A mod p variant of the Andr\'e-Oort conjecture

We state and prove a variant of the Andr\'e-Oort conjecture for the product of 2 modular curves in positive characteristic, assuming GRH for quadratic fields.

1 Introduction 1.1 Definition For a point x in a scheme X we let κ(x) = O X,x /m x be its residue field, and we denote ι x : Spec(κ(x)) → X the induced κ(x)-point of X. So we may view ι x as an element of X(κ(x)), the set of κ(x)-valued points of X. For X = A 2 , we have X(κ(x)) = κ(x) 2 .
By CM-point in A 2 Q we mean a closed point s of the affine plane over Q, such that both coordinates of ι s ∈ κ(s) 2 are j-invariants of CM elliptic curves.
By CM-point in A 2 Z we mean the closure in A 2 Z of a CM-point in A 2 Q . We view such a CM-point {s} as a closed subset, or as a reduced closed subscheme. For any prime number p we then denote by {s} Fp the reduced fibre over p and call it the reduction of s at p. Fp of some Y 0 (n) Fp with n ∈ Z ≥1 .

Remark
If K 1 , . . . , K n are quadratic subfields of Q, then GRH holds for their compositum K if and only if it holds for each quadratic subfield of K (the zeta function of K is the product of the Riemann zeta-function and the L-functions of the quadratic subfields of K).

Some facts on CM elliptic curves
We will need some results on CM elliptic curves and their reduction mod p. For more detail see [3, §2], and references therein. For E over Q an elliptic curve with CM, End(E) is an order in an imaginary quadratic field K, hence isomorphic to O K,f = Z + f O K , with O K the ring of integers in K, and f ∈ Z ≥1 , unique, called the conductor.
For K ⊂ Q imaginary quadratic and f ≥ 1, we let S K,f be the set of isomorphism classes of (E, α), where E is an elliptictic curve over Q and α : O K,f → End(E) is an isomorphism, such that the action of End(E) on the tangent space of E at 0 induces the given embedding K → Q. The group Pic(O K,f ) acts on S K,f , making it a torsor. This action commutes with the action of G K := Gal(Q/K), giving a group morphism G K → Pic(O K,f ) through which G K acts on S K,f . This map is surjective, unramified outside f , and the Frobenius element at a maximal ideal m For p a prime number, and f ′ the prime to p part of f , the map S K,f → S K,f ′ is the quotient by the inertia subgroup at any of the maximal ideals m of O K containing p (to show this, use the adelic description of ramification in class field theory).
Elliptic curves with CM over Q extend uniquely over Z (the integral closure of Z in Q), and their endomorphisms as well.
For K and f as above we define j K,f to be the image of j(E) : Spec(Z) → A 1 Z , where E is an elliptic curve over Z with End(E) isomorphic to O K,f ; this does not depend on the choice of E.
Then j K,f is an irreducible closed subset of A 1 Z . We equip it with its reduced induced scheme structure. Then it is finite over Z of degree #Pic(O K,f ), and in fact j K,f (Z) is in bijection with S K,f and hence is a Pic(O K,f )-torsor (here we use that K has a given embedding into Q). For p prime, we let j K,f,Fp be the fibre of j K,f over F p .
Let p be a prime number, and K and f as above. If p is not split in O K then j K,f,Fp consists of supersingular points, and j K,f can be highly singular above p (by lack of supersingular targets). If p is split in O K then j K,f,Fp consists of ordinary points, and the corresponding elliptic curves over F p have endomorphism ring isomorphic to O K,f ′ , where f ′ is the prime to p part of f , and then j K,f ′ ,Fp = (j K,f,Fp ) red , and for each morphism of rings Z → F p the map j K,f ′ (Z) → j K,f ′ (F p ) is a bijection and it makes j K,f ′ ,Fp (F p ) into a Pic(O K,f ′ )-torsor. Note that every ordinary x in F p belongs to exactly one j K,f ′ (F p ).

