
In this section, we first recall the definition of Hirzebruch–Zagier divisors and their properties and thenwe specify the Hecke orbits that will be used in the rest of the paper.2.1.
The Hilbert modular surface and the Hirzebruch–Zagier divisors.
Recall that we use H todenote the moduli stack over Spec Z that parametrizes abelian surfaces with real multiplication by O F andan a -polarization (see [Pap95, Def. 2.1.1]). It is a Deligne–Mumford stack. A totally positive element a ∈ a gives rise to a polarization on A and a symplectic form ψ on the Betti cohomology group W = H B ( A ( C ) , Q ) (here we choose an embedding K → C ). The set of GSp(
W, ψ )( R ) -conjugate cocharacters of the Hodgecocharacter (from the Hodge decomposition of W ⊗ C ) of A C coincides with H ± , the upper and lower Siegelhalf plane of genus . Let G ⊂ GSp(
W, ψ ) be the subgroup (over Q ) that commutes with O F ⊂ End( W ) and let G = Res F Q GL . Then G is naturally isomorphic to the subgroup of G such that for any Q -algebra R , the set G ( R ) consists of matrices with determinant in R (instead of R ⊗ Q F ). The embedding G ⊂ GSp induces an embedding of the Hilbert modular surface H Q = Sh ( G, X ) into Sh (GSp( W, ψ ) , H ± ) (see [vdG88, Chp. IX.1] and here X is the subset of H ± which consists of cocharacters conjugate to theHodge cocharacter of A C under G ( R ) ).Let H tor be a toroidal compactification of H as in [Rap78] (see also [Cha90, §3]). The stack H tor is regularand proper and H tor \H is a normal crossing divisor (see, for example, [Pap95, 2.1.2, 2.1.3] for the regularityof H and [Cha90, Thm. 3.6, 4.3] for the smoothness of the formal neighborhood of the boundary and theproperty that the boundary is a normal crossing divisor), and hence the arithmetic intersection theorydeveloped by Burgos Gil, Kramer, and Kühn in [BGKK07] applies to H tor (see, for example, [BBGK07, §1,§6] for a summary of their theory). We will use [ A ] (resp. [ A ] ) to denote the unique O K -point (resp. K -point)of H corresponding to A (the stack H tor being proper allows us to do this). . We now summarize some basic facts about H C and the Hirzebruch–Zagier divisors. The facts discussedhere can be found in [vdG88, Chp. I, V, IX], [BBGK07, §2.3, §5.1] and [Gor02, Chp. 2]; however, sinceconventions differ, we will use this subsection to fix our notation. After giving the definition of Hirzebruch–Zagier divisors, we first show that among them, there exists a nonempty compact one and then we give amoduli interpretation of these divisors.Let H denote the upper half plane. The two real embeddings of F induce two embeddings σ , σ :SL ( F ) → SL ( R ) . The action of g ∈ SL ( F ) on H is given by σ ( g ) on the first copy of H and by σ ( g ) on the second copy. Since our Hilbert modular surface H admits a map to S , we have H ( C ) = Γ \ H , where Γ = SL ( F ) ∩ (cid:18) O F ( ad F ) − ad F O F (cid:19) (see, for example, [Gor02, pp. 71]). For any r ∈ Z > , we recall the definition of the Hirzebruch–Zagier divisors T ( r ) in H C (see for example[vdG88, V.1.3] and [BBGK07, sec. 2.3]). Let γ ′ denote the Gal( F/ Q ) -conjugate of a given γ ∈ F . Considerthe lattice L = (cid:26)(cid:18) a γγ ′ b (cid:19) : a ∈ ( D Nm a ) Z , b ∈ Z , γ ∈ a (cid:27) in the rational quadratic space V = (cid:26)(cid:18) a γγ ′ b (cid:19) : a, b ∈ Q , γ ∈ F (cid:27) One needs to choose suitable level structure to ensure that the natural finite morphism is an embedding. In general, one needs to pass to a finite field extension to extend a K -point on a proper Deligne–Mumford stack to an O K point; however, since we have assumed that A has a semistable integral model A over O K , we do not need to pass to a furtherfield extension. The Hilbert modular surface H is connected and hence we may use Res F Q SL instead of G to study the complex points.Notice that our lattice is different from the default choice in [BBGK07]. Using their notation, we work with Γ( O F ⊕ ad F ) . ith the quadratic form given by the determinant. The group Γ acts on V via v.g = ( g ′ ) t · v · g for g ∈ Γ , v ∈ V and this action preserves L . The quadratic space V is of signature (2 , . One may also view H as an orthogonal type Shimura variety defined by SO( V ) . The divisor T ( r ) is defined to be the reduceddivisor in H C whose set of C -points is the image of [ M ∈ L, det( M )= r Nm a { ( z , z ) ∈ H : az z + γz + γ ′ z + b = 0 } . Proposition 2.1.2.
The divisor T ( r ) is nonempty if and only if r Nm a modulo D is − Nm γ for some γ ∈ a . In this case, T ( r ) is defined over Q and is either a modular curve or a Shimura curve defined by theindefinite quaternion algebra (cid:18) D, − r Nm a Q (cid:19) . If r is not the norm of an ideal of O F , then T ( r ) is a Shimuracurve, and hence compact.Proof. The first assertion follows from the definition of a Hirzebruch–Zagier divisor. By the discussionon [vdG88, pp. 89–90], the divisor T ( r ) is the union of Shimura curves defined by the quaternion algebramentioned above and hence is defined over Q . The last assertion follows from [vdG88, Chp. V, 1.7]. (cid:3) Corollary 2.1.3.
Let q denote a rational prime inert in F . Then the divisor T ( qD ) is non-empty andcompact.Proof. As q is inert, qD is not the norm of an ideal of O F . Further, qD is modulo D . It follows fromProposition 2.1.2 that T ( qD ) is compact and nonempty. (cid:3) Hirzebruch–Zagier divisors parametrize abelian surfaces with extra special endomorphisms. After recallingthe definition of special endomorphisms, we give a sketch of the proof of this fact (see Lemma 2.1.6), whichmay be well known to experts. From now on, B denotes an abelian surface over some Z -algebra with an a -polarization and O F ⊂ End( B ) . We fix a totally positive element in a ∩ Q and this provides a fixedpolarization on B and we use this polarization to define the Rosati involution ( − ) ∗ on End( B ) ⊗ Q . Definition 2.1.4 (see also [KR99, Def. 1.2]) . An s ∈ End( B ) is a special endomorphism if a ◦ s = s ◦ a ′ forall a ∈ O F ⊂ End( B ) and s ∗ = s .All the special endomorphisms of B form a sub- Z -module of End( B ) . It is well known that the rank ofthis submodule is at most 4. The following lemma recalls the discussion after [KR99, Def. 1.2]. Lemma 2.1.5.
For a special endomorphism s , there is a Q ( s ) ∈ Z such that s ◦ s = Q ( s ) · Id B andhence also Deg s = Q ( s ) . The function Q is a positive definite quadratic form on the Z -module of specialendomorphisms of B . The Q -vector space in End( B ) ⊗ Q generated by special endomorphisms depends on the choice of thepolarization on B . However, there are natural isomorphisms between the Q -vector spaces of special endo-morphisms defined by different polarizations and the quadratic forms coincide up to multiplying by a fixedscalar determined by the polarizations. Lemma 2.1.6.
The Hirzebruch–Zagier divisor T ( Dr ) defined in 2.1.1 is the locus of H Q where the abeliansurface has a special endomorphism s with Q ( s ) = r Nm a . In particular, the degree of the endomorphism s is ( r Nm a ) .Proof. We only need to check the statement over C . Given a point in H ( C ) corresponding to ( z , z ) ∈ H ,it corresponds to an abelian surface B with B ( C ) = C / O F ( z , z ) + ( ad F ) − . The Riemann form E on H ( B, Z ) is, up to multiplying by a constant ∈ Q > , the pull back of the standard alternating form( Tr F/ Q of (cid:18) − (cid:19) ) on O F ⊕ ( ad F ) − via the isomorphism O F ( z , z ) + ( ad F ) − ∼ = O F ⊕ ( ad F ) − ; see,for example, [vdG88, p. 208] and [BBGK07, the discussion after Thm. 5.1]. Any endomorphism of B is This is the definition in [BBGK07]. The lattice in [vdG88] differs by a multiple of the scalar matrix √ D · I so these twodefinitions of T ( r ) coincide. We only deals with T ( Dr ) since these are the divisors that we will use in the proof of Theorem 1. However, after minormodification, the proof shows that any T ( r ) parametrizes abelian surfaces with an extra special endomorphism. iven by the induced map on B ( C ) of some C -linear map on C . For any endomorphism s , the condition f ◦ s = s ◦ f ′ for all f ∈ O F is equivalent to the condition that the C -linear map corresponding to s is ofthe form (1 , (0 , α ′ z + β ′ ) , (0 , ( αz + β ) where α ∈ ad F , β ∈ O F . This linear map gives rise to anendomorphism of B if and only if the image of ( z , z ) is in the period lattice. In other words, there exists ν ∈ O F , δ ∈ ( ad F ) − such that ( z ( αz + β ) , z ( α ′ z + β ′ )) = ( νz + δ, ν ′ z + δ ′ ) .For every component of T ( Dr ) , there exists M ∈ L in 2.1.1 satisfying det( M ) = Dr Nm a . Write M = (cid:18) a γγ ′ b (cid:19) where a ∈ ( D Nm a ) Z , b ∈ Z , γ ∈ a . Since D Nm a | γγ ′ , we have γ √ D ∈ a ⊂ O F . Moreover a √ D ∈ (Nm a ) d F ⊂ ad F . We take α = a √ D and β = γ ′ √ D . Given ( z , z ) ∈ H such that az z + γz + γ ′ z + b = 0 ,we have αz z + βz = β ′ z − b √ D , α ′ z z + β ′ z = βz + b √ D .
