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arXiv · 2409.14795

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

Jun-Yong Park
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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 $E(K) = \mathbb{Z}$.

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Jun-Yong Park. 2024-09-23. Quantitative rank distribution conjecture over $\mathbb{F}_q(t)$. https://arxiv.org/abs/2409.14795

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