Conjecture: 100% of elliptic surfaces over $\mathbb{Q}$ have rank zero

TL;DR

This paper conjectures that all elliptic surfaces over rationals have rank zero when ordered by coefficient size, supported by heuristics and experimental evidence, and explores implications for $L$-functions and finite fields.

Contribution

It introduces a new conjecture that 100% of elliptic surfaces over $Q$ have rank zero, supported by probabilistic heuristics and experimental data.

Findings

Heuristic supports the conjecture for rank zero.

Experimental evidence aligns with the conjecture.

Discussion on implications for $L$-functions and finite fields.

Abstract

Based on an equation for the rank of an elliptic surface over $\mathbb{Q}$ which appears in the work of Nagao, Rosen, and Silverman, we conjecture that 100% of elliptic surfaces have rank $0$ 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 $L$-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.