Hence (1 , (0 , α ′ z + β ′ ) , (0 , ( αz + β ) is an endomorphism s with f ◦ s = s ◦ f ′ for all f ∈ O F .To check that s = s ∗ , it is equivalent to check that for any u, v ∈ H ( B, Z ) , one has E ( su, v ) = E ( u, sv ) .Since we have already checked that f ◦ s = s ◦ f ′ for all f ∈ O F , one only needs to check the above equalityfor u, v ∈ { e = ( z , z ) , e = (1 , } ⊂ C . By construction, se = β ′ e − b √ D e and se = αe + βe andthen we conclude by the fact that Tr F/ Q β = Tr F/ Q β ′ , Tr F/ Q − b √ D = 0 , and Tr F/ Q α = 0 . Moreover, on C ,the composite s ◦ s = det( M ) D · Id C . Hence s is a special endomorphism with Q ( s ) = r Nm a .On the other hand, the moduli space of B with a special endomorphism is -dimensional. Hence thetwo conditions z ( αz + β ) = νz + δ and z ( α ′ z + β ′ ) = ν ′ z + δ ′ are linearly dependent. Hence either α, δ ∈ Q , β = − ν ′ or α · √ D, δ · √ D ∈ Q , β = ν ′ . In the first case, we have αz z + β ′ z + βz − δ = 0 , ββ ′ + αδ = r > . In this case, there is no ( z , z ) satisfying the above condition (see for example [vdG88, V.4]). In the secondcase, take a = α · √ D, b = − δ · √ D, γ = β ′ · √ D . Then M = (cid:18) a γγ ′ b (cid:19) ∈ L and hence [ B ] ∈ T ( Dr ) . (cid:3) Let T ( r ) be the Zariski closure of T ( r ) in H tor over Spec Z . Corollary 2.1.7.
Assume T ( Dr ) is compact. Then for any finite place v , the points on T ( Dr ) F v correspondto abelian surfaces which are not absolutely simple. The abelian surfaces parametrized by T ( Dr ) admit aspecial endomorphism s such that Q ( s ) = r Nm a .Proof. Since T ( Dr ) is compact, every point parametrized by it has potentially good reduction. For any givenpoint parametrized by T ( Dr ) F v , let [ B ] be a lift of the point on T ( Dr ) over O K for some number field K .By Lemma 2.1.6, the Néron–Severi rank of B K is and hence the Néron–Severi rank of B F v is at least . Bythe classification of the endomorphism algebra of abelian varieties, we see that the Néron–Severi rank of ageometrically simple abelian surface is at most 2 and then conclude that B F v is not geometrically simple. Thelast assertion follows from Lemma 2.1.6 and the fact that the canonical reduction map End( B K ) → End( B F v ) is injective. (cid:3) Hecke orbits.
The idea of the proof of Theorem 1 is to show that the corresponding Hecke orbits of [ A ] intersect certain Hirzebruch–Zagier divisors at more and more places of K as one varies over certainwell-chosen Hecke operators. In this subsection, we specify the Hecke orbits which we will use later.2.2.1 . Recall that p is a prime which splits completely in the narrow Hilbert class field of F and ( p ) = pp ′ ⊂O F . Hence p = ( λ ) , p ′ = ( λ ′ ) with λ, λ ′ ∈ F totally positive and λλ ′ = p . Let G ad1 be the adjoint group of G = Res F Q GL . We denote by G ad1 ( R ) the image of G ( R ) in G ad1 ( R ) and let G ad1 ( Q ) be G ad1 ( Q ) ∩ G ad1 ( R ) .Since λ is totally positive, the image of the diagonal matrix g p := diag(1 , λ ) under G → G ad1 lies in G ad1 ( Q ) , We view every Hecke orbit as a horizontal divisor over K , so the arithmetic intersection number is a sum over the finiteplaces of K . o it induces a correspondence T p on H Z [1 /p ] (defined in [Del79]; see also [Kis10, sec. 3.2]). The followinglemma provides a moduli interpretation of T p . Lemma 2.2.2.
We have T p [ A ] = p + 1 . Over Z [1 /p ] , the set T p [ A ] consists of those points on H thatcorrespond to a quotient of A by an order p subgroup in A [ p ] endowed the with induced O F -structure and asuitable a -polarization. Proof.
The first assertion follows from the definitions: T p [ A ] = / ( g − p Γ g p ∩ Γ) = P ( F p ) = p + 1 . For the second assertion, since the correspondence T p is étale, we only need to show the same statementfor T p [ A ] over C for a fixed embedding of K into C . On the one hand, as O F acts on A [ p ] via O F / p ∼ = F p and Z ⊂ O F surjects onto F p , any subgroup of A [ p ] is O F -invariant. Therefore, any quotient of A by an order p subgroup of A [ p ] has the induced O F -structure. Moreover, by [BBGK07, Lem. 5.9], any such quotient of A is O F -polarizable.On the other hand, by [vdG88, p. 208], if a point ( z , z ) ∈ H corresponds to A C , then A ( C ) is isomorphicto C / ( O F ( z , z )+( ad F ) − ) . Then diag(1 , λ ) z corresponds to C / ( O F ( z /λ, z /λ ′ )+( ad F ) − ) , so the kernelof the isogeny defined by diag(1 , λ ) is contained in ker( λ ) = A [ p ] . On the quotient C / ( O F ( z /λ, z /λ ′ ) +( ad F ) − ) , the O F -structure is the induced one and the choice of λ determines the O F -polarization. Sincethe other elements in T p differ from diag(1 , λ ) by the action of some element in Γ (on H and on A [ p ] ), theset T p injects into the set of order p subgroups of A [ p ] . Since both sets have cardinality p + 1 , this is in facta bijection. (cid:3) Archimedean places and equidistribution of Hecke orbits
Let Ψ be a meromorphic Hilbert modular form of parallel weight k over Q such that Div(Ψ) in H Q isgiven by P c ∈ I c r T ( r ) , where k ∈ N > , I is a finite set, c r ∈ Z , T ( r ) is compact and D | r for all r ∈ I . Inthe proof of Theorem 1, we will use Lemma 5.1.1 to construct such meromorphic Hilbert modular form. Weassume that End( A K ) = O F and hence T p ([ A ]) does not intersect T ( r ) in characteristic zero. This is the keycase in the proof of Theorem 1. Fix an embedding σ : K → C . Given an abelian surface B corresponding toa point [ B ] on H Q , we use σ ([ B ]) to denote the corresponding C -point on H via base change by σ .We set || Ψ( z ) || Pet = | Ψ( z , z )( ℑ z ) k/ ( ℑ z ) k/ | , where z = ( z , z ) ∈ H . This norm is well-definedoutside (the preimage of) S r ∈ I T ( r ) and invariant under Γ (defined in 2.1.1) and hence we will also view || Ψ || Pet as a function on H C \ S r ∈ I T ( r ) . The real analytic function − log || Ψ || Pet is a Green function for P c ∈ I c r T ( r ) and endows it with the structure of an arithmetic divisor \ P c ∈ I c r T ( r ) .The goal of this section is to show that for most p as in 2.2.1, the archimedean contribution (in the heightof T p ([ A ]) with respect to the arithmetic divisor \ P c ∈ I c r T ( r ) ; we will discuss this height in detail in §5.1) − X [ B ] ∈ T p [ A ] log || Ψ( σ ([ B ])) || Pet = o ( p log p ) as p → ∞ . The equidistribution theorems for Hecke orbits on Shimura varieties reduces this goal to a suitable upperbound for − log || Ψ( σ ([ B ])) || Pet for all [ B ] ∈ T p [ A ] which is valid for most p . The proofs are inspired byCharles’ treatment in the case of the modular curve. Recall that all T ( r ) ( r ∈ I ) are compact, so we avoiddealing with estimates around the cusps.Throughout this section, p is a prime as in 2.2.1 and N i ( ∗ ) , C i ( ∗ ) denote constants only depending on ∗ . Inparticular, if there is no ( ∗ ) , it means an absolute constant. After defining the constants, we may abbreviate N i ( ∗ ) , C i ( ∗ ) as N i , C i . Given η ∈ F , we use | η | < C to mean that for any real embedding ι : F → R , theabsolute value | η | ι < C . We also use | · | to denote the absolute value on C .3.1. An upper bound of the values of Green function on Hecke orbits. . Let
F ⊂ H be the fundamental domain for Γ described in [vdG88, I.3] and F its closure (with respectto the complex analytic topology) in H (that is, the cusps of Γ \ H are not included). Let Ω ⊂ F be a The definition of T p depends on the choice of λ if we do not pass to a certain finite quotient of H . However, there areonly finitely many choices: let U be the unit group of O F and U + the subgroup of totally positive units; then the number ofchoices is U + /U . Hence we will not specify our choice of λ as it does not affect the arguments in this paper. The choice of λ determines the polarization. ompact domain containing the preimage of S r ∈ I T ( r ) in Div(Ψ) . Then there exists C ∈ R > such that forany ( z , z ) = ( x + √− y , x + √− y ) ∈ Ω , we have | x i | < C and C − < y i < C .For any Q -point [ B ] in H , we use z ( B ) = ( z ( B ) , z ( B )) = ( x ( B ) + √− y ( B ) , x ( B ) , √− y ( B )) todenote the preimage of σ ([ B ]) in F .Let G be the pull back to H of the Green function − log || Ψ || Pet of P c r T ( r ) . There are only finitely manycomponents of the preimage of T ( r ) in Ω and for each component, we pick ( a, b, γ ) such that (cid:18) a γγ ′ b (cid:19) ∈ L with ab − γγ ′ = r Nm a as in 2.1.1 such that this component is defined by az z + γz + γ ′ z + b = 0 . Weuse M Ω ,r to denote this finite set of ( a, b, γ ) . Then by the definition of the Green function, we have that G + P r ∈ I c r P ( a,b,γ ) ∈M Ω ,r log | az z + γz + γ ′ z + b | is a real analytic function on F .The goal of this subsection is to show that for most p , one has that − log | az ( B ) z ( B ) + γz ( B ) + γ ′ z ( B ) + b | ≤ O (log p ) , ∀ [ B ] ∈ T p ([ A ]) . Proposition 3.1.2.
