Holes in the Infrastructure of Global Hyperelliptic Function Fields

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 $n^{\prime }+1$ 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 $n=n^{\prime }$, and show that asymptotically for $n\to \infty$, the size of a hole next to a reduced divisor $D$ behaves like the function $\frac{n^{g-°D}}{(g-°D)!}$.

arXiv comment30 pages; corrected a problem in the first version