Heegner Points and Elliptic Curves of Large Rank over Function Fields

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 L(E/K, χ, 1) = 0 for all χ : G −→ C×. The product factorisation