Let ( a, b, γ ) be a fixed triple in M Ω ,r . Given C > and ǫ > , there is an N ( ǫ , C ) > such that for every N > N ( ǫ , C ) , the number of primes in [ N / , N ] for which thereexists some [ B ] ∈ T p ( σ [ A ]) such that | az ( B ) z ( B ) + γz ( B ) + γ ′ z ( B ) + b | < p − C is at most ǫ { primes ∈ [ N / , N ] } . We extend the idea in [Cha14] of relating bad primes and degrees of homomorphisms between well-chosenCM elliptic curves to the setting of Hilbert modular surfaces by using the theory of special endomorphisms.A point [ B ] on H Q is called special if there exist T ( n ) and T ( n ) , n , n ∈ N , n n / ∈ ( N ) such that [ B ] ∈ T ( n ) ∩ T ( n ) . If [ B ] is special, then B has complex multiplication. We construct a special point [ A CM ] on H C close to σ ([ A ]) and show that if some point in T p ( σ ([ A ])) is close to Div(Ψ) = P r ∈ I c r T ( r ) ,then A CM has a special endomorphism of certain degree. Proposition 3.1.2 then follows after analysis of thepossible degree of special endomorphisms of A CM . In what follows, we will not specify the dependence ofthe constants C i in this subsection on the fixed triple ( a, b, γ ) .The following lemma shows that if there exists [ B ] ∈ T p ( σ ([ A ])) which is close to T ( r ) , then σ ([ A ]) is closeto T ( pr ) . Lemma 3.1.3.
If there exists [ B ] ∈ T p [ A ] such that | az ( B ) z ( B ) + λz ( B ) + λ ′ z ( B ) + b | < p − C , thenthere exist m ∈ ( D Nm a ) Z , l ∈ Z and η ∈ a such that ml − Nm( η ) = rp Nm a and | mz ( A ) z ( A ) + ηz ( A ) + η ′ z ( A ) + l | < p − C . Moreover, we have | m | , | l | , | η | < C p .Proof. As in 2.2.1, we write p = ( λ ) and after multiplying λ by an element in ( O × ) , we may assume that C − √ p < | λ | < C √ p . We may also assume z ( B ) ∈ Ω (this can be done by letting N be large enough).Let U = (cid:18) u u u u (cid:19) ∈ (cid:18) λ (cid:19) · Γ be the matrix that maps z ( A ) to z ( B ) . The set Γ acts on V (in 2.1.1) via g.M = ( g ′ ) t M g and this action preserves L . Let (cid:18) m ηη ′ l (cid:19) = ( U ′ ) t (cid:18) a γγ ′ b (cid:19) U ∈ L . Then ml − ηη ′ = det( U ′ )( ab − γγ ) det( U ) = rp Nm a and mz ( A ) z ( A ) + ηz ( A ) + η ′ z ( A ) + l = az ( B ) z ( B ) + λz ( B ) + λ ′ z ( B ) + b. This proves the first assertion.For the second assertion, we first bound | u ij | . Consider the real embedding of F corresponding to thefirst coordinate of H and we will still use u ij to denote its image under this embedding. By definitionof U , we have z ( B ) = u z ( A ) + u u z ( A ) + u and hence y ( B ) = λy ( A ) | u z ( A ) + u | . Since y ( B ) > C − and y ( A ) < C , we have | u z ( A ) + u | < C C √ p . Consider the imaginary part of u z ( A ) + u and noticethat y ( A ) > C − . Thus, we have | u | < C C √ p . By considering the real part, we obtain | u | ≤ | u x ( A ) + u | + | u x ( A ) | ≤ | u z ( A ) + u | + | u x ( A ) | ≤ C C √ p + C C √ p. n the other hand, using the bounds of | u | , | u | , we have | u | y ( A ) ≤ | u z ( A ) + u | = | ( z ( B )( u z ( A ) + u )) | ≤ C √ p. Hence we obtain | u | ≤ C C √ p, | u | ≤ | u z ( A ) + u | + | u x ( A ) | ≤ C √ p + C C √ p. The same argument works for the other embedding of F → R by studying z ( B ) , z ( A ) . The bounds of | m | , | l | , | η | follow from the fact that | u ij | is bounded by O ( p / ) . (cid:3) . The following lemma shows that if two Hecke orbits of σ ([ A ]) satisfy the assumption of Lemma 3.1.3,then σ ([ A ]) is close to a special point on H C . Recall that for a special point [ B ] ∈ H ( C ) , one defines aquadratic form Q , up to SL ( Z ) -equivalence, as follows (see, for example, [HZ76, 1.1], [vdG88, V.4]). Let L [ B ] be the sub lattice of L in 2.1.1 such that for any v ∈ L [ B ] , one has ( z ( B ) 1) v ( z ( B ) 1) t = 0 . Thelattice L [ B ] is of rank two and equipped with a natural orientation. The restriction of the quadratic formon L to the rank two lattice L [ B ] is positive definite and it coincides, up to a constant, with the quadraticform on the Z -module of special endomorphisms of B . By choosing an oriented basis, one obtains a positivedefinite binary integral quadratic form Q . Lemma 3.1.5.
Assume that N is large enough and that for primes p , p ∈ [ N / , N ] , p < p , there exist [ B ] ∈ T p [ A ] , [ B ′ ] ∈ T p [ A ] satisfying | az ( B ) z ( B ) + λz ( B ) + λ ′ z ( B ) + b | < p − C and | az ( B ′ ) z ( B ′ ) + λz ( B ′ ) + λ ′ z ( B ′ ) + b | < p − C . Then there exists a special point [ A CM ] on H C such that | z ( A ) − z ( A CM ) |
Assume that N is large enough and that for p i ∈ [ N / , N ] with i = 1 , , , , p = p , p = p , there exists [ B i ] ∈ T p i ([ A ]) such that | az ( B i ) z ( B i ) + λz ( B i ) + λ ′ z ( B i ) + b | < p − C i . Let [ A ] , [ A ] be two special points constructed as in Lemma 3.1.5 by using the assumption on p , p and p , p . Then [ A ] = [ A ] . More precisely, if | z ( A ) − z ( A ) | ≤ C N − C +7 , then z ( A ) = z ( A ) .Proof. Let f ( z ) , f ( z ) ∈ O [ z ] be the quadratic equations defining z ( A ) , z ( A ) in the proof of Lemma 3.1.5.Let α , α ∈ F be the leading coefficients of f , f and − ∆ , − ∆ the discriminants. Since f i has two complexroots for both embeddings of F , one has that ∆ i is totally real. Given a real embedding of F , we may assumeboth α , α are positive with respect to this embedding.By the definition of f i and Lemma 3.1.3, we have | α i | ≤ C N and | ∆ i | ≤ C N . Moreover, since α i , ∆ i ∈ O , we have the nonzero | Nm( α ∆ − α ∆ ) | ≥ and hence | α ∆ − α ∆ | ≥ | ( α ′ ) ∆ ′ − ( α ′ ) ∆ ′ | − ≥ (8 C C ) − N − . Putting these inequalities together, we obtain | z ( A ) − z ( A ) | ≥ | z ( A ) − z ( A ) | ≥ | y ( A ) − y ( A ) | = (cid:12)(cid:12)(cid:12) √ ∆ α − √ ∆ α (cid:12)(cid:12)(cid:12) = | α ∆ − α ∆ | α α ( α √ ∆ + α √ ∆ ) ≥ C N − . This contradicts our assumption when N C − > C C − . (cid:3) Corollary 3.1.7.
Assume that N is large enough as above and that A satisfies the assumption in Lemma 3.1.5.For any p ∈ [ √ N , N ] such that there exists [ B ′′ ] ∈ T p [ A ] satisfying | az ( B ′′ ) z ( B ′′ ) + λz ( B ′′ ) + λ ′ z ( B ′′ ) + b | < p − C , the quadratic form Q N in Lemma 3.1.5 represents p r .Proof. By Lemma 3.1.5, we construct special points [ A ] by using p , p and [ A ] by using p , p . ByLemma 3.1.6, we have [ A ] = [ A ] and hence they have the same quadratic form Q N . Since [ A ] lies on T ( p r ) , then Q N represents p r . (cid:3) Lemma 3.1.8.
