Conjecture: 100% of elliptic surfaces over $\mathbb{Q}$ have rank zero
Number Theory
2020-09-23 v2
Abstract
Based on an equation for the rank of an elliptic surface over which appears in the work of Nagao, Rosen, and Silverman, we conjecture that 100% of elliptic surfaces have rank when ordered by the size of the coefficients of their Weierstrass equations, and present a probabilistic heuristic to justify this conjecture. We then discuss how it would follow from either understanding of certain -functions, or from understanding of the local behaviour of the surfaces. Finally, we make a conjecture about ranks of elliptic surfaces over finite fields, and highlight some experimental evidence supporting it.
Keywords
Cite
@article{arxiv.2009.08622,
title = {Conjecture: 100% of elliptic surfaces over $\mathbb{Q}$ have rank zero},
author = {Alex Cowan},
journal= {arXiv preprint arXiv:2009.08622},
year = {2020}
}