整体超椭圆函数场基础结构中的漏洞
Holes in the Infrastructure of Global Hyperelliptic Function Fields
摘要：
证明了亏格为$g$的超椭圆函数域$K$与有限常数域$\F_q$和$n + 1$的构造中的“洞元”$H(K)$的个数为无穷大，其中$n' + 1$为1次，满足$|\frac{H(K)}{\abs{\Pic^0(K)}} - \frac{n'}{q}| = O(16^g n q^{-3/2}).$我们得到一个明确的公式，孔的数量只使用信息的无限的地方和系数的$L$ -超椭圆函数域的多项式这证明了E. Landquist和作者关于一个全局函数场的基础结构的孔的数量。此外，我们调查的情况下$n = n'$孔的大小，并表明，渐近$n \to \infty$，孔的大小旁边的一个减少的除数$D$的行为像功能$\frac{n^{g - °D}}{(g - °D)!}$。
英文摘要：
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 $\frac{n^{g - °D}}{(g - °D)!}$.
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