Fix C > and let ∆ N denote the discriminant of Q N in Lemma 3.1.5. As N → ∞ , wehave | ∆ N | → ∞ .Proof. Fix a bound X of | ∆ N | , then there are only finitely many equivalent classes of integral binary quadraticforms of discriminant ≤ X . For each class, [HZ76, Thm. 1] shows that are only finitely many special pointscorresponding to the given class of quadratic forms. As N → ∞ , the CM approximation [ A CM ,N ] is closerto σ [ A ] and hence | ∆ N | cannot be bounded. (cid:3) In the proof, we give a constant N ( C ) such that being large enough means N > N . Although in [HZ76], they assume that D is a prime, their method still works in general. See for example [vdG88, V.6]. roof of Proposition 3.1.2. Let p , p be the smallest primes in [ √ N , N ] such that there exists B ∈ T p i A such that | az ( B ) z ( B ) + λz ( B ) + λ ′ z ( B ) + b | < p − C i . Then by Lemma 3.1.5 and Corollary 3.1.7, weobtain a quadratic form Q N associated to a special point [ A CM ,N ] which represents p r for any prime p in [ √ N , N ] which satisfies the condition that there exists [ B ′′ ] ∈ T p [ A ] satisfying | az ( B ′′ ) z ( B ′′ ) + λz ( B ′′ ) + λ ′ z ( B ′′ ) + b | < p − C . There exists a positive definite integral binary quadratic form Q ′ N such that a prime p is represented by Q ′ N if and only if pr is represented by Q N . One also has that the absolute value of thediscriminant ∆ ′ N of Q ′ N is at least | ∆ N | /r .It remains to show that the density { prime p ∈ [ N / , N ] , p is represented by Q ′ N }{ p ∈ [ N / , N ] } → as N → ∞ . When | ∆ ′ N | ≤ (log N ) , [TZ16, Corollary 1.3] shows that { prime p ∈ [ N / , N ] , p is represented by Q N } ≪ Li( N ) h N ,where h N is the number of SL ( Z ) -equivalence classes of primitive positive definite integral binary quadraticforms of discriminant ∆ ′ N . Since ∆ ′ N → −∞ by Lemma 3.1.8, one has h N → ∞ . When | ∆ ′ N | > (log N ) ,[Cha14, Lemma 5.2] shows that { integer n ∈ [ N / , N ] , n is represented by Q ′ N } ≤ √ N + 8 N/ | ∆ ′ N | / = O (cid:18) N (log N ) (cid:19) . We get the desired property by putting these two cases together. (cid:3)
From equidistribution to an upper bound of archimedean contribution.
This is the maintheorem of this section. We use Proposition 3.1.2 and equidistribution theorem for Hecke orbits to showthat for most p , the archimedean contribution in the height of T p ([ A ]) is o ( p log p ) . Theorem 3.2.1.
For any ǫ , ǫ > , there is an N ( ǫ , ǫ ) > such that for every N > N ( ǫ , ǫ ) , the numberof primes in the interval [ N / , N ] for which − X [ B ] ∈ T p [ A ] log || Ψ( σ ([ B ])) || Pet ≥ ǫ p log p is at most ǫ { ℓ ∈ [ N / , N ] prime } .Proof. Notation as in 3.1.1. We first show that a fixed triple ( a, b, γ ) , − X B ∈ T p A log | az ( B ) z ( B ) + γz ( B ) + γ ′ z ( B ) + b | = o ( p log p ) . For any ǫ > , by the equidistribution theorem of Hecke orbits (see for example [COU01]), there existconstants N ( ǫ, C ) > and C ( ǫ, C ) < such that for any p > N , { [ B ] ∈ T p [ A ] : log | az ( B ) z ( B ) + γz ( B ) + γ ′ z ( B ) + b | < C } < ǫp/C . Let I ′ = { r ∈ I | c ( r ) > } and M = P r ∈ I ′ M Ω ,r . Taking ǫ = ǫ /M and applying Proposition 3.1.2 forall triples ( a, b, γ ) ∈ S r : c ( r ) > M Ω ,r , we have that, for every N > N ( ǫ , C ) := max ( a,b,γ ) { N ( a, b, γ, ǫ , C ) , N } , − X [ B ] ∈ T p [ A ] log | az ( B ) z ( B ) + γz ( B ) + γ ′ z ( B ) + b | < − ( p + 1) C + ( ǫp/C ) · C log p holds for primes p ∈ [ N / , N ] outside a set B N of density ǫ (this set is the union of the exceptional sets forall ( a, b, γ ) ∈ ∪ r : c ( r ) > M Ω ,r ).Let φ be a smooth function which is in Ω with compact support in F . Then by 3.1.1, the function f = G + P r ∈ I c ( r ) P ( a,b,γ ) ∈M Ω ,r φ ( z ) log | az z + γz + γ ′ z + b | is smooth on F . Since G and hence f go to −∞ as y y goes to ∞ , we see that f is bounded above on F . On the other hand, since φ ( z ) has compactsupport, φ ( z ) log | az z + γz + γ ′ z + b | is also bounded above. Therefore, since T p [ A ] = p + 1 , we have X [ B ] ∈ T p [ A ] (cid:0) G ( z ( B )) + X r ∈ I ′ c ( r ) X ( a,b,γ ) ∈M Ω ,r φ ( z ) log | az ( B ) z ( B ) + γz ( B ) + γ ′ z ( B ) + b | (cid:1) < C · ( p + 1) . ake ǫ = ǫ / (2 M ) , then we have, for N > N , for p ∈ [ N / , N ] \ B N , X [ B ] ∈ T p [ A ] G ( z ( B )) < C ( p + 1) + ( ǫ p log p ) / . Then by taking N ( ǫ , ǫ ) > N large enough so that C N ( ǫ , ǫ ) / < ( ǫ N ( ǫ , ǫ ) / log N ( ǫ , ǫ )) / , thetheorem follows. (cid:3) Special endomorphisms and contributions at finite places
In this section, we will bound the local intersection multiplicities of ( T p ([ A ]) , T ( r )) at non-archimedeanplaces for r ∈ I , where I is a fixed finite set such that D | r and T ( r ) is compact in H for all r ∈ I . The set I will be chosen by Lemma 5.1.1. Throughout, ℓ denotes a prime, and B, B ′ , B , B ′ denote abelian surfaces with O F -multiplication and a -polarization. We fix a finite place v of a number field K over ℓ . Recall that e v isthe ramification degree of K at v . The abelian surfaces may be defined over K, O K , O K v , or O K v /v n . Recallthat we use B , B ′ to denote abelian surfaces defined over O K v and we use B v,n , B ′ v,n to denote their reductionmodulo v n . We will use B ℓ,n and B ′ ℓ,n to denote surfaces over O K v /ℓ n (note that the abelian varieties aredefined modulo ℓ n , not v n ) which do not a priori come with lifts to O K v . Let M v,n and M ℓ,n denote themodule of special endomorphisms of B v,n and B ℓ,n respectively. Finally, we let Λ v = End( B [ ℓ ∞ ]) ∩ M v, ⊗ Z ℓ ,where the intersection takes place in End( B v, ) ⊗ Z ℓ = End( B v, [ ℓ ∞ ]) . We call the Z ℓ -module Λ v the set ofspecial endomorphisms of B [ ℓ ∞ ] Deformation theory.
The following result is crucial to bounding the local intersections:
Theorem 4.1.1.
Let m be the Z -rank of M v, and m ′ be the Z ℓ -rank of Λ v . Then there exists a positiveinteger n such that M n ′ + ke v ,v = (Λ v + ℓ k M v,n ′ ⊗ Z ℓ ) ∩ M n ,v where n ′ ≥ n and k is allowed to be anypositive integer. In an earlier version of the paper, we had in an earlier version of this paper implicitly assumed that Λ v = 0 (this assumption simplified the geometry-of-numbers arguments), and we are very grateful to Davesh Maulikfor pointing this out to us. We will need the following result on homomorphisms between abelian surfaces: Lemma 4.1.2.
Let α ∈ Hom( B ℓ,n − , B ′ ℓ,n − ) for any n ≥ .(1) The homomorphism ℓα lifts uniquely to Hom( B ℓ,n , B ′ ℓ,n ) .(2) If ℓα lifts to Hom( B ℓ,n +1 , B ′ ℓ,n +1 ) , then α lifts to Hom( B ℓ,n , B ′ ℓ,n ) . Let G i = B ℓ,i [ ℓ ∞ ] , and G ′ i = B ′ ℓ,i [ ℓ ∞ ] . By the Serre–Tate lifting theorem, it suffices to prove the analogousresult for ℓ -divisible groups. This is a straightforward application of Grothendieck–Messing theory. Beforeproceeding to the proof, we recall some facts from Grothendieck–Messing theory ([Mes72] contains everyresult that we need). All the reduction maps between the O K v /ℓ i for i = n, n ± are canonically equippedwith nilpotent divided powers (in fact, as n > , all the ideals in play are square-zero). Let D and D ′ denotethe Dieudonne-crystals associated to G n − and G ′ n − (see [Mes72, §2.5 of Chapter IV]). Any homomorphismbetween G n − and G ′ n − canonically induces a map of crystals D → D ′ .Let D i and D ′ i denote D and D ′ evaluated at O K v /ℓ i for i = n, n ± (these are free O K v /ℓ i -moduleswhose ranks equal the heights of G and G ′ ). Grothendieck–Messing theory associates canonical filtrations F i ⊂ D i and F ′ i ⊂ D ′ i to the groups G i and G ′ i for i = n, n ± . Note that the F n +1 reduces to F n and F n − under the canonical quotient maps (the analogous statement holds for F ′ n +1 ). The filtrations are directsummands of the crystals evaluated at the O K v /ℓ i . Suppose that W n +1 ⊂ D ′ n +1 is some submodule suchthat D ′ n +1 = F ′ n +1 ⊕ W n +1 . Let W n ⊂ D ′ n denote the mod- ℓ n reduction of W n +1 . Clearly, W n ⊕ F ′ n = D ′ n .By [Mes72, Theorem 1.6 of Chapter V], a homomorphism between G n − and G ′ n − lifts to Hom( G i , G ′ i ) (for i = n, n + 1 ) if and only if the associated map of crystals evaluated at O K v /ℓ i maps the filtration F i to F ′ i .Let α i : D i → D ′ i ( i = n, n ± ) denote the maps induced by α . We now proceed to the proofs of the twostatements. Proof of Lemma 4.1.2.
