The $p$-adic Riemann Hypothesis For Expnonential Sums

TL;DR

This paper investigates the $p$-adic Riemann hypothesis for exponential sums' $L$-functions, defining Frobenius polygons and proving the hypothesis in specific cases, with general results on Newton polygons for large primes.

Contribution

It introduces the Frobenius polygon for generic polynomials and proves the $p$-adic Riemann hypothesis for certain cases, advancing understanding of exponential sums over finite fields.

Findings

Proves the $p$-adic Riemann hypothesis for $n=2$ and $p ot\equiv -1 mod d$.

Shows the Newton polygon lies above the Frobenius polygon for large $p$.

Defines the Frobenius polygon for generic polynomials.

Abstract

The $L$-function of exponential sums associated to the generic polynomial of degree $d$ in $n$ variables over a finite field of characteristic $p$ is studied. A polygon called the Frobenius polygon of the generic polynomial of degree $d$ in $n$ variables over a finite field of characteristic $p$ is defined. A $p$-adic Riemann hypothesis is formulated. It asserts that the Newton polygon of the $L$-function coincides with the Frobenius polygon when $p$ is large enough. This $p$-adic Riemann hypothesis is proved when $n=2$ and $p\equiv-1({\rm mod }\ d)$. In general, it is proved that the Newton polygon of the $L$-function lies above the Frobenius polygon with coincide endpoints when $p$ is large enough.