Some facts about pairs of CM elliptic curves
. Then x 1 is in j K 1 ,f 1 (Q) for a unique imaginary quadratic subfield K 1 of Q, and similarly for x 2 , and Let p be a prime. Let s be a finite closed subset of A 2 Fp that is the reduction at p of a CM-point in A 2 Z (see Def. 1.1). Then s(F p ) is a finite subset of F p × F p that is stable under G Fp := Gal(F p /F p ). For each of the 2 projections, the image of s(F p ) consists entirely of ordinary points or entirely of supersingular points (this follows from the facts recalled in §2). If for all (x 1 , x 2 ) in s(F p ) both x 1 and x 2 are ordinary, then the x 1 form a Pic(O K 1 ,f 1 )-orbit, and the x 2 form a Pic(O K 2 ,f 2 )-orbit, with f 1 and f 2 prime to p.

Images under suitable Hecke correspondences
For ℓ a prime number, T ℓ denotes the correspondence on the j-line, over any field not of characteristic ℓ, sending an elliptic curve E over an algebraically closed field k to the sum of its ℓ+1 quotients by the subgroups of E(k) of order ℓ. Similarly, T ℓ ×T ℓ is the correspondence on the j-line times itself that sends a pair of elliptic curves (E 1 , E 2 ) to the sum of all (E 1 /C 1 , E 2 /C 2 ) with C 1 and C 2 subgroups of order ℓ.

Theorem
Assumptions as in Theorem 1.2, and assume that all irreducible components of Z are of dimension 1, and are not a fibre of any of the 2 projections. There are infinitely many Proof There are only finitely many points (x 1 , x 2 ) in Z(F p ) such that x 1 or x 2 is not ordinary. Therefore we can replace Σ by its subset of s's whose images under both projections are ordinary.
At this point we combine the arguments of [4] with reduction modulo p. Let d 1 and d 2 be the degrees of the projections from Z to A 1 Fp . For s in Σ and (x 1 , x 2 ) in s(F p ), let O 1,s and O 2,s be the endomorphism rings of the elliptic curves E 1 and E 2 over F p corresponding to x 1 and x 2 .
We claim that for all but finitely many s there is a prime number ℓ such that ℓ is split in both O 1,s and O 2,s , and #s(F p ) > 2d 1 d 2 (ℓ + 1) 2 , and ℓ > log(#s(F p )). This claim follows, as in the proof of [4, Lemma 7.1], from the (conditional) effective Chebotarev theorem of Lagarias and Odlyzko [8] as stated in Theorem 4 of [11], and Siegel's theorem on class numbers of imaginary quadratic fields, [14] and [9, Ch. XVI]. Now let s, (x 1 , x 2 ) and ℓ be as in the claim above. Let ϕ : Z → F p be a morphism of rings. Then there are unique embeddings of O 1,s and O 2,s into Z that composed with ϕ give the actions on the tangent spaces at 0 of E 1 and E 2 . Let m be a maximal ideal of index ℓ in O 1,s O 2,s ⊂ Z, and m 1 and m 2 the intersections of m with O 1,s and O 2,s . By the facts recalled at the end of § 2, there are canonicalx 1 andx 2 in Z lifting E 1 and E 2 toẼ 1 andẼ 2 with End(Ẽ 1 ) = End(E 1 ) and End(Ẽ 2 ) = End(E 2 ). Let σ be a Frobenius element in G K 1 K 2 at m. ThenẼ 1 = [m 1 ] −1 [m 1 ]Ẽ 1 shows thatẼ 1 is ℓ-isogenous to [m 1 ]Ẽ 1 which is the conjugate ofẼ 1 by σ −1 , and similarly Fp are (ℓ + 1) 2 d 1 and (ℓ + 1) 2 d 2 , so the intersection number (in (P 1 × P 1 ) Fp ) of Z and (T ℓ × T ℓ )Z is 2d 1 d 2 (ℓ + 1) 2 . But the intersection contains s(F p ), which has more points than this intersection number, so the intersection is not of dimension 0.