1) Let v n ∈ F n ⊂ D n , whose image in F n − is denoted by v n − . It suffices to prove that ℓα n ( v ) ∈ F ′ n .Let α n ( v ) = v n + w n , where v ′ n ∈ F ′ n and w n ∈ W n . As α n − ( F n − ) ⊂ F ′ n − , we have w n modulo ℓ n − is zero. It follows that ℓw n = 0 . Therefore, ℓα n ( v n ) = ℓv ′ n ∈ F ′ n . Thus ℓα n preserves filtrations,as requried.(2) As above, let v n ∈ F n . Let v n +1 ∈ F n +1 , whose mod- ℓ n reduction is v n . Suppose that α n +1 ( v n +1 ) = v ′ n +1 + w n +1 , where v ′ n +1 ∈ F ′ n +1 and w n +1 ∈ W ′ n +1 . As ℓα lifts to Hom( G n +1 , G ′ n +1 ) , it follows that ℓw n +1 = 0 . Therefore, w n +1 = 0 modulo ℓ n . It follows that α n +1 ( v n +1 ) modulo ℓ n - which equals α n ( v n ) - is an element of F ′ n . (cid:3) We now prove Theorem 4.1.1
Proof of Theorem 4.1.1.
For ease of notation, denote by Λ v,i the Z ℓ -module M v,i ⊗ Z ℓ . By the Serre–Tatetheorem, it suffices to prove the existence of n such that Λ v,n ′ + ke v = Λ v + ℓ k Λ v,n ′ . First, note thatLemma 4.1.2 implies that Λ v ⊂ Λ v, e v is co-torsion free. Let Λ ′ ⊂ Λ v, e v denote a direct summand of Λ v .As the Z ℓ -module of special endomorphisms of B [ ℓ ∞ ] = Λ v , it follows that Therefore, T n (Λ ′ ∩ Λ v,n ) = 0 .The theorem follows directly from the following claim. Claim.
We have that Λ ′ ∩ Λ v,n + e v ⊂ Λ ′ ∩ ℓ Λ v,n for large enough n .To prove the claim, we fix any n ′ > e v . Since T n (Λ ′ ∩ Λ v,n ) = 0 , then Λ ′ ∩ Λ v,n ′ + ke v ⊂ ℓ (Λ ′ ∩ Λ v,n ′ ) for largeenough k . We now prove by contradiction that Λ ′ ∩ Λ v,n ′ +( k +1) e v ⊂ ℓ (Λ ′ ∩ Λ v,n ′ + ke v ) for such k . Assume thatthere exists a special endomorphism α ∈ (Λ ′ ∩ Λ v,n ′ +( k +1) e v ) \ ℓ (Λ ′ ∩ Λ v,n ′ + ke v ) . If α ∈ ℓ (Λ ′ ∩ Λ v,n ′ +( k − e v ) ,we write α = ℓβ , where β ∈ Λ ′ ∩ Λ v,n ′ +( k − e v . By assumption, ℓβ ∈ Λ ′ ∩ Λ v,n ′ +( k +1) e v and then byLemma 4.1.2, β ∈ Λ ′ ∩ Λ v,n ′ + ke v . This contradicts that α / ∈ ℓ (Λ ′ ∩ Λ v,n ′ + ke v ) and hence we have shownthat α / ∈ ℓ (Λ ′ ∩ Λ v,n ′ +( k − e v ) . By iterating this argument, it follows that α / ∈ ℓ (Λ ′ ∩ Λ v,n ′ ) , which is acontradiction. (cid:3) Geometry of numbers and applications to counting special endomorphisms.
For an m -dimensional lattice M with a positive definite quadratic form Q , let µ ( M ) ≤ µ ( M ) . . . ≤ µ m ( M ) denotethe successive minima of M (see [EK95, Definition 2.2] for the definition of the term successive minima).We will need the following lemma due to Schmidt: Lemma 4.2.1.
Then { s ∈ M | Q ( s ) ≤ N } = O m X j =0 N j/ µ ( M ) . . . µ j ( M ) , where the implied constantdepends only on m .Proof. Equations (5) and (6) on page 518 of [EK95] imply that Lemma 2.4 of loc. cited implies the statedresult (the authors refer to [Sch68] for a proof of Lemma 2.4). (cid:3)
We now prove a elementary lemma that will allow us to use Lemma 4.2.1 to bound special endomorphismsof B v,n . We refer to the beginning of §4 for notation. Lemma 4.2.2.
Let m be the Z -rank of M v, and m ′ be the Z ℓ -rank of Λ v . Then j Y i =1 µ i ( M v,n ) ≫ ℓ n ( j − m ′ ) /e v .Proof. We may assume that m ′ < j . It suffices to prove that j Y i =1 µ i ( M v,n +( k +1) e v ) ≥ ℓ j − m ′ j Y i =1 µ i ( M v,n + ke v ) .For a lattice M , let d ( M ) denote the square root of its discriminant. Theorem 4.1.1 implies that d ( M n +( k +1) e v ) ≥ ℓ m − m ′ d ( M n + ke v ) . Thus,(4.2.1) m Y i =1 µ i ( M v,n +( k +1) e v ) ≥ ℓ m ′ m Y i =1 µ i ( M v,n + ke v ) by [EK95, Eqn. (5),(6)]. This is the desired result for j = m . oreover, if M ⊂ M ′ are lattices, then µ i ( M ) ≥ µ i ( M ′ ) . Therefore, Theorem 4.1.1 implies that µ i ( M n +( k +1) v ) ≤ µ i ( ℓM n + kv ) = ℓµ i ( M n + kv ) . The lemma follows from multiplying (4.2.1) with theinequality m Y i = j +1 µ i ( M v,n +( k +1) e v ) − ≥ m Y i = j +1 ℓ − µ i ( M v,n + ke v ) − . (cid:3) Lemma 4.2.1 and Lemma 4.2.2 immediately yield the following corollary:
Corollary 4.2.3.
Suppose that the Z ℓ -rank of Λ v is ≤ and the rank of M v, is m . Then { s ∈ M v,n | Q ( s ) ≤ N } = O (cid:16) N / + m X j =2 N j/ ℓ ( j − n/e v (cid:17) . Proof of the non-archimidean local results.
In what follows, we consider T p as in 2.2.1 with p = ℓ v and recall that we write ( p ) = pp ′ ⊂ O F . We always use N to denote a large enough integer. Lemma 4.3.1.
Over Z [1 /pr ] , we have T p T ( r ) = T ( pr ) = T p ′ T ( r ) . Moreover, for any n , if there exists [ B ] ∈ T p [ A ] such that [ B v,n ] ∈ T ( r ) , then [ A v,n ] ∈ T ( pr ) .Proof. By checking on complex points, we have T p T ( r ) = T ( pr ) = T p ′ T ( r ) . By definition, T ( m ) is theZariski closure of T ( m ) and hence T p T ( r ) = T ( pr ) . The second assertion then follows from the étaleness ofHecke orbits. (cid:3) For the rest of this section, we assume
End( A Q ) = O F and A has good reduction at v . Further, the norm of a special endomorphism s denotes the integer Q ( s ) .The following lemma is well-known and follows directly from the crystalline realization of the module ofspecial endomorphisms. We record a proof here for completeness. Lemma 4.3.2.
Let m be the Z -rank of M v, and m ′ be the Z ℓ -rank of Λ v . Then m ≤ and m ′ ≤ .Proof. Since s ∗ = s for any s ∈ M v, (resp. Λ v ), we have that Λ v ⊂ M v, ⊗ Z ℓ ⊂ H ( B v, /W (¯ F ℓ )) .On the other hand, since O F ⊂ End( B v, ) is stable under Rosati involution, we have a natural embedding O F ⊂ H ( B v, /W (¯ F ℓ )) . Since H ( B v, /W (¯ F ℓ ))[1 /ℓ ] is a W (¯ F ℓ ))[1 /ℓ ] -vector space of dimension , thenthe Frobenius invariant part of H ( B v, /W (¯ F ℓ ))[1 /ℓ ] ϕ =1 is a Q ℓ -vector space of dimension at most .Since s ◦ f = f ′ ◦ s for any s ∈ M v, ⊗ Q ℓ and f ∈ F , then O F ⊗ Q ℓ ∩ M v, ⊗ Q ℓ = 0 in H ( B v, /W (¯ F ℓ ))[1 /ℓ ] ϕ =1 .Since O F ⊗ Q ℓ has dimension , we have that M v, ⊗ Q ℓ is at most of dimension . Since M v, ֒ → M v, ⊗ Q ℓ ,we have that m ≤ .On the other hand, the de Rham cohomology of B induces a (decreasing) Hodge filtration Fil • on H ( B v, /W (¯ F ℓ )) ⊗ ¯ Q ℓ with dim Fil = 5 and dim Fil = 1 . Hence Fil ∩ H ( B v, /W (¯ F ℓ ))[1 /ℓ ] ϕ =1 isa Q ℓ -vector space of dimension at most . By Grothendieck–Messing theory, both O F and Λ v lie in Fil andhence m ′ + 2 ≤ . If m ′ = 3 , then Span {O F , Λ v } = Fil . By Mazur’s weak admissibility theorem, since both O F and Λ v lie in H ( B v, /W (¯ F ℓ ))[1 /ℓ ] ϕ =1 , then Span {O F , Λ v } only has trivial filtration. This contradictsthat = Fil ⊂ Fil and we conclude that m ′ ≤ . (cid:3) Theorem 4.3.3.
