A mod p variant of the André–Oort conjecture

We state and prove a variant of the André–Oort conjecture for the product of 2 modular curves in positive characteristic, assuming GRH for quadratic fields.


Theorem 1.2 Assume the generalised Riemann hypothesis for quadratic fields. Let p be a prime number. Let be a set of finite closed subsets s of A 2 F p that are reductions of CMpoints in A 2 Z . Let Z be the Zariski closure of the union of all s in . Then every irreducible component of dimension 1 of Z is special: a fibre of one of the 2 projections, or an irreducible component of the image in A 2
F p of some Y 0 (n) F p with n ∈ Z ≥1 . Remark 1. 3 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 [4,Sect. 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 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 ,F p 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 ,F p 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 ,F p 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 ,F p = ( j K , f ,F p ) 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 ,F p (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
Let s be a CM-point in A 2 Q as in Definition 1.1. Then s(Q) is a G Q -orbit. Let (x 1 , x 2 ) be in s(Q). 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 G K 1 K 2 acts through Pic(O K 1 , f 1 ) × Pic(O K 2 , f 2 ), and s(Q) decomposes into at most 4 orbits under G K 1 K 2 .
Let p be a prime. Let s be a finite closed subset of A 2 F p that is the reduction at p of a CM-point in A 2 Z (see Definition 1.1). Then s(F p ) is a finite subset of F p × F p that is stable under G F p := 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 Sect. 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 a prime number, T denotes the correspondence on the j-line, over any field not of characteristic , sending an elliptic curve E over an algebraically closed field k to the sum of its + 1 quotients by the subgroups of E(k) of order . Similarly, T × T 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 . 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 by its subset of s's whose images under both projections are ordinary.
At this point we combine the arguments of [5] with reduction modulo p. Let d 1 and d 2 be the degrees of the projections from Z to A 1 F p . For s in 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 such that is split in both O 1,s and O 2,s , and #s(F p ) > 2d 1 d 2 ( + 1) 2 , and > log(#s(F p )). This claim follows, as in the proof of [5, Lemma 7.1], from the (conditional) effective Chebotarev theorem of Lagarias and Odlyzko [9] as stated in Theorem 4 of [12], and Siegel's theorem on class numbers of imaginary quadratic fields [14] and [10, Chap. XVI]. Now let s, (x 1 , x 2 ) and 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 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 Sect. 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.
Now the degrees of the projections from (T × T )Z to A 1 F p are ( + 1) 2 d 1 and ( + 1) 2 d 2 , so the intersection number (in (P 1 × P 1 ) F p ) of Z and (T × T )Z is 2d 1 d 2 ( + 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 5.1 Let C be an irreducible reduced closed curve in A 2
F p , not a fibre of one of the 2 projections, such that there are infinitely many prime numbers for which (T × T )(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 : 2 ), with projections G ,1 and G ,2 . Because of the Weil pairing, G acts on det(V ,1 ) and det(V ,2 ) by the cyclotomic character χ : G → F × l = Aut(μ (K sep )). For all but finitely many , G ,1 contains SL(V ,1 ) and similarly for E 2 (this follows, as in [2], 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 [6, Theorem 3] and [7, Corollary 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 , G ,1 is the subgroup of elements in GL(V ,1 ) whose determinant is a power of q, and similarly for G ,2 . Let L be the set of prime numbers = 2 for which G ,1 and G ,2 are as in the previous sentence, and such that (T × T )(C) is reducible. Then L is infinite.
Let be in L. Let N ,1 := ker(G → G ,2 ) and N ,2 := ker(G → G ,1 ). Then l is a character, and γ is an isomorphism from V ,1 to the twist of V ,2 by ε .
Let U ⊂ C be the open subscheme where C is regular and where E 1 and E 2 have good reduction. Then for all in L, and all closed x in U , ε 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. Then both E 1 and E 2 have semistable reduction over C by [3,Corollary 5.18]. The Galois criterion for semi-stability in [13, Example IX, Proposition 3.5] tells us that all ε become unramified on C . This shows that also for p = 2 there are only finitely many distinct ε . The conclusion is that, for general p, there are only finitely many distinct ε , 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 are trivial. Now Zarhin's result [17, Corollary 2.7] tells us that there is a non-zero morphism α : E 1 → E 2 .

Remark 5.2
Up to sign, there is a unique isogeny α : E 1 → E 2 of minimal degree n. Then C is an irreducible component of the image of Y 0 (n) F p . We write n = p k m with m prime to p. Then C is the image of the image of Y 0 (m) F p 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 5.3 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: and therefore 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 F p 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 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 with pr 1 (s) disjoint from B 1 and pr 2 (s) disjoint from B 2 are contained in Z . Let be the set of these s. The s in − 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 is dense in Z . We replace Z by Z , and by . 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 by its subset of s's whose image under both projections is ordinary.
Theorem 4.1 gives us an infinite set L of primes such that Z ∩(T ×T )Z is of dimension 1. Let (Z i ) i∈I be the set of irreducible components of Z . Then for each in L there are i and j in I such that Z i is in (T × T )Z j . If moreover > 12d 1 then (T × T )Z j is reducible, because if not, then (T × T )Z j equals Z i (as closed subsets of A 2 F p ), but for any ordinary (x, y) in Z j (F p ), T (y) consists of at least ( + 1)/12 > d 1 distinct points.
There is a j 0 ∈ I such that for infinitely many ∈ L, (T ×T )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 F p of an irreducible component of Y 0 (n) F p . 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) F p , 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 that meets T (n) F p is contained in T (n) F p . 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, and ϕ :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) F p ∩ Z is finite, the set of s in that do not meet T (n) F p is dense in Z and our proof is finished by induction on the number of irreducible components of Z .

Remark 6.1
We think that Theorem 1.2 remains true if E ⊂ Q is a finite extension of Q and we work with A 2 F p 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 such that Z consists of one irreducible component of Y 0 ( p) F p .
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