Based on an NSF-CBMS lecture series given by the author at the University of Central Florida in Orlando from August 8 to 12, 2001, this monograph surveys some recent developments in the arithmetic of modular elliptic curves, with special emphasis on the Birch and Swinnerton-Dyer conjecture, the construction of rational points on modular elliptic curves, and the crucial role played by modularity in shedding light on these questions.

constant j-invariant

Abstract. For r = 6,7,..., 11 we find an elliptic curve E/Q of rank at least r and the smallest conductor known, improving on the previous records by factors ranging from 1.0136 (for r = 6) to over 100 (for r = 10 and r = 11). We describe our search methods, and tabulate, for each r = 5,6,..., 11, the five curves of lowest conductor, and (except for r = 11) also the five of lowest absolute discriminant, that we found. 1

Let E --> C be an elliptic surface defined over a number field K. For a finite covering C' --> C defined over K, let = E C C be the corresponding elliptic surface over C . In this paper we give a strong upper bound for the rank of E in the case of unramified abelien coverings C ! C and under the assumption that the Tate conjecture is true for E =K. In the case that C is an elliptic curve and the map C = C ! C is the multiplication-by-n map, the bound for rank(E =K)) takes the form O n , which may be compared with the elementary bound of O(n ).

Abstract not found

- a survey-

Abstract. We conjecture the true rate of growth of the maximum size of the Riemann zetafunction and other L-functions. We support our conjecture using arguments from random matrix theory, conjectures for moments of L-functions, and also by assuming a random model for the primes.

This survey paper contains two parts. The first one is a written version of a lecture given at the “Random Matrix Theory and L-functions ” workshop organized at the Newton Institute in July 2004. This was meant as a very concrete and down to earth introduction to elliptic curves with some description of how random matrices become a tool for the (conjectural) understanding of the rank of Mordell-Weil groups by means of the Birch and Swinnerton-Dyer Conjecture; the reader already acquainted with the basics of the theory of elliptic curves can certainly skip it. The second part was originally the write-up of a lecture given for a workshop on the Birch and Swinnerton-Dyer Conjecture itself, in November 2003 at Princeton University, dealing with what is known and expected about the variation of the rank in families of elliptic curves. Thus it is also a natural continuation of the first part. In comparison with the original text and in accordance with the focus of the first part, more details about the input and confirmations of Random Matrix Theory have been added. Acknowledgments. I would like to thank the organizers of both workshops for

Abstract. This note presents a connection between Ulmer’s construction [Ulm02] of non-isotrivial elliptic curves over Fp(t) with arbitrarily large rank, and the theory of Heegner points (attached to parametrisations by Drinfeld modular curves, as sketched in Section 3 of Ulmer’s article (see page??). This ties in the topics in Section 4 of that article more closely to the main theme of this volume. A review of the number field setting. Let K be a quadratic imaginary extension of F = Q, and let E /Q be an elliptic curve of conductor N. When all the prime divisors of N are split in K/F, the Heegner point construction (in the most classical form that is considered in [GZ], relying on the modular parametrisation X0(N) − → E) produces not only a canonical point on E(K), but also a norm-coherent system of such points over all abelian extensions of K which are of “dihedral type”. (An abelian extension H of K is said to be of dihedral type if it is Galois over Q and the generator of Gal(K/Q) acts by −1 on the abelian normal subgroup Gal(H/K).) The existence of this construction is consistent with the Birch and Swinnerton-Dyer conjecture, in the following sense: an analysis of the sign in the functional equation for L(E/K, χ, s) = L(E/K, ¯χ, s) shows that this sign is always equal to −1, for all complex characters χ of G:= Gal(H/K). Hence The product factorisation implies that L(E/K, χ, 1) = 0 for all χ: G − → C ×. L(E/H, s) = ∏ L(E/K, χ, s) χ

points and elliptic curves of large rank