Let M ( N, n, ǫ ) denote the number of primes p ∈ [ N / , N ] such that { T p ([ A v,n ]) ∩ ( S r ∈ I T ( r )) } ≥ ǫp . Then M ( N, ⌈ e v log log N ⌉ , ǫ ) = o ( N/ (log N )) .Proof. The number of primes in the interval [ N / , N/ log( N )] is o ( N/ log( N ) , so we will restrict ourselvesto primes p ∈ [ N/ log( N ) , N ] . For each prime p , each [ B v,n ] ∈ T p ([ A v,n ]) ∩ ( ∪ r ∈ I T ( r )) induces a specialendomorphism of A v,n whose norm is pr Nm a /D . For all p ∈ [ N/ log N, N ] , the quantity pr Nm a /D = O ( N ) . Notice that distinct [ B v,n ] ∈ T p ([ A v,n ]) ∩ ( ∪ r ∈ I T ( r )) induce distinct special endomorphisms of A v,n .Therefore, A v,n has at least M ( N, n, ǫ ) ǫN/ log N special endomorphisms with norm bounded by N .Applying the crudest bounds that Theorem 4.1.1, Lemma 4.2.1 and Lemma 4.3.2 yield, the number ofspecial endomorphisms A v,n has with norm bounded by N is O (cid:18) N ℓ n/ev + N / (cid:19) . Therefore M ( N, n, ǫ ) = O (cid:18) N log Nℓ n/ev + N / log N (cid:19) ( ǫ gets absorbed in the O () ). Substituting n = ⌈ e v log log N ⌉ yields M ( N, n, ǫ ) = o ( N/ (log N )) as required. (cid:3) he following theorem shows that one can choose a sequence of p such that the largest v -adic intersectionmultiplicity of a point in T p ([ A ]) with S r ∈ I T ( r ) is O (log p ) . Theorem 4.3.4.
Set n = ⌈ e v log N log ℓ ⌉ . Then the number of primes p ∈ [ N / , N ] for which there exists [ B ] ∈ T p ([ A ]) and r ∈ I with [ B v,n ] ∈ T ( r ) is o ( N/ log N ) .Proof. Let p ∈ [ N / , N ] such that there exists [ B ] ∈ T p ([ A ]) and r ∈ I as in the statement. Then A v,n has a special endomorphism, say s p , of norm pr Nm a /D . Clearly, s p = s p ′ where p ′ = p also satisfies theconditions in the statement. Therefore, each such p induces a distinct special endomorphism of A v,n havingnorm O ( N ) and it suffices to bound the number of special endomorphisms of norm ≤ N .Let Λ v denote the module of special endomorphsims of A O Kv [ ℓ ∞ ] . If Λ v has Z ℓ -rank ≤ , then Corol-lary 4.2.3 yields the desired result. Therefore, we assume that the rank is at least 2. By Lemma 4.3.2, therank of Λ v is at most two, so we assume that the rank equals two. Let n = n ′ + ke v , where n ′ − n < e v with n as in Theorem 4.1.1. We have M v,n ⊂ ℓ k ∩ M v,n + P ′ n , where P ′ n is a rank-two sublattice of M v,n .There is no unique choice of P ′ n , so we choose P n to be any one with minimal root-discriminant d ( P n ) . As B has no special endomorphisms generically, it follows that d ( P n ) → ∞ .We first deal with the case when d ( P n ) ≥ log( N ) . Since µ ( M v,n ) µ ( M v,n ) is of the same order ofmagnitude as d ( P n ) , then { v ∈ M v,n : Q ( v ) ≤ N } = O ( N ℓ n/ev + N / ℓ n/ev + Nd ( P n ) + N / ) by Lemma 4.2.1. This quantity is o ( N/ log N ) and so the result follows in this case.Suppose now that d ( P n ) ≤ log( N ) . By Lemma 4.3.5 below, if v ∈ M v,n has norm bounded by N , itfollows that v ∈ P n . As d ( P n ) → ∞ , the same argument used to finish the proof of Proposition 3.1.2 appliesto prove that the proportion of primes p such that there exists a B ∈ T p ( A ) modulo v n goes to zero. (cid:3) Lemma 4.3.5.
Notation as above. Suppose that d ( P n ) ≤ log( N ) . If Q ( v ) ≤ N , then v ∈ P n .Proof. For brevity set d = d ( P n ) . Fix a constant n as in Theorem 4.1.1. Let P ′ n denote the intersection with M v,n of the orthogonal complement of P n ⊗ Q in M v,n ⊗ Q . We have that Cd M v,n ⊂ P n + P ′ n , where C is a positive constant only depending on the discriminant of Q on M v,n . Indeed, let P ∨ n (resp. P ′∨ n ) denotethe dual lattice of P n (resp. P ′ n ) in P n ⊗ Q (resp. P ′ n ⊗ Q ) with respect to the restriction of the quadraticform Q to P n ⊗ Q (resp. P ′ n ⊗ Q ). Then d P ∨ n ⊂ P n . On the other hand, there is a constant C ′ dependingonly on disc Q such that P ′ n is spanned by two vectors x, y such that Q ( x ) , Q ( y ) ≤ C ′ d (since they are givenby solving linear equations with coefficients bounded by O ( d ) ). Therefore, there exists a constant C suchthat Cd P ′∨ n ⊂ P ′ n . Since M v,n ⊂ ( P n + P ′ n ) ∨ = P ∨ n + P ′∨ n , we have Cd M v,n ⊂ P n + P ′ n .Let v ∈ M v,n satisfy Q ( v ) ≤ N . Suppose that v = u + ℓ ⌊ ( n − n ) /e v ⌋ w with u ∈ P n and w ∈ M v,n , and let Cd w = w + w ′ , where w ∈ P n and w ′ ∈ P ′ n . Then, Cd v = ( Cd u + ℓ ⌊ ( n − n ) /e v ⌋ w ) + ℓ ⌊ ( n − n ) /e v ⌋ w ′ , and thus C d N ≥ Q ( Cd v ) ≥ ℓ ⌊ ( n − n ) /e v ⌋ Q ( w ′ ) . As ℓ ⌊ ( n − n ) /e v ⌋ > C d N , it follows w ′ = 0 as required. (cid:3) Proof of the main theorem
The goal of this section is to deduce our main theorem from the results in §§3-4 which provide upperbounds of the local intersection numbers. Recall that A is an abelian surface over K with an a -polarizationand O F ⊆ End( A ) . As in 2.1, we will assume the existence of a semi-abelian scheme A over O K withsemistable reduction everywhere, whose generic fiber is A . Recall that p denotes a prime which is totallysplit in the narrow Hilbert class field of F . To prepare for our proof, we first use Borcherds’ theory to choosea suitable Hirzebruch–Zagier divisor in the Hilbert modular surface and then compute the asymptotic ofFaltings heights on the Hecke orbits. Note that k differs from n/e v by a quantity bounded independent of n , so /ℓ k = O (1 /ℓ nev ) . .1. Borcherds’ theory and the Faltings height.
We devote this subsection to applying arithmeticBorcherds’ theory to choose a rational section of certain tensor powers of the Hodge line bundle. We theninterpret the Faltings height of an abelian surface as a certain Arakelov intersection number.We use A univ-sa to denote the universal family of semi-abelian schemes over H tor (with suitable levelstructure). In [BBGK07, §6], the authors explain a way to define the arithmetic intersection independentlyof the choice of a level structure. We recall their definition in 5.1.3. Let e : H tor → A univ-sa be the identitysection and let ω = det( e ∗ Ω A univ-sa / H tor ) over H tor be the Hodge line bundle. We endow ω with a Hermitianmetric || · || F (only on H ( C ) ) as in [Fal86, sec. 3] and denote by ω the Hermitian line bundle with logsingularity along the boundary. By definition, we have h F ( A ) = ht ω ([ A ]) , where h F denotes the stable Faltings height and ht is the height function of subvarieties of an arithmeticvariety with respect to certain arithmetic cycles (see, for example, [BBGK07, §1.5, eqn. (1.17)] and wenormalize h F and ht ω to be independent on the choice of K ; more specifically, || ℓ || v = ℓ − [ Kv : Q ℓ ][ K : Q ] ).It is well known that the space of global sections of the line bundle ω ⊗ k over H tor C (resp. H tor Q ) is thespace of Hilbert modular forms of parallel weight k over C (resp. Q ); see, for example, [FC90, Chp. V.1]and [Cha90, sec. 4]. Up to a constant, the Hermitian metric || · || F on ω ⊗ k is defined by || f ( z ) || Pet = | f ( z , z )( ℑ z ) k/ ( ℑ z ) k/ | , where f is a Hilbert modular form of parallel weight k and z = ( z , z ) ∈ H :indeed, this follows from the G ( R ) -invariance of both metrics. Lemma 5.1.1.
There exist a positive integer k and a meromorphic Hilbert modular form Ψ over Q of parallelweight k such that the divisor Div(Ψ) defined by Ψ on H Q is given by P r ∈ I c r T ( r ) , where c r ∈ Z and I is afinite subset of J = { qD | q is a rational prime inert in F } . In particular,
Div(Ψ) is a weighted sum of compact Shimura curves.
Borcherds’ theory lifts weakly holomorphic modular forms on modular curves to meromorphic Hilbertmodular forms on H C . By lifting, it means that the divisor defined the resulting Hilbert modular form isdetermined by the principal part of the Fourier expansions of the given modular form at cusps of the modularcurve. Borcherds and Bruinier showed that the existence such lift of certain modular form can be verifiedby certain explicit conditions on the Fourier coefficients of its principal part. The Fourier expansions ofBorcherds lifts (sometimes also called Borcherds products) have also been studied by many people, whichleads to an arithmetic theory of these lifts. One may see [BBGK07, §4] for a summary of relevant resultswhen the discriminant of F is a prime. Proof of Lemma 5.1.1.
