Special points on the product of two modular curves.

Bas Edixhoven^{∗}

September 1996

1 Introduction.

It is well known that the j-invariant establishes a bijection between ℂ and the set of isomorphism classes of elliptic curves over ℂ, see for example [10]. The endomorphism ring of an elliptic curve E over ℂ is either ℤ or an order in an imaginary quadratic extension of ℚ; in the second case E is said to be a CM elliptic curve (CM meaning complex multiplication). A complex number x is said to be CM if the corresponding elliptic curve over ℂ is CM. A point (x_{1}, x_{2}) in ℂ^{2} is defined to be CM if both x_{1} and x_{2} are CM. The aim of this article is to determine all irreducible algebraic curves C in ℂ^{2} containing infinitely many CM points. In other words, we want to determine all irreducible polynomials f in ℂ[x_{1}, x_{2}] that vanish at infinitely many CM points. The motivation for doing this comes from a conjecture of Frans Oort (see [7, Chapter IV, §1] for a precise statement), saying roughly that the irreducible components of the Zariski closure of any set of CM points in any Shimura variety are sub Shimura varieties. For the irreducible components of dimension zero this is trivially true. For those of dimension one Oort’s conjecture was in fact stated earlier by Yves André as a problem in [2, Chapter X, §1].

We view ℂ^{2} as the Shimura variety which is the moduli space of pairs of elliptic curves. Then the irreducible sub Shimura varieties of dimension one are the following: ℂ × {x_{2}} with x_{2} a CM point, {x_{1}} × ℂ with x_{1} a CM point, or the image in ℂ^{2}, under the usual map, of the modular curve Y_{0}(n) for some integer n ≥ 1. Recall that, for n ≥ 1, Y_{0}(n) is the modular curve classifying elliptic curves with a cyclic subgroup of order n, or, equivalently, cyclic isogenies of degree n between elliptic curves. The usual map from Y_{0}(n) to ℂ^{2} sends an isogeny to its source and target, i.e., ϕ: E_{1} → E_{2} is sent to (j(E_{1}), j(E_{2})). We will prove the following result, giving evidence for the conjecture just mentioned.

1.1 Theorem. Assume the generalized Riemann hypothesis for imaginary quadratic fields. Let C be an irreducible algebraic curve in ℂ^{2} containing infinitely many CM points and such that neither of its projections to ℂ is constant. Then C is the image of Y_{0}(n) for some n ≥ 1.

1.2 Remark. In the proof of Theorem 1.1 we will see that the state of the art in analytic number theory is such that the Riemann hypothesis is “almost not needed” (see Remark 5.4). It is clear that Theorem 1.1 implies similar statements for curves contained in the product of two modular curves. In particular, if one assumes GRH, Oort’s conjecture is true for curves contained in the product of two modular curves. □

∗partially supported by the Institut Universitaire de France

1.3 Remark. Ben Moonen has proved Oort’s conjecture for the sets of CM points in moduli spaces of abelian varieties such that there exists a prime number p at which all the CM points are canonical in the sense that they have an ordinary reduction of which they are the Serre-Tate canonical lift (see [7, Chapter IV, §1]). Yves André has proved the conclusion of Theorem 1.1 with the Riemann hypothesis replaced by the assumption that the Zariski closure of C in ℙ^{1}×ℙ^{1} meets {∞} × ℂ only in points (∞, x_{2}) with x_{2} a CM point (see [1]). In the case where C meets the union of {∞} × ℂ and ℂ × {∞} only in ∞ × ∞ he has a very simple proof. □

The idea of the proof of Theorem 1.1 is the following. We use the Galois action on the set of CM j-invariants to show that for all but finitely many CM points (x_{1}, x_{2}) on C the CM fields of x_{1} and x_{2} coincide. Then we consider intersections of C with its images under certain Hecke operators. The Riemann hypothesis implies that C is actually contained in some of these images. To finish, we consider an irreducible component X of the inverse image of C in ℍ×ℍ, the product of the complex upper half plane by itself, and show that the stabilizer of X in SL_{2}(ℝ) × SL_{2}(ℝ) is of the kind it should be.

1.4 Remark. At the time this article came back from the referee (June 1997), Yves André succeeded in proving the conclusion of Theorem 1.1 unconditionally, using a result of Masser on Diophantine approximation and the j-function. □

2 Some facts about CM elliptic curves.

Before we start with the proof of Theorem 1.1, we need to recall some facts about CM elliptic curves. These facts can be found for example in [10, Appendix C, §11]. First of all, CM elliptic curves are defined over \overline{ℚ}. Let K be an imaginary quadratic extension of ℚ, with a given embedding in \overline{ℚ}. Let O_{K} ⊂ K be the ring of integers. Every subring of O_{K} of finite index is of the form O_{K,f} := ℤ + fO_{K} for a unique integer f ≥ 1. For f ≥ 1 let S_{K,f} be the set of isomorphism classes of pairs (E, α), with E an elliptic curve over \overline{ℚ} and α: O_{K,f} → End(E) an isomorphism of rings inducing the given embedding of K into \overline{ℚ} via the action on Lie(E). The group G_{K} := Gal(\overline{ℚ}/K) acts on S_{K,f}. But also the Picard group Pic(O_{K,f}) acts on S_{K,f} by the following formula:

