Skip to main content Accessibility help
×
Hostname: page-component-6bb9c88b65-6scc2 Total loading time: 0 Render date: 2025-07-24T08:48:57.743Z Has data issue: false hasContentIssue false

Heegner Points and Elliptic Curves of Large Rank over Function Fields

Published online by Cambridge University Press:  06 July 2010

Henri Darmon
Affiliation:
McGill University, Montréal
Shou-wu Zhang
Affiliation:
Columbia University, New York
Get access

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 ords=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

Information

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2004

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Book purchase

Temporarily unavailable

Accessibility standard: Unknown

Accessibility compliance for the PDF of this book is currently unknown and may be updated in the future.

Save book to Kindle

To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×