By [Bru16, Thm. 1.1], in which we take the infinite admissible set to be J , thereexists a Borcherds product Ψ ′ of non-zero weight k whose divisor is supported on ∪ r ∈ J T ( r ) . In other words, Ψ ′ is a Hilbert modular form of parallel weight k over C . We may assume k > , since otherwise we justtake Ψ ′− . By [Bru16, Prop. 3.1], the weakly holomorphic modular f whose Borcherds lift is Ψ has integralFourier coefficients. [Hör14, 3.2.14] shows that, after multiplying by a suitable scalar, the Borcherds lift of amodular form with Fourier coefficients in Q is defined over Q . In particular, if we take Ψ to be Ψ ′ multipliedby a suitable scalar, then Ψ is a rational section of ω ⊗ k over H tor Q . For r ∈ J , the divisor T ( r ) is compact byCorollary 2.1.3. (cid:3) . We view Ψ as a rational section of ω ⊗ k over H tor O K ′ , where K ′ is a large enough number field such that Ψ is defined. Hence Div(Ψ) = P r c r T ( r ) + P p E p , where the second sum is over finitely many p and E p is afinite (weighted) sum of irreducible components of H tor F p . We also use || · || F to denote the metric on ω ⊗ k given by the tensor product of the Hermitian metric || · || F on ω . [Bru16, Thm. 1.1] is a generalization of [BBGK07, Lem. 4.11]. The proof of this lemma, which only deals with the casewhen D is a prime, contains the main idea of the proof for the general case. .1.3 . Given an arithmetic divisor D and a horizontal -cycle Z intersecting properly on H tor O K ′ , one defines the(arithmetic) intersection number as follows: (see, for example, [BGKK07, Thm. 1.33] for regular schemesand [Yan10, eqn. (2.1)] for regular Deligne–Mumford stacks.) D . Z = X v X x ∈ ( Z∩D )( F v ) log( O Z∩D ,x ) x ) = X v X x ∈ ( Z∩D )( F v ) Length( ˜ O Z∩D ,x ) log( k ( x )) x ) , where v ranges over the through finite places of K ′ , the intersection Z ∩D = Z × H tor D is a Deligne–Mumfordstack of dimension , the ring e O Z∩D ,x is the strictly Henselian local ring of Z ∩ D at x , and k ( x ) is theresidue field of e O Z∩D ,x . By definition, D . Z [ K ′ : Q ] is independent of the choice of K ′ .In [BBGK07, sec. 6.3], they define the arithmetic intersection number on H tor as the arithmetic intersectionnumber of the pull back of arithmetic cycles to H tor ( N ) , the Hilbert modular surface with full level N -structure with N ≥ , divided by the degree of the map H tor ( N ) → H tor . This is the idea behind the aboveformula. Remark . For D and Z as above, let n denote the largest integer such that Z is contained in D modulo v n (here, we consider Z and D as subschemes of the coarse Hilbert modular surface). In our applications,we will only consider the intersection at finitely many places. Therefore, we may pass to a suitable levelstructure étale at these finitely many places so that T ( r ) are regular ([Car86]). Then the length at v referredto in 5.1.3 differs from n by an absolutely bounded factor. As we are only concerned with bounds, we willin the sequel restrict ourselves with controlling the growth of n . Lemma 5.1.5.
Assume that
End( A K ) = O F . Following the notation as in Lemma 5.1.1, there exists aconstant C independent of A such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h F ( A ) − k [ K : Q ] [ A ] . X r ∈ I c r T ( r ) − k [ K : Q ] X σ : K֒ → C A K ) log || Ψ( σ ([ A ])) || Pet (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < C . Proof.
By Lemma 2.1.6, if [ A ] lies on T ( r ) , then End( A K ) is strictly larger than O F . Hence our assumptionimplies that [ A ] does not lie on any T ( r ) . It follows that the -cycle [ A ] intersects P r ∈ I c r T ( r ) properly. Set E ( A ) = K : Q ] [ A ] . ( P p E p ) , where P p E p was defined in 5.1.2. By definition, E ( A ) is bounded by an absoluteconstant independent of A . Moreover, h F ( A ) = ht ω ([ A ]) = 1 k [ K : Q ] [ A ] . X r ∈ I c r T ( r ) − k [ K : Q ] X σ : K֒ → C A K ) log || Ψ( σ ([ A ])) || F + 1 k E ( A ) , (see, for example, [Yan10, eqn. (2.3)]). The lemma then follows from the fact that || · || F and || · || Pet differby an absolute constant independent of A . (cid:3) We end this subsection with a formula for the average Faltings height of abelian surfaces correspondingto points in T p [ A ] when A has good reduction at all the primes of K above p . The idea of proof builts on useAutissier’s idea in [Aut05]. From now on, we say that A has good reduction at p (resp. ordinary reductionat p ) if A has good reduction (resp. ordinary reduction) at all the primes of K above p . Proposition 5.1.6.
Let p be a prime as in 2.2.1. If A has good reduction at p , then X [ B ] ∈ T p [ A ] h F ( B ) = ( p + 1) h F ( A ) + p −
12 log p. Proof.
The proof consists two parts. We first show that P [ B ] ∈ T p [ A ] h F ( B ) − ( p + 1) h F ( A ) is independent of A . Then we compute this quantity in a particular case.(1) Let H ( p ) denote the Hilbert modular surface over Z with Γ ( p ) level structure. The stack H ( p ) parameterizes degree p isogenies φ : A → A between abelian varieties with O F -multiplication suchthat ker( φ ) ⊂ A [ p ] (see for example [Pap95, sec. 2.2]). Let π i : H ( p ) Z ( p ) → H Z ( p ) for i = 1 , be theforgetful map that sends φ to A i . We claim that each π i is finite flat. The higher tor group vanishes since we work with a Cartier divisor which intersects the -cycle properly. e first show that π i is quasi-finite. Let v be any finite place of K over p . The group scheme A [ p ∞ ] of p -power torsions of A is a p -divisible group of height 2, whose mod v reduction has dimension 1.There are two cases: the mod- v reduction of A [ p ∞ ] is either ordinary, or supersingular. If ordinary,[FC90, §VII.4] shows that there are only finitely many degree p subgroups of the mod- v reduction of A [ p ] . Therefore, by the modular interpretation of H ( p ) , the map π is quasi-finite. Now we assumethat the mod- v reduction of A [ p ∞ ] is supersingular. Since the number of degree p subgroups of thereduction of A [ p ] only depends on the isomorphism class of the p -divisible group, we may assumethat A has supersingular reduction at v . Since p is split in F , the supersingular locus of H F v is -dimensional and hence there are only finitely many mod- v points on H corresponding to abeliansurfaces isogenous to A mod v . In particular, π is quasi-finite. The same argument applies to π when we study the kernel of the Rosati involution of φ .By [Pap95, 2.1.3, Cor. 2.2.3], the stack H ( p ) is Cohen–Macaulay and H is regular. Since all thefibers of π i are -dimensional, then by [EGAIV, II.6.1.5], each π i is flat. On the other hand, the π i are proper by [Pap95, the discussion after Def. 2.2.1]. Therefore, each π i is finite flat.By the argument in [Aut05, Theorem 5.1], one observes that the independence of the quantity P [ B ] ∈ T p [ A ] h F ( B ) − ( p + 1) h F ( A ) on A is a formal consequence of the finite-flatness of π i , thenormality of H , and the irreducibility of H F v for every v above p .(2) We now compute P [ B ] ∈ T p [ A ] h F ( B ) − ( p + 1) h F ( A ) when A has ordinary reduction at p . Such an A always exists: indeed, for a CM field K containing F such that K is Galois over Q of degree and p splits completely in K , there exist abelian surfaces with CM by K that correspond to pointson H and these abelian surfaces are ordinary at p .Now assume that A has good ordinary reduction at p and we prove the result for such A . Wefirst enlarge K so that all [ B ] ∈ T p [ A ] are defined over K . Since A [ p ∞ ] over O K v is -dimensional,it only contains a unique degree p subgroup which is multiplicative (equivalently, in the ordinarycase, not étale) for any v above p . Then by Lemma 2.2.2, there exists only one element in T p [ A ] that corresponds to an isogeny with multiplicative kernel. Now we apply [Fal86, Lem. 5]. By thestandard calculation on Ω of finite flat groups of degree p , at each v there are p out of p + 1 elementsin T p [ A ] such that the term log( e ∗ (Ω φ/ O Kv )) in Faltings’ formula is , and one element suchthat log( e ∗ (Ω φ/ O Kv )) = − [ K v : Q p ] log p . We obtain the desired formula by summing up all thelocal contributions. (cid:3) Proof of Theorem 1. . We first sketch the proof of Theorem 1. First, we use Theorem 1 to choose a good Hirzebruch–Zagier divisor P r ∈ I c r T ( r ) = Div(Ψ) . By Proposition 5.1.6, we have P [ B ] ∈ T p ([ A ]) h F ( B ) is O ( p log p ) . Onthe other hand, the local results in §§4-3 show that each local term in Lemma 5.1.5 is o ( p log p ) . Bylocal term, we mean either − P [ B ] ∈ T p [ A ] log || Ψ( σ ([ B ])) || Pet for all σ : K ֒ → C or the v -adic intersectionnumber ( P [ B ] ∈ T p [ A ] [ B ] , P r ∈ I c r T ( r )) v for finite places v . This implies that T p ([ A ]) intersects P r ∈ I c r T ( r ) at infinitely many places as p → ∞ and then Theorem 1 follows from Corollary 2.1.7.5.2.2 . If O F ( End( A K ) , then by the classification of the endomorphism ring of absolutely simple abeliansurfaces over a characteristic zero field, End( A K ) ⊗ Q is either an indefinite quaternion algebra over Q ora degree CM field. In the first case, A F v is not simple if the quaternion algebra splits at char F v . In thesecond case, there exists a positive density set of primes ℓ so that A F v is supersingular for all v | ℓ and hencenot geometrically simple. Therefore, to prove Theorem 1, we assume that End( A Q ) = O F from now on andhence for any [ B ] ∈ T p ([ A ]) , we also have End( B K ) = O F . Therefore, all T p ([ A ]) intersect Hirzebruch–Zagierdivisors properly. Proof of Theorem 1.
