Year: 2024
Type: preprint
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 $E(K) = \mathbb{Z}$.
Source: arXiv (Cornell University)
Author: Park, Jun-Yong
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4403780336