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

TL;DR

This paper investigates the distribution of $q$-adic valuations of reciprocal roots of $L$-functions associated with $Z_p$-covers over the projective line, revealing uniform distribution and arithmetic progression structures under certain genus growth conditions.

Contribution

It establishes the uniform distribution of valuations and describes their structure as finite unions of arithmetic progressions for a broad class of $Z_p$-covers with quadratic genus growth.

Findings

Valuations are uniformly distributed in [0,1] for certain covers.

Valuations form finite unions of arithmetic progressions.

Results apply to covers with quadratic polynomial genus growth.

Abstract

Let $\mathcal{P}: \cdots \rightarrow C_2\rightarrow C_1\rightarrow {\mathbb P}^1$ 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 $\mathcal{P}$. We show that for all covers $\mathcal{P}$ 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 $\mathcal{P}$, the valuations of the reciprocal roots in fact form a finite union of arithmetic progressions.