(2.1)  (E, [L]) ↦ E ⊗_{O_{K,f}} L,

where L is an invertible O_{K,f}-module, [L] its equivalence class and E ⊗_{O_{K,f}} L the cokernel of the map p: E^{2} → E^{2} if p: O_{K,f}^{2} → O_{K,f}^{2} has cokernel L (view p as a matrix with coefficients in O_{K,f}). If we choose an embedding of \overline{ℚ} in ℂ and write E(ℂ) as ℂ modulo a lattice Λ, then (E ⊗_{O_{K,f}} L)(ℂ) is the quotient of ℂ ⊗_{O_{K,f}} L by Λ ⊗_{O_{K,f}} L. The actions by G_{K} and Pic(O_{K,f}) on S_{K,f} commute.

2.2 Proposition. The set S_{K,f} is a Pic(O_{K,f})-torsor, i.e., the action of Pic(O_{K,f}) is free and has exactly one orbit.

Proof. (Sketch.) For every (E, α) and Λ as above, End_{O_{K,f}} (Λ) = O_{K,f}. Moreover, O_{K,f} is of the form ℤ[x]/(g). It follows that Λ is an invertible O_{K,f}-module. □

It follows that G_{K} acts on S_{K,f} via a morphism G_{K} → Pic(O_{K,f}). This morphism is surjective and unramified outside f. The Frobenius element at a maximal ideal m not containing f is the element [m]^{−1} of Pic(O_{K,f}) (all this can be seen from deformation theory, using the theorem of Serre-Tate, or from class field theory). Let H_{K,f} be the Galois extension of K corresponding to this quotient Pic(O_{K,f}) of G_{K}. We remark that we have H_{K,f} = K(j(E)) for all (E, α) in S_{K,f}.

3 The two CM fields are almost always equal.

Let C_{ℂ} ⊂ ℂ^{2} be as in Theorem 1.1 (i.e., it is irreducible, it contains infinitely many CM points and its two projections to ℂ are not constant). Since all CM points have coordinates in \overline{ℚ}, C_{ℂ} is defined over \overline{ℚ}, in the sense that it is the locus of zeros of an irreducible polynomial, call it f, with coefficients in \overline{ℚ}. It will be convenient for us to work with a curve defined over ℚ, hence we let C be the union of the finitely many conjugates of C_{ℂ}. Then C is defined by the product F of the Galois conjugates of f, if we take f such that it has a non-zero coefficient in ℚ. Let d_{1} and d_{2} be the degrees of F with respect to the second and first variable. Then d_{i} is the degree of the ith projection from C to ℂ. For x in ℂ we will denote the endomorphism ring of the corresponding elliptic curve by End(x). For a CM point x in ℂ we will call ℚ ⊗ End(x) the CM field of x. Note that the isogeny class of a CM elliptic curve over \overline{ℚ} consists of all elliptic curves with the same CM field. We want to prove that C is the image in ℂ^{2} of some Y_{0}(n). Our first step in this direction is the following proposition.

3.1 Proposition. Let C be as above. For all but finitely many CM points (x_{1}, x_{2}) in C the CM fields of x_{1} and x_{2} coincide.

Proof. Suppose that (x_{1}, x_{2}) is a CM point in C(\overline{ℚ}) such that the two CM fields K_{1} and K_{2} are different. Since C is defined over ℚ, ℚ(x_{1}, x_{2}) has degree at most d_{2} over ℚ(x_{1}) and degree at most d_{1} over ℚ(x_{2}). Let L be the field generated by K_{1} and K_{2}, and M the intersection of L(x_{1}) and L(x_{2}). Let us write End(x_{i}) = O_{K_{i},f_{i}} for i = 1 and 2. The field L(x_{i}) is an abelian Galois extension of L, of degree at least |Pic(O_{K_{i},f_{i}})|/2. The degrees of L(x_{1}, x_{2}) over L(x_{2}) and L(x_{1}) are equal to those of L(x_{1}) and L(x_{2}) over M, respectively. This gives us:

(3.2)  |Pic(O_{K_{i},f_{i}})| ≤ 2d_{i}[M : L].

We will now work to get a suitable upper bound for [M : L]. The group Gal(L(x_{1}, x_{2})/ℚ) is an extension of Gal(L/ℚ) by the abelian group Gal(L(x_{1}, x_{2})/L). Hence the action of Gal(L(x_{1}, x_{2})/ℚ) on Gal(L(x_{1}, x_{2})/L) by conjugation factors through an action of Gal(L/ℚ).

In the same way, Gal(L/ℚ) acts on the two groups Gal(L(x_{i})/L), which we view as subgroups of Gal(K_{i}(x_{i})/K_{i}). Now Gal(L/ℚ) is equal to Gal(K_{1}/ℚ) × Gal(K_{2}/ℚ), hence equal to ℤ/2ℤ × ℤ/2ℤ. The action of Gal(L/ℚ) on Gal(L(x_{i})/L) factors through Gal(K_{i}/ℚ) and as such coincides with the restriction of the action of Gal(K_{i}/ℚ) on Gal(K_{i}(x_{i})/K_{i}) = Pic(O_{K_{i},f_{i}}).

