Abstract. Elliptic curves over Q are equipped with a systematic collection of Heegner points arising from the theory of complex multiplication and defined over abelian extensions of imaginary quadratic fields. These points are the key to the most decisive progress in the last decades on the Birch and Swinnerton-Dyer conjecture: an essentially complete proof for elliptic curves over Q of analytic rank ≤ 1, arising from the work of Gross-Zagier and Kolyvagin. In [Da2], it is suggested that Heegner points admit a host of conjectural generalisations, referred to as Stark-Heegner points because they occupy relative to their classical counterparts a position somewhat analogous to Stark units relative to elliptic or circular units. A better understanding of Stark-Heegner points would lead to progress on two related arithmetic questions: the explicit construction of global points on elliptic curves (a key issue arising in the Birch and Swinnerton-Dyer conjecture) and the analytic construction of class fields sought for in Kronecker’s Jugendtraum and Hilbert’s twelfth problem. The goal of this article is to survey Heegner points, Stark-Heegner points, their arithmetic applications and their relations (both proved, and conjectured) with special values of L-series attached to modular forms.

Abstract. Let E/Q be an elliptic curve defined over Q with conductor N and Gal(Q/Q) the absolute Galois group of an algebraic closure Q of Q. We prove that for every σ ∈ Gal(Q/Q), the Mordell-Weil group E(Q σ) of E over the fixed subfield of Q under σ has infinite rank. Our approach uses the modularity of E/Q and a collection of algebraic points on E – the so-called Heegner points – arising from the theory of complex multiplication. In particular, we show that for some integer r, the rank of E over all the ring class fields of conductor of the form rm, and of the form rpn is unbounded, as m goes to infinity, as m and as n goes to infinity respectively, where m is a square-free integer and p is a prime such that (m, rN) = 1 and (p, rN) = 1. This paper is motivated by the following conjecture of M. Larsen [6]: Conjecture. Let K be a number field and E/K an elliptic curve over K. Then, for every σ ∈ Gal(K/K), the Mordell-Weil group E(K σ) of E over K σ = {x ∈ K | σ(x) = x} has infinite rank. In [3], we have proved this conjecture in certain cases: for a number field K and an elliptic curve E/K over K, • if 2-torsion points of E/K are K-rational, or • if E/K has a K-rational point P such that 2P ̸ = O and 3P ̸ = O, then for every automorphism σ ∈ Gal(K/K), the rank of the Mordell-Weil group E(K σ) is infinite. In this paper, we prove that the conjecture is true for elliptic curves over Q without any hypothesis on rational points of E/Q, i.e. if E/Q is an elliptic curve over Q, then,

Abstract. We exhibit algorithms to compute systems of Hecke eigenvalues for spaces of Hilbert modular forms over a totally real field. We provide many explicit examples as well as applications to modularity and Galois representations. Contents 1. Classical (elliptic) modular forms 2

This is a summary of a three-part lecture series given at the meeting on “Explicit methods in number theory ” that was held in Oberwolfach from July 12 to 18, 2009. The theme of this lecture series was the explicit construction of algebraic points on elliptic curves from cycles on modular varieties. Given a fixed elliptic curve E over Q, the goal is to better understand the group E ( ¯ Q) of algebraic points on E by focusing on the following question: Which points in E ( ¯ Q) can be accounted for by a “modular construction”? Heegner points arising from CM points on modular curves are the prototypical example of such a modular construction. While we do not dispose of a completely satisfactory general definition of modular points, fulfilling the conflicting requirements of flexibility and mathematical precision, several “test cases ” that go beyond the setting of Heegner points have been studied over the last 10 years (cf. [Da01], [DL], [BDG], [Da04], [Tr], [Gre], [BDP2]). Three illustrative examples were touched upon in these lectures: 1. [BDP1], [BDP2]. “Chow-Heegner points ” arising from algebraic cycles on higher dimensional varieties. The existence and key properties of Chow-Heegner points are typically conditional on the Hodge or Tate conjectures on algebraic cycles. 2. [DL], [BDG], [CD]. “Stark-Heegner points ” arising from ATR (“Almost Totally Real”) cycles on Hilbert modular varieties parametrising elliptic curves over totally real fields. These ATR cycles are not algebraic, and the expected algebraicity properties of the associated Stark-Heegner points do not seem (for now) to be part of a systematic philosophy. 3. [Da01], [DP]. Stark-Heegner points attached to real quadratic cycles on the “mock Hilbert modular surface ” SL2(Z[1/p])\(Hp ×H) parametrising an elliptic curve E over Q of prime conductor p. These real quadratic cycles are indexed by ideal classes of orders in a real quadratic field K, and are topologically isomorphic to R/Z. By an analytic process that combines complex and p-adic integration, they can be made to yield p-adic points on E which are expected to be defined over class fields of K. This setting leads to convincing experimental evidence for the existence of a theory of “complex multiplication for real quadratic fields”.

Abstract. Let k be a global field, k a separable closure of k, and Gk the absolute Galois group Gal(k/k) of k over k. For every σ ∈ Gk, let k σ be the fixed subfield of k under σ. Let E/k be an elliptic curve over k. We show that for each σ ∈ Gk, the Mordell-Weil group E(k σ) has infinite rank in the following two cases. Firstly when k is a global function field of odd characteristic and E is parametrized by a Drinfeld modular curve, and secondly when k is a totally real number field and E/k is parametrized by a Shimura curve. In both cases our approach uses the non-triviality of a sequence of Heegner points on E defined over ring class fields. 1.

Abstract. Let K be a real quadratic field, let p be a prime number which is inert in K and let Kp be the completion of K at p. As part of a Ph.D. thesis, we constructed a certain p-adic invariant u ∈ K × p, and conjectured that u is, in fact, a p-unit in a suitable narrow ray class field of K. Inthispaperwegive numerical evidence in support of that conjecture. Our method of computation is similar to the one developed by Dasgupta and relies on partial modular symbols attached to Eisenstein series. 1.

Abstract. In the predecessor to this article, we used global equidistribution theorems to prove that given a correspondence between a modular curve and an elliptic curve A, the intersection of any finite rank subgroup of A with the set of CM-points of A is finite. In this article we apply local methods, involving the theory of arithmetic differential equations, to prove quantitative versions of a similar statement. The new methods apply also to certain infinite rank subgroups, and to the situation where the set of CM-points is replaced by certain isogeny classes of points on the modular curve. Finally, we prove Shimura curve analogues of these results. 1.

Abstract. Let

We prove a general subconvex bound in the level aspect for Rankin–Selberg L-functions associated with two primitive holomorphic or Maass cusp forms over Q. We use this bound to establish the equidistribution of incomplete Galois orbits of Heegner points on Shimura curves associated with indefinite quaternion algebras over Q.

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