Quantitative rank distribution conjecture over ${\mathbb{F}}_q(t)$

Abstract

We combine the exact counting of all elliptic curves over $K={\mathbb{F}}_q(t)$ with $\mathrm{char}(K)>3$ by Bejleri, Satriano and the author, together with the torsion-free nature of most elliptic curves over global function fields proven by Phillips, and the overarching conjecture of Goldfeld and Katz-Sarnak regarding the ``Distribution of Ranks of Elliptic Curves''. Consequently, we arrive at the quantitative statement which naturally renders even finer conjecture regarding the lower order main terms differing for the number of $E/K$ with $|E(K)|=1$ and $E(K)=\mathbb{Z}$.

This paper is superseded by a newer paper, entitled "100% of elliptic curves with a marked point have positive rank" arXiv:2504.01965 (by J. Park, and T. Phillips)