3.3 Lemma. Let K be a quadratic imaginary field and f ≥ 1. Then the non-trivial element σ of Gal(K/ℚ) acts as −1 on Pic(O_{K,f}).

Proof. The endomorphism σ + 1 of Pic(O_{K,f}) factors through the norm map from Pic(O_{K,f}) to Pic(ℤ). □

Now note that Gal(M/L) is a quotient of both Gal(L(x_{i})/L), so the action of Gal(L/ℚ) on it is by the non-trivial character given by the first projection, but also by the second projection. This implies that Gal(M/L) is killed by multiplication by two.

3.4 Lemma. Let K be an imaginary quadratic field and f ≥ 1. Then the dimension of the 𝔽_{2}-vector space Pic(O_{K,f}) ⊗ 𝔽_{2} is at most the number of odd primes dividing the discriminant discr(O_{K,f}) of O_{K,f} plus ten.

Proof. (Sketch.) The exact bound we give does not matter so much, so we just give some indications. First one notes that there is an exact sequence:

(3.4.1)  (K ⊗ ℚ_{2})^{∗} → Pic(O_{K,f}) → Pic(O_{K,f}[1/2]) → 0.

Let S := Spec(O_{K,f}[1/2]) and T := Spec(ℤ[1/2]). The Kummer sequence gives a surjection from H^{1}(S_{et}, 𝔽_{2}) onto the 2-torsion subgroup of Pic(S), which has the same dimension as Pic(S) ⊗ 𝔽_{2}. One deals with H^{1}(S_{et}, 𝔽_{2}) by projecting to T_{et}. □

Since Gal(M/L) is killed by 2 and a quotient of a subgroup of Pic(O_{K_{i},f_{i}}), we have:

(3.5)  log_{2}[M : L] ≤ |{2 ≠ p|discr(O_{K_{i},f_{i}})}| + 10,  i ∈ {1, 2}.

On the other hand, we have Siegel’s theorem (see [8]), stating that:

(3.6) log |Pic(O_{K_{i},f_{i}})| = (1/2 + o(1)) log |discr(O_{K_{i},f_{i}})|,  (|discr(O_{K_{i},f_{i}})| → ∞).

Combining equations (3.5) and (3.6) shows that |Pic(O_{K_{i},f_{i}})|/[M : L] tends to infinity as the discrminiant of O_{K_{i},f_{i}} tends to infinity. But then equation (3.2) can hold for only finitely many (x_{1}, x_{2}). This ends the proof of Proposition 3.1. □

3.7 Remark. The proof of Proposition 3.1 shows actually more: the function on the set of CM points on C that sends (x_{1}, x_{2}) to f_{1}/f_{2} takes only finitely many values. Using this, one can reduce the proof of Theorem 1.1 to the case where there are infinitely many CM points (x_{1}, x_{2}) on C with End(x_{1}) = End(x_{2}) (one replaces C by its image under a suitable Hecke correspondence). As we do not know how to exploit this, we do not go into further detail. □

3.8 Remark. Proposition 3.1 was also proved by Yves André in [1], and also by Ching-Li Chai (not published). □

4 Intersecting C with something.

We continue the proof of Theorem 1.1. So we let C be as before. At this point we already know that we have infinitely many CM points (x_{1}, x_{2}) on C for which x_{1} and x_{2} are isogeneous because they have the same CM field. We have to prove that there is an integer n ≥ 1 such that for infinitely many (x_{1}, x_{2}) there exists an isogeny of degree n between x_{1} and x_{2}. A direct approach for this is the following. Consider a CM point (x_{1}, x_{2}) such that x_{1} and x_{2} have the same CM field, say K, and an isogeny from x_{1} to x_{2} of minimal degree, say n. One can get an upper bound for n in terms of the discriminants of the End(x_{i}). By Remark 3.7, one can assume that End(x_{1}) = End(x_{2}) = O_{K,f} and get an upper bound for n from Minkowski’s theorem on ideals of small norm representing elements of the class group; the bound is a constant times |discr(O_{K,f})|^{1/2}. Then one considers the intersection of C with Y_{0}(n). The degrees of both projections from Y_{0}(n) to ℂ are equal to ψ(n), where ψ(n) = n ∏_{p|n}(1+1/p). The Picard group of ℙ^{1} × ℙ^{1} (over a field, say ℚ) is isomorphic to ℤ × ℤ, the isomorphism sending an effective divisor to the degrees of its two projections to ℙ^{1}. The intersection form is the following: (a, b) · (c, d) = ad + bc. Hence the intersection number of the Zariski closures in ℙ^{1} × ℙ^{1} of C and Y_{0}(n) is ψ(n)(d_{1} + d_{2}). Since both curves we intersect are defined over ℚ, the intersection contains all Galois conjugates of (x_{1}, x_{2}), of which there are |Pic(O_{K,f})|. So if |Pic(O_{K,f})| exceeds ψ(n)(d_{1} + d_{2}), the proof is finished, since then the intersection is not proper. Unfortunately, equation (3.6) does not imply such an inequality.

