#### 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 c_{1}X^{19/24}(log X)^{3/8} curves with |Delta|< X with even parity and positive (analytic) rank; since Brumer and McGuinness predict cX^{5/6} 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.

#### Recommended Citation

Watkins, Mark, "Some Heuristics about Elliptic Curves" (2006). *Mathematics Faculty Scholarship.* Paper 110.

http://surface.syr.edu/mat/110

#### Source

Harvested from arXiv.org

#### Creative Commons License

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

## Additional Information

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