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

TL;DR

This paper combines exact counting of elliptic curves over function fields with conjectures on rank distribution, providing a refined quantitative conjecture on the number of elliptic curves with specific group structures over $_q(t)$.

Contribution

It integrates recent counting results with rank distribution conjectures to formulate a more precise quantitative conjecture on elliptic curve distributions over function fields.

Findings

Derived a quantitative statement on elliptic curve counts over $_q(t)$

Refined conjecture on lower order terms for elliptic curves with specific group structures

Connected exact counts with rank distribution conjectures

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 $E(K) = \mathbb{Z}$.