Nevertheless, the idea of intersecting C with something is a good one. Natural “somethings” to take are images of C itself under Hecke correspondences. Again, we consider a CM point (x_{1}, x_{2}) on C such that the CM fields of x_{1} and x_{2} coincide. Let K, f_{1} and f_{2} be defined by: End(x_{i}) = O_{K,f_{i}}. Let f be the least common multiple of f_{1} and f_{2}. The field generated by H_{K,f_{1}} and H_{K,f_{2}} is contained in H_{K,f}, and one easily checks that H_{K,f} has degree at most three over it. Hence the orbit of (x_{1}, x_{2}) under the action of G_{K} has at least |Gal(H_{K,f}/K)|/3 elements. Recall from §2 that we can identify Gal(H_{K,f}/K) with Pic(O_{K,f}). For σ in Gal(H_{K,f}/K) corresponding to the class [I] of an invertible ideal I of O_{K,f}, there are isogenies from x_{1} to σ(x_{1}) and from x_{2} to σ(x_{2}) whose kernels are isomorphic, as O_{K,f}- modules, to O_{K,f}/I. Hence if we take I such that O_{K,f}/I is a cyclic group of some order n, then σ(x_{i}) is in T_{n}(x_{i}) for i equals 1 and 2, where T_{n} is the correspondence on ℂ that sends an elliptic curve to the sum (as divisors) of its quotients by its cyclic subgroups of order n. (Let us note that this T_{n} is not the same as the correspondence on ℂ that is usually called T_{n} if n is not square free, since the usual one involves a sum over all subgroups of order n.) Let T_{n} × T_{n} be the correspondence on ℂ × ℂ that is the product of T_{n} on each factor: it sends a pair (E_{1}, E_{2}) of elliptic curves to the sum of the (E_{1}/G_{1}, E_{2}/G_{2}), where G_{i} is a cyclic subgroup of order n in E_{i}. Then (x_{1}, x_{2}) is in the intersection of C and (T_{n} × T_{n})C, because x_{i} is in T_{n}(σ(x_{i})) and (σ(x_{1}), σ(x_{2})) is in C. Since both C and (T_{n} × T_{n})C are defined over ℚ, their intersection contains all Galois conjugates of (x_{1}, x_{2}). Hence the intersection has at least |Pic(O_{K,f})|/3 elements. Let us now calculate the degrees of the projections of (T_{n} × T_{n})C to ℂ. By definition, (T_{n} × T_{n})C consists of the (x, y) such that there exist u and v in ℂ with (u, v) in C, and cyclic isogenies of degree n from u to x and from v to y. Let x be in ℂ. Then there are ψ(n) u’s with x ∈ T_{n}(u). For each such a u there are d_{1} v’s with (u, v) on C. For each such a v there are ψ(n) y’s in T_{n}(v). This shows that the degree of the first projection of (T_{n} × T_{n})C is ψ(n)^{2}d_{1}. Of course, for the second projection one has the analogous result. So, for the intersection number of C and (T_{n} × T_{n})C we find 2d_{1}d_{2}ψ(n)^{2}. We conclude that if |Pic(O_{K,f})| is bigger than 6d_{1}d_{2}ψ(n)^{2}, then C is contained in (T_{n} × T_{n})C. The next thing to do is to see if there do exist ideals I with the required properties.

Let x_{1}, x_{2}, K and f be as above. Let p be a prime number that splits in O_{K,f}, i.e., such that O_{K,f} ⊗ 𝔽_{p} is isomorphic to 𝔽_{p} × 𝔽_{p}. For I we take one of the two maximal ideals containing p. As explained above, we have the following implication:

(4.1)  6d_{1}d_{2}(p + 1)^{2} < |Pic(O_{K,f})| implies C ⊂ (T_{p} × T_{p})C.

Equation (3.6) tells us that |Pic(O_{K,f})| = |discr(O_{K,f})|^{1/2+o(1)}. So we want p to be at most something as |discr(O_{K,f})|^{1/4}. More precisely:

4.2 Proposition. Suppose that there exists ε > 0 such that, when K ranges through all imaginary quadratic fields and f through all positive integers, the number of primes p less than |discr(O_{K,f})|^{1/4−ε} that are split in O_{K,f} tends to infinity as |discr(O_{K,f})| tends to infinity. Then there are infinitely many primes p such that C is contained in (T_{p} × T_{p})C.

Proof. Because we have infinitely many CM points (x_{1}, x_{2}) on C, we know that the discriminants |discr(O_{K,f})| associated to them as above tend to infinity. The implication (4.1) and equation (3.6) give us the infinitely many required primes. □

5 Existence of small split primes.

The aim of this section is to prove the hypothesis in Proposition 4.2. It turns out that this is no problem at all if one assumes GRH for imaginary quadratic fields and uses the resulting effective Chebotarev theorem of Lagarias, Montgomery and Odlyzko as stated in [9].

For K an imaginary quadratic field, f a positive integer and x ≥ 2 a real number, let π_{K,f}(x) be the number of primes p ≤ x that are split in O_{K,f}, let d_{K} := |discr(O_{K})| and let d_{K,f} := |discr(O_{K,f})|. Note that d_{K,f} = f^{2}d_{K}. As usual, let Li(x) := ∫_{2}^{x} dt/ log(t). Theorem 4 of [9] and the second remark following it say that, for x sufficiently big and for all K as above for which GRH holds, one has:

(5.1)  |π_{K,1}(x) − \frac{1}{2}Li(x)| ≤ \frac{1}{6}x^{1/2} (log(d_{K}) + 2 log(x)) .

