Holes in the Infrastructure of Global Hyperelliptic Function Fields

TL;DR

This paper provides an explicit formula and bounds for the number of 'holes' in the infrastructure of hyperelliptic function fields, confirming a conjecture and analyzing the asymptotic size of holes.

Contribution

It derives an explicit formula for the number of holes in hyperelliptic function fields and proves a special case of a conjecture regarding their distribution.

Findings

The number of holes approximates n'/q with bounded error.

An explicit formula for holes depends only on infinite places and L-polynomial coefficients.

Asymptotic size of a hole near a reduced divisor behaves like n^{g - deg D}/(g - deg D)!.

Abstract

We prove that the number of "hole elements" $H(K)$ in the infrastructure of a hyperelliptic function field $K$ of genus $g$ with finite constant field $\F_q$ with $n + 1$ places at infinity, of whom $n' + 1$ are of degree one, satisfies $|\frac{H(K)}{\abs{\Pic^0(K)}} - \frac{n'}{q}| = O(16^g n q^{-3/2}).$ We obtain an explicit formula for the number of holes using only information on the infinite places and the coefficients of the $L$-polynomial of the hyperelliptic function field. This proves a special case of a conjecture by E. Landquist and the author on the number of holes of an infrastructure of a global function field. Moreover, we investigate the size of a hole in case $n = n'$, and show that asymptotically for $n \to \infty$, the size of a hole next to a reduced divisor $D$ behaves like the function $\frac{n^{g - \deg D}}{(g - \deg D)!}$.