Compositio Math. 141 (2005) 561–572 DOI: 10.1112/S0010437X04001022 + r Towards the triviality of X0 (p )(Q)forr>1 Pierre J. R. Parent Abstract We give a criterion to check if, given a prime power pr with r>1, the only rational points + r of the modular curve X0 (p ) are trivial (i.e. cusps or points furnished by complex multi- plication). We then prove that this criterion is verified if p satisfies explicit congruences. This applies in particular to the modular curves Xsplit(p), which intervene in the problem of Serre concerning uniform surjectivity of Galois representations associated to division points of elliptic curves. 1. Introduction r r Let p beapowerofaprimenumberp,withr>1, and X0(p ) be the classical modular curve + r + r over Q.LetX0 (p ) be its quotient by the Atkin–Lehner involution. We say that a point of X0 (p )(C) is trivial if it is a cusp, or if the underlying elliptic curves have complex multiplication. In this + r paper, we state a criterion to check whether X0 (p )(Q) is trivial (Proposition 3.2). Then we prove that this criterion is verified if p satisfies some congruences. Explicitly, set A := {primes that are simultaneously a square mod 3, mod 4, mod 7, and a square mod at least five of the following: 8, 11, 19, 43, 67, 163}. Our main theorem is the following. Theorem 1.1. If pr is a prime power such that r>1, p 11, p =13 ,andp does not belong to the + r above set A,thenX0 (p )(Q) is trivial. + r+2 + r (Note that the existence of Q-morphisms X0 (p ) → X0 (p )(see[Mom86, p. 443]) shows that this result actually boils down to the case r =2andr =3.) Theorem 1.1 applies in particular to the modular curve Xsplit(p), which is isomorphic over Q + 2 to X0 (p ). This special result was our first motivation for this work, because of its relation with the following problem of Serre. Let E be an elliptic curve over a number field K without complex mul- tiplication over K. The Galois action induces a representation Gal(K/K) → GL(E[p]) GL2(Fp). In his famous article [Ser72], Serre proved that there exists an integer CE such that this repre- sentation is surjective if p>CE. In the same paper Serre asked if the integer CE can be chosen to depend only on K, not on E ([Ser72, p. 299]; see also [Maz77, Introduction]). This question boils down to determining whether the K-rational points of several modular curves of level p are trivial (in the above sense) for large enough p. These curves are X0(p), Xsplit(p), Xnon-split(p), and maybe ‘exceptional’ ones. The latter case (of exceptional curves) was ruled out by Serre (see [Maz77, Introduction]), and it is a celebrated theorem of Mazur [Maz78]thatX0(p)(Q) is made of cusps for p>163 (and is trivial for p>37). In the cases of Xsplit(p)andXnon-split(p), a new difficulty arises from the fact that elliptic curves over Q with complex multiplication always provide rational points on one of those two modular curves. From the above, our criterion for Xsplit(p)(Q)tobetrivial (Proposition 3.2) can be verified for p 11, p ∈{13, 37},wherep does not belong to the set A. Received 6 November 2003, accepted in final form 1 March 2004, published online 21 April 2005. 2000 Mathematics Subject Classification 11G18, 11G05, 14G10. Keywords: elliptic curves, modular curves, rational points, Gross–Heegner points. This journal is c Foundation Compositio Mathematica 2005. Downloaded from https://www.cambridge.org/core. IP address: 170.106.202.226, on 30 Sep 2021 at 23:05:22, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1112/S0010437X04001022 P. J. R. Parent The density of the prime numbers p in Theorem 1.1 is (1−7.2−9) 0.986 .... At the moment, we are unable to prevent a positive density of primes from escaping our methods, which use quadratic imaginary orders of trivial class number (as one guesses from the shape of A). Still in § 6 we indicate a procedure that could asymptotically improve this density. For general p we also prove a + r quantitative result giving explicit upper bounds for the number of non-trivial points in X0 (p )(Q) (Theorem 6.2). Our approach is based on the well-known method introduced by Mazur. We also make use of + r previous works by Momose on X0 (p )(Q), Kolyvagin–Logachev’s (or now Kato’s) theorem on the Birch and Swinnerton-Dyer conjecture, and a recent application by Merel of the graph method for X0(p) of Mestre and Oesterl´e. Our criterion is in fact almost the same as that of Merel [Mer01, Proposition 4], which arose in a different context. A new tool we use is a formula of Gross, generalized by Zhang, on special values of L-functions, which allows us to describe the cotangent space of Je (conjecturally the largest quotient of J0(p) having rank 0 over Q) in terms of Heegner points (Proposition 4.2). More precisely, the text is organized as follows. In § 2 we reduce our problem to a question on the fiber of X0(p)atp.In§ 3 we give our criterion (Propositions 3.1 and 3.2). In § 4 we concentrate on the cotangent space of Je in order to reformulate our criterion with the help of the Gross formula. In § 5 we restrict to primes satisfying the congruences of Theorem 1.1 and apply the graph method to build special elements that make the criterion work. Finally, in § 6 we prove our + r quantitative improvement (Theorem 6.2), we describe an algorithm to verify triviality of X0 (p )(Q) + r for any specified prime p, and we discuss the example of X0 (37 ). In the course of this paper we also give new elementary proofs of a result by Ahlgren and Ono on Weierstrass points of X0(p)(Q) (Theorem 3.3), and of a theorem by Vatsal for the equidistribution of Heegner points (Theorem 4.3). 2. Reducing to the bad fiber of X0(p) For any positive integer N, recall that X0(N) is the modular curve over Q corresponding to the congruence subgroup a b Γ0(N):= ∈ SL2(Z),c≡ 0modN . c d This curve deprived of its cusps is the coarse moduli space over Q of the isomorphism classes of elliptic curves equipped with an N-isogeny. If M|N,wewriteπN,M : X0(N) → X0(M)forthede- generacy morphism, which is defined functorially as (E,CN ) → (E,CM ), where CM := E[M] ∩ CN . In this paper, the model of X0(N)overZ that we consider is the modular one, which is obtained 1 1 by taking the normalization of PZ in X0(N)Q via the morphism πN,1 : X0(N)Q → X0(1)Q PQ. Models over arbitrary schemes of this modular curve will be deduced by base change. We denote by sm X0(p)Z the smooth part of X0(p)Z, obtained by removing the singular points in the fiber at p.IfM is a divisor of N such that M and N/M are relatively prime, we write wM for the corresponding + Atkin–Lehner involution, and X0 (N):=X0(N)/wN .Asusual,wewriteJ0(N) for the Jacobian − over Q of X0(N), and J0 (N):=J0(N)/(1+wN )J0(N). The models we use for abelian varieties over rings of integers are N´eron models. One defines similarly the curve Xsp.C.(N) associated to the split Cartan subgroup ∗ 0 Γsp C (N):= γ ∈ SL2(Z),γ≡ mod N , . 0 ∗ and the curve Xsplit(N) corresponding to the normalizer of the above group: ∗ 0 0 ∗ Γsplit(N):= γ ∈ SL2(Z),γ≡ or mod N . 0 ∗ ∗ 0 562 Downloaded from https://www.cambridge.org/core. IP address: 170.106.202.226, on 30 Sep 2021 at 23:05:22, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1112/S0010437X04001022 + r Towards the triviality of X0 (p )(Q) for r>1 The curve Xsp.C.(N) (respectively, Xsplit(N)) parametrizes elliptic curves endowed with an ordered (respectively, unordered) pair of independent N-isogenies. There is an involution w on Xsp.C.(N), defined functorially by (E,(A, B)) → (E,(B,A)), such that Xsplit(N)=Xsp.C.(N)/w.Themap 2 z → Nz on the upper half-plane induces a Q-isomorphism X0(N ) Xsp.C.(N)(an‘exotic’mapin 2 the language of [KM85]), and Xsplit(N) X0(N )/wN 2 . r + r We now fix a prime power p , r>1. Let P be a non-cuspidal point in X0 (p )(Q)andz ∈ r X0(p )(K) a lifting of P ,forK a quadratic number field (or for K = Q, but this case is possible only for p 163 by Mazur’s theorem). The point z corresponds to a couple (E,Cpr )overK,by[DR73, − Proposition VI.3.2]. Set π := πpr,p, x := wpπ(z)andx0 := πwpr (z) ∈ X0(p)(K). Let y ∈ J0 (p)(K) be the image of the divisor class of (x) − (x0)inJ0(p)(K). Lemma 2.1. The point y is Q-rational. Proof. This is a straightforward calculation: if σ is the non-trivial element of Gal(K/Q), then cl(y − σ(y)) = cl((wpπ(z)) − (πwpr (z)) − (wpπwpr (z)) + (π(z))) =(1+wp)cl((π(z)) − (πwpr (z))).