Since the number of primes dividing f is at most log_{2}(f), equation (5.1) implies: (5.2)

π_{K,f}(x) ≥ \frac{x}{2 log(x)} (Li(x)\frac{log(x)}{x} − \frac{log(x)}{3x^{1/2}} (log(d_{K}) + 2 log(x)) − \frac{2 log(x) log(f)}{x log(2)}) .

If x tends to infinity, Li(x) log(x)/x tends to 1 and log(x)^{2}/x^{1/2} tends to 0. One checks easily that for x sufficiently big (i.e., bigger than some absolute constant), and bigger than log(d_{K,f})^{2}(log(log(d_{K,f}))^{2}, one has log(x) log(d_{K})/3x^{1/2} < c < 1, with c independent of K and f. Under the same conditions, log(x) log(f)/x tends to zero if x tends to infinity. This means that we have proved the following proposition.

5.3 Proposition. Let C be as before (i.e., as in the beginning of §3). Assume GRH for all imaginary quadratic fields. Then there exist infinitely many primes p such that C is contained in (T_{p} × T_{p})C. □

5.4 Remark. Of course, the question remains whether one can prove the hypothesis of Proposition 4.2 without assuming GRH. Etienne Fouvry tells me the following. He shows that for r > 0 and all n, the set of d_{K,f} such that the number of primes p < d_{K,f}^{r} that are split in O_{K,f} is at most n, has density zero (i.e., the number of such d_{K,f} < x is o(x) for x → ∞). Moreover, he says that the exponent 1/4 is critical, in the sense that one can prove that for all ε > 0, the number of primes p < d_{K,f}^{1/4+ε} that are split in O_{K,f} tends to infinity as d_{K,f} tends to infinity. To prove this, he uses a result of Linnik and Vinogradov in [6], see also [4]. The central point in [6] is an upper bound for short character sums by Burgess, in which the exponent 1/4 + ε appears. This 1/4 has not moved in the last 30 years. □

6 Some topological arguments.

In this section we finish the proof of Theorem 1.1 by combining Proposition 5.3 with the following theorem, which gives yet another characterization of modular curves.

6.1 Theorem. Let C in ℂ^{2} be an irreducible algebraic curve. Let d_{1} and d_{2} be the degrees of its two projections to ℂ. Suppose that d_{1} and d_{2} are both non-zero, and that we have C ⊂ (T_{n} × T_{n})C for some square free integer n > 1 that is composed of primes p ≥ max{5, d_{1}}. Then C is the image of Y_{0}(m) in ℂ^{2} for some m ≥ 1.

Let us first show that this theorem and Proposition 5.3 imply Theorem 1.1. So let C_{ℂ} and C be as in the beginning of §3. Recall that C is the union of the finitely many Galois conjugates of the irreducible component C_{ℂ} of it. We know that there are infinitely many primes p such that C is contained in (T_{p} × T_{p})C. For such a prime p, let T_{C,p} denote the correspondence on C induced by T_{p} ×T_{p}. By this we mean the following. The correspondence T_{p} ×T_{p} on ℂ^{2} is given by the map from Y_{0}(p) × Y_{0}(p) to ℂ^{2} × ℂ^{2} that sends a point (ϕ, ψ) to (s(ϕ), s(ψ), t(ϕ), t(ψ)), where s and t stand for source and target, respectively. Take the inverse image of C × C in Y_{0}(p) × Y_{0}(p), and delete its zero-dimensional part; that, together with its two maps to C, is T_{C,p}. We have to show that a suitable product T_{C,p_{1}} ⋯ T_{C,p_{r}} with r ≥ 1 and the p_{i} distinct induces a non-trivial correspondence from C_{ℂ} to itself, because then we can apply Theorem 6.1 to C_{ℂ} with n = p_{1} ⋯ p_{r}. Let S be the finite set of irreducible components of C. Then each T_{C,p} induces a correspondence T_{S,p} on S that is surjective in the sense that both maps from T_{S,p} to S are surjective. Moreover, the Galois group G_{ℚ} acts transitively on S, and all T_{S,p} are compatible with this action. Let x_{0} in S correspond to C_{ℂ}. If there is some T_{S,p} such that x_{0} is in T_{S,p}x_{0}, we can take n = p. So suppose that for all T_{S,p} we have x_{0} ∉ T_{S,p}x_{0}. Then we have for all T_{S,p} and all x that x ∉ T_{S,p}x. One now easily sees that there are p_{1}, …, p_{r} distinct with 1 ≤ r ≤ |S| and x_{0} ∈ T_{S,p_{1}} ⋯ T_{S,p_{r}}x_{0}.

Proof. (Of Theorem 6.1.) We take an integer n as in the theorem we are proving. Let T_{C,n} be the correspondence on C induced by T_{n} × T_{n}, in the sense explained above. (In fact, for everything that follows we could also replace T_{C,n} by one of its irreducible components, but it is useful to see how to exploit all of it.) We view T_{C,n} as a subset of C × C. The image of T_{C,n} under the map (pr_{1}, pr_{1}) from C × C to ℂ × ℂ is the image T_{n} of Y_{0}(n) in ℂ × ℂ. Consider the commutative diagram:

(6.2)  \begin{aligned}C & → & ℂ \\ ↑ & & ↑ \\ T_{C,n} & → & T_{n} \end{aligned}

in which the vertical maps are induced by the projections from C × C and ℂ × ℂ on the first factor.