Theorem Let C be an irreducible reduced closed curve in A 2
Fp , not a fibre of one of the 2 projections, such that there are infinitely many prime numbers ℓ for which (T ℓ × T ℓ )(C) is reducible. Then there is an n ∈ Z >0 such that C is the image of an irreducible component of Proof Let K denote the function field of C, and let E 1 and E 2 be elliptic curves over K with j-invariants the projections π 1 and π 2 , viewed as functions on C; these E 1 and E 2 are unique up to quadratic twist. We must prove that E 1 is isogeneous to a twist of E 2 . Let K → K sep be a separable closure and let G := Gal(K sep /K). For ℓ = p a prime number, let V ℓ,1 := E 1 (K sep )[ℓ] and V ℓ,2 := E 2 (K sep )[ℓ] and let G ℓ be the image of G in GL(V ℓ,1 ) × GL(V ℓ,2 ), with projections G ℓ,1 and G ℓ,2 . Because of the Weil pairing, G acts on det(V ℓ,1 ) and det(V ℓ,2 ) by the cyclotomic character χ ℓ : G → F × l = Aut(µ ℓ (K sep )). For all but finitely many ℓ, G ℓ,1 contains SL(V ℓ,1 ) and similarly for E 2 (this follows, as in [1], from the fact that for n prime to p the geometric fibres of the modular curve over Z[ζ n , 1/n] parametrising elliptic curves with symplectic basis of the n-torsion are irreducible, [5,Thm. 3] and [6, Cor. 10.9.2]). Let q be the number of elements of the algebraic closure of F p in K. Then, for all but finitely many ℓ, G ℓ,1 is the subgroup of elements in GL(V ℓ,1 ) whose determinant is a power of q, and similarly for G ℓ,2 . Let L be the set of prime numbers ℓ = 2 for which G ℓ,1 and G ℓ,2 are as in the previous sentence, and such that (T ℓ × T ℓ )(C) is reducible. Then L is infinite.
Let U ⊂ C be the open subscheme where C is regular and where E 1 and E 2 have good reduction. Then for all ℓ in L, and all closed x in U, ε ℓ is unramified at x. As U is a smooth curve over a finite field, there are only finitely many characters ε : G → {±1} unramified on U, if p = 2 (this uses Kummer theory). For p = 2, one has to be more careful; we argue as follows. There are infinitely many characters ε : G → {±1} unramified on U, but only finitely many with bounded conductor on the projective smooth curve C with function field K. Let K ′ ⊂ K sep be the extension cut out by V 3,1 × V 3,2 , and let C ′ → C be the corresponding cover. shows that also for p = 2 there are only finitely many distinct ε ℓ . The conclusion is that, for general p, there are only finitely many distinct ε ℓ , and therefore we can assume (by shrinking L to an infinite subset) that they are all equal to some ε. Then we replace E 2 by its twist by ε, and then ε ℓ are trivial. Now Zarhin result [16, Cor. 2.7] tells us that there is a non-zero morphism α : E 1 → E 2 .

Remark
Up to sign, there is a unique isogeny α : E 1 → E 2 of minimal degree. Then C is an irreducible component of the image of Y 0 (n) Fp . We write n = p k m with m prime to p. Then C is the image of the image of Y 0 (m) Fp by the p k -Frobenius map on the first or on the second coordinate, and C is also an irreducible component of the images of all Y 0 (p 2i n) with i ∈ Z ≥0 .

Lemma
Let G be a group, N a normal subgroup of G and Q the quotient. Let α be an automorphism of G inducing the identity on N and on Q, and suppose that G acts trivially by conjugation on the center of N, and that there is no non-trivial morphism from Q to the center of N. Then α is the identity on G.
Proof We write, for all g ∈ G: α(g) = gβ(g), with β a map (of sets!) from G to itself.
As α induces the identity on Q, β takes values in N. As α is the identity on N, we have β(n) = 1 for all n ∈ N. For all g 1 and g 2 in G we have: g 1 g 2 β(g 1 g 2 ) = α(g 1 g 2 ) = α(g 1 )α(g 2 ) = g 1 β(g 1 )g 2 β(g 2 ) , and therefore β(g 1 g 2 ) = g −1 2 β(g 1 )g 2 β(g 2 ) . For g 1 in N, this gives that for all g 2 in G, β(g 1 g 2 ) = β(g 2 ). Hence β factors through β : Q → N: β(g) = β(g). Now, for g 1 in G and g 2 in N, we have hence β takes values in the center of N. Now let g 1 and g 2 be in G. As g −1 2 β(g 1 )g 2 = β(g 1 ), β is a morphism of groups from Q to the center of N and therefore trivial.

