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arXiv · 2301.11437

Densities for Elliptic Curves over Global Function Fields

Andrew Yao
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Abstract

Let $K$ be a global function field. We obtain a set of formulas for the densities of the Kodaira types and Tamagawa numbers of elliptic curves over a completion of $K$ that is independent of the field's characteristic. Furthermore, for a finite field $F$ and real numbers $s$ and $\epsilon$ such that $s>1$ and $\epsilon>0$, we prove that there exists a global function field $K$ such that the full constant field of $K$ is $F$ and the value of the zeta function of $K$ at $s$ is less than $1+\epsilon$.

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Andrew Yao. 2023-01-26. Densities for Elliptic Curves over Global Function Fields. https://arxiv.org/abs/2301.11437

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