6.3 Lemma. The map from T_{C,n} to the fibred product C ×_{ℂ} T_{n} induced by (6.2) is surjective.

Proof. By construction, all four maps in (6.2) are finite as morphisms of (possibly reducible) algebraic curves. Therefore, the map from T_{C,n} to C ×_{ℂ}T_{n} is also a finite morphism of algebraic curves. Hence to show that it is surjective, it suffices to show that C ×_{ℂ} T_{n} is irreducible, or, equivalently, that the tensor product of the function fields of C and Y_{0}(n) over ℂ(j) is a field. For this, it is enough to prove that the tensor product with Y_{0}(n) replaced by Y (n) is a field (Y (n) is the modular curve parametrizing elliptic curves with a symplectic basis of their ntorsion). The function field of Y (n) is Galois over ℂ(j) with Galois group SL_{2}(ℤ/nℤ)/{±1}. The group SL_{2}(ℤ/nℤ) is isomorphic to the product of the SL_{2}(𝔽_{p_{i}}), 1 ≤ i ≤ r; one checks easily that it has no non-trivial subgroup of index at most d_{1}. This means that the function fields of C and Y (n) are linearly disjoint. □

For reasons to become clear soon, we now first prove the following lemma.

6.4 Lemma. The orbits in C of T_{C,n} are not discrete for the strong topology.

Proof. The morphism pr_{1} from C to ℂ is proper, hence the image of a closed subset of C is closed in ℂ. In particular, the image of the closure of any subset of C is the closure of its image. Hence it is enough to see that the images in ℂ of the orbits of T_{C,n} are not closed. Let x be in C, and let y be its image in ℂ. Lemma 6.3 implies that pr_{1}T_{C,n}x = T_{n}y, hence we just have to show that the orbits in ℂ of T_{n} are not closed. For this we view ℂ as the quotient of the complex upper half plane ℍ by the group SL_{2}(ℤ) via the map π: τ ↦ j(ℂ/(ℤ + ℤτ)). Let x be in ℂ, and choose τ in π^{−1}x. Then for all a and b in ℤ, π(τ + a) and π(n^{b}τ) are in the orbit of x under T_{n}. By composing these operations, we see that π(n^{b}τ + a) and π(τ + n^{−b}a) are in the orbit of x. Taking a non-zero and b big shows that the orbit is not closed. (In fact, it is easy to show, using τ ↦ −τ^{−1}, that all orbits in ℂ of T_{n} are dense.) □

We view ℂ × ℂ as the quotient of ℍ × ℍ by the group Γ := SL_{2}(ℤ) × SL_{2}(ℤ), via the map:

(6.5)  π: ℍ × ℍ → ℂ × ℂ, (τ_{1}, τ_{2}) ↦ (j(ℂ/(ℤ + ℤτ_{1})), j(ℂ/(ℤ + ℤτ_{2}))).

Let X be an irreducible component of the analytic subvariety π^{−1}C of ℍ × ℍ. The group G := SL_{2}(ℝ) × SL_{2}(ℝ) acts transitively on ℍ × ℍ. We will study its subgroup G_{X}, the stabilizer of X. What we have to prove is that G_{X} is the graph of an inner automorphism of SL_{2}(ℝ); this automorphism then tells us for which m our curve C is the image of Y_{0}(m). The decisive step in the proof of this is to see that G_{X} is not discrete (if C is an arbitrary curve in ℂ^{2}, then G_{X} is typically discrete).

6.6 Lemma. The group G_{X} is an analytic subgroup of G.

Proof. The action of G on ℍ × ℍ is algebraic (it is given by fractional linear transformations). The subgroup G_{X} consists of exactly those elements g in G that satisfy, for all x in X, the two conditions gx ∈ X and g^{−1}x ∈ X. All these conditions are analytic. □

6.7 Lemma. The kernels of the two projections from G_{X} to SL_{2}(ℝ) are discrete.

Proof. This kernel K, say for the second projection, is the same as the stabilizer of X in the subgroup SL_{2}(ℝ) × {1} of G. For all τ in ℍ, it stabilizes X_{τ} := X ∩ (ℍ × {τ}), which is discrete since d_{2} > 0; hence the connected component K^{o} of K stabilizes every element of X_{τ}. We conclude that K^{o} acts trivially on X. Now the stabilizer in SL_{2}(ℝ) of the element i of ℍ is SO_{2}(ℝ). Because d_{1} > 0, K^{o} is contained in all conjugates of SO_{2}(ℝ), the intersection of which is {±1}. □

6.8 Lemma. The image in SL_{2}(ℤ) of Γ_{X}, the stabilizer of X in Γ, under the ith projection, has index at most d_{i}.

Proof. We do the proof for i = 2. We factor the map π: ℍ × ℍ → ℂ × ℂ as follows:

(6.8.1)  ℍ × ℍ → ℂ × ℍ → ℂ × ℂ.

