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

Author
Kosters, Michiel · Zhu, Hui June
Year 2017
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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.

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Details

Title
On slopes of $L$-functions of $\mathbb{Z}_p$-covers over the projective line
Author
Kosters, Michiel · Zhu, Hui June
Year
2017
Type
Research Article
Language
eng
Comment
20 pages
History
2017-01-30 00:00:00
Categories
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.