We now show that there are infinitely many primes v of K such that A F v is notsimple. Assume, for the sake of contradiction, that there is a finite set of places Σ of K such that A has geometrically simple, or bad reduction modulo v for v / ∈ Σ . Corollary 2.1.3 and Lemma 5.1.1 givea meromorphic Hilbert modular form Ψ such that Div(Ψ) is a compact special divisor P r ∈ I c r T ( r ) with One may also choose any CM abelian surface on H and apply the formula for the Faltings height in [Moc17] to computethis difference. | r for all r ∈ I . The intersection ( T p ([ A ]) , P r ∈ I c r T ( r )) has a nonzero v -adic term only when v ∈ Σ byCorollary 2.1.7. Throughout the proof, p will denote a prime which is totally split in the narrow Hilbertclass field of F and v ∤ p for all v ∈ Σ . We now explain how to choose an increasing sequence of primes p such that we can bound the local terms as described in 5.2.1 by ǫp log p for arbitrary ǫ > .By Theorems 4.3.3 and 4.3.4 and Remark 5.1.4, outside a density-zero set of primes p , the v -adic inter-section (cid:16) X [ B ] ∈ T p [ A ] [ B ] , X r ∈ I c r T ( r ) (cid:17) v ≤ C (cid:18) X r ∈ I | c r | · (cid:16) ( p + 1)(3 e v log(2 log p )) + ǫp (2 e v log p + 1)) (cid:17)(cid:19) , where C is the absolute constant mentioned in Remark 5.1.4. Notice that p ≥ log N .Let C be the density of primes splitting completely in the narrow Hilbert class field of F . By taking ǫ = ǫ, ǫ = C K : Q ] in Theorem 3.2.1, we have that, for N ≫ and for p ∈ [ N / , N ] in a set of density atleast C/ , for all σ : K ֒ → C , the archimedean term − X [ B ] ∈ T p [ A ] log || Ψ( σ ([ B ])) || Pet < ǫp log p. We have shown that for N ≫ , there exists a positive density set of primes p ∈ [ N / , N ] such that all thelocal terms are o ( p log p ) . On the other hand, by Proposition 5.1.6, X [ B ] ∈ T p ([ A ]) h F ( B ) has order of magnitude p log p . We then obtain the desired contradiction by applying Lemma 5.1.5 to all [ B ] ∈ T p ([ A ]) . (cid:3) References [Aut05] Pascal Autissier,
Hauteur moyenne de variétés abéliennes isogènes , Manuscripta Math. (2005), no. 1, 85–92(French, with English and French summaries).[Bru16] Jan Hendrik Bruinier,
Borcherds products with prescribed divisor (2016). Available on arXiv: 1607.08713.[BBGK07] Jan H. Bruinier, José I. Burgos Gil, and Ulf Kühn,
Borcherds products and arithmetic intersection theory on Hilbertmodular surfaces , Duke Math. J. (2007), no. 1, 1–88.[BGKK07] J. I. Burgos Gil, J. Kramer, and U. Kühn,
Cohomological arithmetic Chow rings , J. Inst. Math. Jussieu (2007),no. 1, 1–172.[Car86] Henri Carayol, Sur la mauvaise réduction des courbes de Shimura , Compositio Math. (1986), no. 2, 151–230(French).[Cha90] C.-L. Chai, Arithmetic minimal compactification of the Hilbert-Blumenthal moduli spaces , Ann. of Math. (2) (1990), no. 3, 541–554.[Cha14] Francois Charles,
Exceptional isogenies between reductions of pairs of elliptic curves (2014). Available atarXiv:1411.2914.[Cha97] Nick Chavdarov,
The generic irreducibility of the numerator of the zeta function in a family of curves with largemonodromy , Duke Math. J. (1997), no. 1, 151–180.[COU01] Laurent Clozel, Hee Oh, and Emmanuel Ullmo, Hecke operators and equidistribution of Hecke points , Invent. Math. (2001), no. 2, 327–351.[Del79] Pierre Deligne,
Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques ,Automorphic forms, representations and L -functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis,Ore., 1977), Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 247–289 (French).[Elk87] Noam D. Elkies, The existence of infinitely many supersingular primes for every elliptic curve over Q , Invent.Math. (1987), no. 3, 561–567, DOI 10.1007/BF01388985.[Elk89] , Supersingular primes for elliptic curves over real number fields , Compositio Math. (1989), no. 2, 165–172.[EK95] Alex Eskin and Yonatan R. Katznelson, Singular symmetric matrices , Duke Math. J. (1995), no. 2, 515–547,DOI 10.1215/S0012-7094-95-07913-7.[Fal86] Gerd Faltings, Finiteness theorems for abelian varieties over number fields , Arithmetic geometry (Storrs, Conn.,1984), Springer, New York, 1986, pp. 9–27. Translated from the German original [Invent. Math. (1983), no. 3,349–366; ibid. (1984), no. 2, 381; MR 85g:11026ab] by Edward Shipz.[FC90] Gerd Faltings and Ching-Li Chai, Degeneration of abelian varieties , Ergebnisse der Mathematik und ihrer Grenzge-biete (3) [Results in Mathematics and Related Areas (3)], vol. 22, Springer-Verlag, Berlin, 1990. With an appendixby David Mumford.[FKRS12] Francesc Fité, Kiran S. Kedlaya, Víctor Rotger, and Andrew V. Sutherland,
Sato-Tate distributions and Galoisendomorphism modules in genus 2 , Compos. Math. (2012), no. 5, 1390–1442, DOI 10.1112/S0010437X12000279.[Gor02] Eyal Z. Goren,
Lectures on Hilbert modular varieties and modular forms , CRM Monograph Series, vol. 14, AmericanMathematical Society, Providence, RI, 2002. With the assistance of Marc-Hubert Nicole. EGAIV] A. Grothendieck,
Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas.I , Inst. Hautes Études Sci. Publ. Math. (1964), 259 (French).[HZ76] F. Hirzebruch and D. Zagier, Intersection numbers of curves on Hilbert modular surfaces and modular forms ofNebentypus , Invent. Math. (1976), 57–113.[Hör14] Fritz Hörmann, The geometric and arithmetic volume of Shimura varieties of orthogonal type , CRM MonographSeries, vol. 35, American Mathematical Society, Providence, RI, 2014.[KS99] Nicholas M. Katz and Peter Sarnak,
Random matrices, Frobenius eigenvalues, and monodromy , American Mathe-matical Society Colloquium Publications, vol. 45, American Mathematical Society, Providence, RI, 1999.[Ked15] Kiran S. Kedlaya,
Sato-Tate groups of genus 2 curves , Advances on superelliptic curves and their applications,NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., vol. 41, IOS, Amsterdam, 2015, pp. 117–136.[Kis10] Mark Kisin,
Integral models for Shimura varieties of abelian type , J. Amer. Math. Soc. (2010), no. 4, 967–1012.[KR99] Stephen S. Kudla and Michael Rapoport, Arithmetic Hirzebruch-Zagier cycles , J. Reine Angew. Math. (1999),155–244.[Mes72] William Messing,
The crystals associated to Barsotti-Tate groups: with applications to abelian schemes , LectureNotes in Mathematics, Vol. 264, Springer-Verlag, Berlin-New York, 1972.[MP08] V. Kumar Murty and Vijay M. Patankar,
Splitting of abelian varieties , Int. Math. Res. Not. IMRN (2008), Art.ID rnn033, 27, DOI 10.1093/imrn/rnn033.[Moc17] Lucia Mocz, A new Northcott property for Faltings height (2017). preprint.[Ogu82] Arthur Ogus,
Hodge cycles and crystalline cohomology , Hodge cycles, motives, and Shimura varieties, Lecture Notesin Mathematics, vol. 900, Springer-Verlag, Berlin-New York, 1982.[Pap95] Georgios Pappas,
Arithmetic models for Hilbert modular varieties , Compositio Math. (1995), no. 1, 43–76.[Rap78] M. Rapoport, Compactifications de l’espace de modules de Hilbert-Blumenthal , Compositio Math. (1978), no. 3,255–335 (French).[Saw16] William F. Sawin, Ordinary primes for Abelian surfaces , C. R. Math. Acad. Sci. Paris (2016), no. 6, 566–568,DOI 10.1016/j.crma.2016.01.025 (English, with English and French summaries). MR3494322[Ser12] Jean-Pierre Serre,
Lectures on N X ( p ) , Chapman & Hall/CRC Research Notes in Mathematics, vol. 11, CRC Press,Boca Raton, FL, 2012.[TZ16] Jesse Thorner and Asif Zaman, A Chebotarev variant of the Brun-Titchmarsh theorem and bounds for the Lang-Trotter conjectures (2016). Available on arXiv:1606.09238.[vdG88] Gerard van der Geer,
Hilbert modular surfaces , Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results inMathematics and Related Areas (3)], vol. 16, Springer-Verlag, Berlin, 1988. MR930101[Sch68] Wolfgang M. Schmidt,
Asymptotic formulae for point lattices of bounded determinant and subspaces of boundedheight , Duke Math. J. (1968), 327–339.[Yan10] Tonghai Yang, An arithmetic intersection formula on Hilbert modular surfaces , Amer. J. Math. (2010), no. 5,1275–1309.[Zyw14] David Zywina,
The splitting of reductions of an abelian variety , Int. Math. Res. Not. IMRN (2014), 5042–5083.MR3264675(2014), 5042–5083.MR3264675