Let Y be the image of X in ℂ × ℍ. Then Y is an irreducible component of the inverse image Z of C in ℂ × ℍ. Let S be the set of c in C such that every x in π^{−1}c is contained in more than one irreducible component of π^{−1}C. Then S is contained in the finite subset of C consisting of singular points and points of which at least one of the coordinates is in {0, 1728}. Let C′ be C − S, and let X′ and Y′ be the inverse images, in X and Y, respectively, of C′. The map from X′ to C′ is the quotient for the action of Γ_{X}, hence the map from Y′ to C′ is the quotient for the action of pr_{2}Γ_{X}. It follows that pr_{2}Γ_{X} is the stabilizer in SL_{2}(ℤ) of Y in Z, so the set SL_{2}(ℤ)/pr_{2}Γ_{X} is the set of irreducible components of Z. But Z is also the fibred product of pr_{2}: C → ℂ and ℍ → ℂ, which implies that Z has at most d_{2} irreducible components. □

Lemmas 6.6, 6.7 and 6.8 are in fact valid for any curve C in ℂ^{2} for which d_{1} and d_{2} are non-zero. The next one crucially exploits that C ⊂ (T_{n} × T_{n})C.

6.9 Lemma. The topological group G_{X} is not discrete.

Proof. The subgroup G_{X} of G is analytic, hence closed. It contains Γ_{X}. The inclusion C ⊂ (T_{n} × T_{n})C implies that it contains some less trivial elements as well. The correspondence T_{n} on ℂ can be described as follows. Take z in ℂ; take its inverse image in ℍ; apply the map τ ↦ nτ = ({n \atop 0} {0 \atop 1})τ to it and take its image in ℂ; that is T_{n}z. Another way to say this is: take representatives t_{i} in GL_{2}(ℚ) (there are ψ(n) of them) for the quotient set SL_{2}(ℤ)({n \atop 0} {0 \atop 1})SL_{2}(ℤ)/SL_{2}(ℤ); then for z in ℂ and τ in ℍ mapping to it, T_{n}z is the image of the sum of the t_{i}τ. It follows that for each (i, j) such that (t_{i}, t_{j})X is contained in π^{−1}C we get an element g_{i,j} in G_{X} of the form

g_{i,j} = γ_{i,j,1} · (n^{−1/2} ({n \atop 0} {0 \atop 1}) , n^{−1/2} ({n \atop 0} {0 \atop 1})) · γ_{i,j,2},

with γ_{i,j,1} and γ_{i,j,2} in Γ. For c in C and x in X mapping to c, T_{C,n}c is the image of the sum of the g_{i,j}x. Let H be the subgroup of G_{X} generated by Γ_{X} and these elements g_{i,j}. We will prove that H is not discrete. Let \overline{H} be the closure of H. We take an element x in X. The map from G to ℍ × ℍ sending g to gx is proper, because the stabilizers of elements of ℍ × ℍ are compact. Hence \overline{H}x is also the closure of Hx. The subset Hx of X is discrete if and only if its image in C is discrete, since H contains Γ_{X} and the map X → C is the quotient for the action of Γ_{X}. By construction, the image of Hx in C is the orbit of x for T_{C,n}, which, by Lemma 6.4, is not discrete. This proves that G_{X} is not discrete. □

We can now quickly finish the proof of Theorem 6.1. Consider the Lie algebra Lie(G_{X}), which by Lemma 6.9 is non-zero. Lemma 6.7 tells us that the two projections pr_{i}Lie(G_{X}) are nonzero. But pr_{i}Lie(G_{X}) is normalized by pr_{i}Γ_{X}, which is Zariski dense in SL_{2}(ℝ) by Lemma 6.8. Since Lie(SL_{2}(ℝ)) is simple, it follows that pr_{i}Lie(G_{X}) is equal to Lie(SL_{2}(ℝ)) for both i. So, since SL_{2}(ℝ) is connected, G_{X} projects surjectively on both factors SL_{2}(ℝ) of G. Now we apply what is called Goursat’s lemma: let H be a subgroup of a product G_{1} × G_{2}, such that the projections p_{1} and p_{2} from H to G_{1} and G_{2} are surjective, then ker(p_{1}) and ker(p_{2}) are normal subgroups of G_{2} and G_{1}, respectively, and H is the inverse image of the graph of an isomorphism between G_{1}/ ker(p_{2}) and G_{2}/ ker(p_{1}). The kernel of pr_{2}: G_{X} → SL_{2}(ℝ) is a normal subgroup of SL_{2}(ℝ), viewed as SL_{2}(ℝ)×{1}. Since it is discrete and contains {1, −1}, it is {1, −1}. The same holds for the other projection, and G_{X} is the inverse image in G of the graph of an analytic automorphism, σ say, of SL_{2}(ℝ)/{±1}. Every such automorphism is inner. Since the pr_{i}Γ_{X} have finite index in SL_{2}(ℤ), it follows that σ is induced from an inner automorphism of the algebraic group SL_{2,ℚ}. The algebraic group of automorphisms of SL_{2,ℚ} is PGL_{2,ℚ}. Since the map GL_{2}(ℚ) → PGL_{2}(ℚ) is surjective (for example by Hilbert 90), σ is given by conjugation by some element g in GL_{2}(ℚ). So G_{X} is the set {(h, ±ghg^{−1}) | h ∈ SL_{2}(ℝ)}. Let x be an element of X, and write it as x = (τ, hτ) with τ in ℍ and h in SL_{2}(ℝ).

