$p$-adic Theory for Partial Toric Exponential Sums

TL;DR

This paper introduces a $p$-adic approach to analyze partial toric $L$-functions, proving their rationality and establishing a cohomological framework, with explicit formulas for their unique $p$-adic unit root.

Contribution

It provides a novel $p$-adic proof of rationality for partial toric $L$-functions and constructs a corresponding cohomology theory, extending previous $ extit{ ext{l}}$-adic methods.

Findings

Partial $L$-functions are $p$-adic meromorphic.

A $p$-adic cohomology theory for partial toric $L$-functions is developed.

The unique $p$-adic unit root is explicitly expressed via $A$-hypergeometric series.

Abstract

Wan proved the rationality of partial toric $L$-functions using $\ell$-adic techniques. In this paper, we present a $p$-adic proof in the spirit of Dwork. We demonstrate that partial $L$-functions can be expressed as an alternating product of twisted Fredholm determinants. These twisted determinants appear to be intrinsic to the analytic structure of partial $L$-functions, and unlike their classical counterparts, twisted Fredholm determinants of completely continuous operators are not automatically $p$-adic entire functions. However, for partial $L$-functions they will be $p$-adic meromorphic. After proving rationality, we construct a $p$-adic cohomology theory and give a $p$-adic cohomological formula for partial toric $L$-functions. Last, we show they have a unique $p$-adic unit root which may be explicitly written in terms of $A$-hypergeometric series.