Proof of the main theorem
We are now ready to prove Theorem 1.2.
If Z = A 2 Fp or is finite, then Z has no irreducible components of dimension 1. Now assume that Z has dimension 1. We write Z = V ∪ H ∪ F ∪ Z ′ with V the union in Z of fibers of the 1st projection pr 1 , H the union in Z of fibers of pr 2 , and F the set of isolated points in Z, and Z ′ the union of the remaining irreducible components of Z. Let B 1 be the image of V ∪ F under pr 1 , and B 2 the image of H ∪ F under pr 2 .
Let s be in Σ such that pr 1 (s) meets B 1 . Then either pr 1 (s)(F p ) consists of supersingular points, or it consists of ordinary points with the same endomorphism ring as an ordinary point in B 1 . Hence for such a pr 1 (s) there are only finitely many possibilities. Similarly for the pr 2 (s). It follows that the s in Σ with pr 1 (s) disjoint from B 1 and pr 2 (s) disjoint from B 2 are contained in Z ′ . Let Σ ′ be the set of these s. The s in Σ − Σ ′ lie on a finite union of fibres of pr 1 and pr 2 , and the intersection of this union with Z ′ is finite. Therefore the union of the s in Σ ′ is dense in Z ′ . We replace Z by Z ′ , and Σ by Σ ′ . Then all irreducible components of Z are of dimension 1 and are not a fibre of pr 1 or pr 2 . Let d i (i in {1, 2}) be the degree of pr i restricted to Z.
There are only finitely many points (x 1 , x 2 ) in Z(F p ) such that x 1 or x 2 is not ordinary. Therefore we can replace Σ by its subset of s's whose image under both projections is ordinary.
Theorem 4.1 gives us an infinite set L of primes ℓ such that Z ∩ (T ℓ × T ℓ )Z is of dimension 1. Let (Z i ) i∈I be the set of irreducible components of Z. Then for each ℓ in L there are i and j in I such that Z i is in (T ℓ × T ℓ )Z j . If moreover ℓ > 12d 1 then (T ℓ × T ℓ )Z j is reducible, because if not, then (T ℓ × T ℓ )Z j equals Z i (as closed subsets of A 2 Fp ), but for any ordinary (x, y) in Z j (F p ), T ℓ (y) consists of at least (ℓ + 1)/12 > d 1 distinct points.
There is a j 0 ∈ I such that for infinitely many ℓ ∈ L, (T ℓ × T ℓ )Z j 0 is reducible. Theorem 5.1 then tells us that there is an n ≥ 1 such that Z j 0 is the image in A 2 Fp of an irreducible component of Y 0 (n) Fp . We let T (n) be the reduced closed subscheme of A 2 Z whose geometric points correspond to pairs (E 1 , E 2 ) of elliptic curves that admit a morphism ϕ : E 1 → E 2 of degree n. Let J be the set of j ∈ I such that Z j is an irreducible component of T (n) Fp , let Z(n) be their union, and and let Z ′ be the union of the Z i with i ∈ J.
We claim that any s in Σ that meets T (n) Fp is contained in T (n) Fp . So let (j(E 1 ), j(E 2 )) be in s(F p ), and ϕ : E 1 → E 2 of degree n. Let Z → F p be a morphism of rings, andφ :Ẽ 1 →Ẽ 2 be the canonical lift over Z. Thenφ is of degree n, and so are all its conjugates by G Q , and so s(F p ), consisting of all reductions of these conjugates, lies in T (n)(F p ).
As T (n) Fp ∩ Z ′ is finite, the set Σ ′ of s in Σ that do not meet T (n) Fp is dense in Z ′ and our proof is finished by induction on the number of irreducible components of Z.

Remark
We think that Theorem 1.2 remains true if E ⊂ Q is a finite extension of Q and we work with A 2 Fp and consider reductions of G E -orbits of CM-points in A 2 (Z). However, the case E = Q has a special feature: up to fibres of the projections, the Z are invariant under switching the coordinates. This comes from the dihedral nature of the Galois action. As soon as E contains an imaginary quadratic field, there are Σ such that Z consists of one irreducible component of Y 0 (p) Fp .