Authors/Contributors

Mark Watkins, Syracuse University

Document Type

Article

Date

8-30-2006

Embargo Period

11-18-2011

Disciplines

Mathematics

Description/Abstract

We give some heuristics for counting elliptic curves with certain properties. In particular, we re-derive the Brumer-McGuinness heuristic for the number of curves with positive/negative discriminant up to X, which is an application of lattice-point counting. We then introduce heuristics (with refinements from random matrix theory) that allow us to predict how often we expect an elliptic curve E with even parity to have L(E,1)=0. We find that we expect there to be about c1X19/24(log X)3/8 curves with |Delta|< X with even parity and positive (analytic) rank; since Brumer and McGuinness predict cX5/6 total curves, this implies that asymptotically almost all even parity curves have rank 0. We then derive similar estimates for ordering by conductor, and conclude by giving various data regarding our heuristics and related questions.

Additional Information

This manuscript is from arXiv.org, for more information see http://arxiv.org/abs/math/0608766

Source

Harvested from arXiv.org

Creative Commons License


This work is licensed under a Creative Commons Attribution 3.0 License.

Included in

Mathematics Commons

Share

COinS
 
 

To view the content in your browser, please download Adobe Reader or, alternately,
you may Download the file to your hard drive.

NOTE: The latest versions of Adobe Reader do not support viewing PDF files within Firefox on Mac OS and if you are using a modern (Intel) Mac, there is no official plugin for viewing PDF files within the browser window.