Since G_{X}x is in X, which is of dimension two, the stabilizer of x in G_{X} has dimension at least one. Let H be the stabilizer of τ in the connected component of identity G_{X}^{o}, for the action of G_{X}^{o} on the first factor ℍ; then the stabilizer of hτ for the action on the second factor is the conjugate g^{−1}hHh^{−1}g of H. Since H is of dimension one and connected (it is isomorphic to SO_{2}(ℝ)) we must have H = g^{−1}hHh^{−1}g, i.e., g^{−1}h normalizes H. Since the normalizer of SO_{2}(ℝ) in SL_{2}(ℝ) is just SO_{2}(ℝ) itself, this means that g^{−1}h is in H, or, equivalently, that hτ = gτ. This means that X = {(τ, gτ) | τ ∈ ℍ}. We may replace g by multiples ag of it, with a a non-zero rational number. So we can and do suppose that gℤ^{2} is contained in ℤ^{2} and that ℤ^{2}/gℤ^{2} is cyclic, say of order m. It is now clear that C is Y_{0}(m). □

7 Some remarks.

7.1 Remark. Our proof of Theorem 1.1 shows in fact that, assuming GRH, for each pair (d_{1}, d_{2}) of positive integers there exists an effectively computable number B(d_{1}, d_{2}), such that on every irreducible curve C in ℂ^{2} of bi-degree (d_{1}, d_{2}) that is defined over ℚ and not a modular curve there are at most B(d_{1}, d_{2}) CM points. (Note that under GRH, the statement that |Pic(O_{K})|/|Pic(O_{K})[2]| → ∞ is effective.) □

7.2 Remark. It is not true that all irreducible curves C in ℂ^{2} with C ⊂ (T_{n} × T_{n})C for some n > 1 are the image of some Y_{0}(m). Here we construct some examples. Let n > 1. Let w_{n} be the Atkin-Lehner involution of Y_{0}(n): it sends an isogeny to its dual. The correspondence T_{n} on ℂ has the following description. For z in ℂ, take its inverse image in Y_{0}(n), take the image of that under w_{n} and then the image in ℂ. It follows that for an irreducible curve C in ℂ^{2} such that at least one of the irreducible components of its inverse image in Y_{0}(n) × Y_{0}(n) is stable under the involution (w_{n}, w_{n}) we have C ⊂ (T_{n} ×T_{n})C. Let Z be the quotient of Y_{0}(n)×Y_{0}(n) by that involution. Bertini’s theorem, see for example [5, Theorem 6.3], gives the existence of whole families of curves in Z with irreducible inverse image in Y_{0}(n) × Y_{0}(n). Take C to be the image in ℂ^{2} of such an inverse image. □

7.3 Remark. The condition that n be square free in Theorem 6.1 should not be necessary; it is due to the laziness of the author. □

7.4 Remark. It is very tempting to try to generalize the methods of this article to the general case of Oort’s conjecture. □

Acknowledgements. I would like to thank Rutger Noot for interesting discussions on this subject that motivated me enough to work on it, and for his remarks on previous versions of this article. I thank Johan de Jong for his interest and the reference to [9]. I am very grateful to Etienne Fouvry for a letter in which he explains in detail the results mentioned in Remark 5.4. Tim Dokshitzer pointed out a gap in a previous version of this article. I want to thank Fabrice Rouillier for helping me installing the necessary software on the computer with which this article is written. Finally, I am very grateful for an invitation to the Centre for Research in Mathematics at the Institut d’Estudis Catalans in Barcelona, where I could compile my somewhat chaotic and incomplete notes into this article.

References

[1] Y. André. Talk given at the IHP in Paris on January 28, 1996.

[2] Y. André. G-functions and geometry. Aspects of mathematics, Vol. E13, Vieweg, Braunschweig, 1989.

[3] J.W.S. Cassels and A. Frohlich, eds. Algebraic number theory. Academic Press, 1967.

[4] J. Friedlander. Certain hypotheses concerning L-functions. Pacific Journal of Mathematics 69 (1977), 37–44

[5] J.P. Jouanolou. Théorèmes de Bertini et applications. Progress in Mathematics, Vol. 42, Birkhauser, Boston, 1983.

[6] Yu.V. Linnik and A.I. Vinogradov. Hyperelliptic curves and the least prime quadratic residue. Doklady Nauk. Akad. SSSR 168 (1966), 259–261.

[7] B.J.J. Moonen. Special points and linearity properties of Shimura varieties. Thesis, Utrecht, September 1995.

[8] J. Oesterlé. Nombres de classes des corps quadratiques imaginaires. Séminaire Bourbaki, exposé 631, juin 1984. Astérisque 121–122 (1985), 309–323.

[9] J-P. Serre. Quelques applications du théorème de densité de Chebotarev. Publications Mathématiques de l’IHES 54 (1981), 123–202.

[10] J.H. Silverman. The arithmetic of elliptic curves. Graduate Texts in Mathematics 106, Springer-Verlag, 1986.

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