Densities for Elliptic Curves over Global Function Fields
Andrew Yao
TL;DR
The paper develops a comprehensive, characteristic-free framework for densities of Kodaira types and Tamagawa numbers for elliptic curves over completions of a global function field, using Tate's algorithm and Haar-measure techniques. It proves that local density formulas depend only on $Q_P$ and extends these to global densities via a product-measure approach, including a precise multiplicativity formula $\delta_K(\mathfrak{r},n,k;P)=Q_P^{-10k}\delta_K(\mathfrak{r},n,0;P)$. The author provides explicit local density values for a wide range of Kodaira types and their starred variants across primes $p\ge 5$, $p=3$, and $p=2$, and shows how to assemble these into global Tamagawa-number densities with nontrivial lower bounds (e.g., $\ge \zeta_K(2)^{-1}$). Finally, the paper constructs global function fields with zeta-functions arbitrarily close to $1$ for any $s>1$, by engineering curves with controlled point counts, thereby demonstrating the practical reach of the density framework in arithmetic statistics and its potential applicability to number-field analogues.
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 $ε$ such that $s>1$ and $ε>0$, 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 $1+ε$.