Arithmetic Conjectures Suggested by the Statistical Behavior of Modular Symbols

Abstract

Suppose E is an elliptic curve over $\mathbb{Q}$ and χ is a Dirichlet character. We use statistical properties of modular symbols to estimate heuristically the probability that $L(E,\chi ,1)=0$. Via the Birch and Swinnerton-Dyer conjecture, this gives a heuristic estimate of the probability that the Mordell–Weil rank grows in abelian extensions of $\mathbb{Q}$. Using this heuristic, we find a large class of infinite abelian extensions F where we expect E(F) to be finitely generated. Our work was inspired by earlier conjectures (based on random matrix heuristics) due to David, Fearnley, and Kisilevsky. Where our predictions and theirs overlap, the predictions are consistent.

2010 Mathematics Subject Classification:

• Primary: 11G05
• Secondary: 11F67
• 11G40
• 14G05

Acknowledgments

We thank Jon Keating and Asbjørn Nordencroft for helpful conversations about the random matrix theory conjectures.

Declaration of Interest

No potential conflict of interest was reported by the authors.

Additional information

Funding

This material is based upon work supported by the National Science Foundation under grants DMS-1302409 and DMS-1500316.