Densities for Elliptic Curves over Global Function Fields

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 $1+\epsilon$.