Heegner Points and Elliptic Curves of Large Rank over Function Fields

Summary

Abstract. This note presents a connection between Ulmer's construction [Ulm02] of non-isotrivial elliptic curves over 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 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

so that the Birch and Swinnerton-Dyer conjecture predicts that

In fact, the G-equivariant refinement of the Birch and Swinnerton-Dyer conjecture leads one to expect that the rational vector spacecontains a copy of the regular representation of G.

It is expected in this situation that Heegner points account for the bulk of the growth of E(H), as H varies over the abelian extensions of K of dihedral type. For example we have:

Lemma 1. If ord_s=1L(E/H, s) ≤ [H : K], then the vector space E(H) ⊗ Q has dimension [H : K] and is generated by Heegner points.

Proof. For V any complex representation of G, let