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 ).

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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

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Abstract: The goal of this paper is to give a quick survey of some important recent results and open problems in the area of function field arithmetic, which studies geometric analogs of arithmetic questions. We will sketch related developments and try to trace the multiple influences of works of John Tate in this context.

Classical automorphic functions are complex valued functions on the upper half plane left invariant under a subgroup of finite index of the modular group SL(2, Z). We consider the analogue of this classical setting in characteristic p. In particular, we investigate the analogue of the upper half plane and analyze the structure of the fundamental domain of a Hecke congruence group of level A over a function field over a finite field. We describe two algorithms to compute spaces of cusp forms of level A and analyze their complexity. For the second algorithm we extend Manin’s classical Modular Symbols method to the function field case.

Abstract. We prove quantitative upper bounds for the number of quadratic twists of a given elliptic curve E/Fq(C) over a function field over a finite field that have rank � 2, and for their average rank. The main tools are constructions and results of Katz and uniform versions of the Chebotarev density theorem for varieties over finite fields. Moreover, we conditionally derive a bound in some cases where the degree of the conductor is unbounded. Let first E/Q be an elliptic curve over Q, and for fundamental quadratic discriminants d, let Ed denote the curve E twisted by the associated Kronecker character χd. Goldfeld conjectured that Ed is most of the time of minimal rank compatible with the root number of Ed, which in this case means 1 ∑ lim rankEd(Q) =

Higher Heegner points on elliptic curves over function fields

Higher Heegner points on elliptic curves over function fields