$T$-adic Exponential Sums over Affinoids

TL;DR

This paper develops a new $( au,p)$-adic Dwork theory for $L$-functions of exponential sums over affinoids, generalizing existing trace formulas and analyzing the $C$-function's properties.

Contribution

It introduces a novel $( au,p)$-adic framework for exponential sums, extending Dwork theory and providing new analytic and geometric insights.

Findings

Proves a generalized trace formula for $( au,p)$-adic exponential sums.

Establishes analytic continuation of the $C$-function $C_f(s,\pi)$.

Computes the lower $(\tau,p)$-adic bound and Hodge polygon for the $C$-function.

Abstract

We introduce and develop $(\pi,p)$-adic Dwork theory for $L$-functions of exponential sums associated to one-variable rational functions, interpolating $p^k$-order exponential sums over affinoids. Namely, we prove a generalization of the Dwork-Monsky-Reich trace formula and apply it to establish an analytic continuation of the $C$-function $C_f(s,\pi)$. We compute the lower $(\pi,p)$-adic bound, the Hodge polygon, for this $C$-function. Along the way, we also show why a strictly $\pi$-adic theory will not work in this case.