Densities for Elliptic Curves over Global Function Fields

TL;DR

This paper derives formulas for the distribution of Kodaira types and Tamagawa numbers of elliptic curves over global function fields, and constructs fields with prescribed zeta function values at specific points.

Contribution

It provides explicit formulas for densities of elliptic curve invariants over global function fields and shows the existence of fields with controlled zeta function values.

Findings

Formulas for densities of Kodaira types and Tamagawa numbers.

Existence of global function fields with specified zeta function values.

Independence of formulas from the characteristic of the field.

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 $1+\epsilon$.