# On slopes of $L$-functions of $\mathbb{Z}_p$-covers over the projective line

Author Kosters, Michiel · Zhu, Hui June 2017 See all Details

#### Abstract

Let $P: \cdots \rightarrow C_2\rightarrow C_1\rightarrow {\mathbb P}^1$ be a $\mathbb{Z}_p$-cover of the projective line over a finite field of cardinality $q$ and characteristic $p$ which ramifies at exactly one rational point, and is unramified at other points. In this paper, we study the $q$-adic valuations of the reciprocal roots in $\mathbb{C}_p$ of $L$-functions associated to characters of the Galois group of $P$. We show that for all covers $P$ such that the genus of $C_n$ is a quadratic polynomial in $p^n$ for $n$ large, the valuations of these reciprocal roots are uniformly distributed in the interval $[0,1]$. Furthermore, we show that for a large class of such covers $P$, the valuations of the reciprocal roots in fact form a finite union of arithmetic progressions.

### Details

Title On slopes of $L$-functions of $\mathbb{Z}_p$-covers over the projective line Kosters, Michiel · Zhu, Hui June 2017 Research Article eng 20 pages 2017-01-30 00:00:00 Algebraic Geometry · Number Theory
This is Version 1 of this record. We added this version on February 1, 2017. This version is based on an original data import from arXiv.org e-Print archive.