Generic Newton Slopes for Artin-Schreier-Witt Tower in two variables

TL;DR

This paper proves that for generic two-variable polynomials over the rationals, the Newton slopes of associated character series and L-functions at large primes are independent of characters and follow a predictable pattern.

Contribution

It establishes the independence of Newton slopes from characters and describes their progression for generic polynomials in two variables over the rationals.

Findings

Newton slopes are independent of the character for large primes.

Newton slopes of L-functions form a weighted arithmetic progression.

Results hold for generic polynomials of specified degrees.

Abstract

We prove that for any generic polynomial $f$ in two variables of degree $(d_1,d_2)$ over the rationals, for $p$ large enough the Newton slopes of the character power series $C_f^*(\chi_m,s)$ of $f$ at $p$ is independent of the choice of the character $\chi_m$ (of conductor $p^m$); and the Newton slopes of the $L$-function $L_f^*(\chi_m,s)$ of $f$ at $p$ is in weighted arithmetic progression.