On unit root formulas for toric exponential sums

TL;DR

This paper generalizes the computation of p-adic unit roots for exponential sums on tori using Dwork's dual theory and hypergeometric functions, extending previous results to more general, nondegenerate sums without cohomological methods.

Contribution

It introduces a noncohomological approach to express p-adic unit roots of exponential sums on tori via hypergeometric functions, broadening applicability beyond prior nondegeneracy assumptions.

Findings

Expressed p-adic unit roots in terms of hypergeometric functions

Extended results to arbitrary exponential sums without nondegeneracy conditions

Used Dwork's relative dual theory and classical Bessel function series

Abstract

Starting from a classical generating series for Bessel functions due to Schlomilch, we use Dwork's relative dual theory to broadly generalize unit-root results of Dwork on Kloosterman sums and Sperber on hyperkloosterman sums. In particular, we express the (unique) p-adic unit root of an arbitrary exponential sum on the torus in terms of special values of the p-adic analytic continuation of a ratio of A-hypergeometric functions. In contrast with the earlier works, we use noncohomological methods and obtain results that are valid for arbitrary exponential sums without any hypothesis of nondegeneracy.