p-adic variation of unit root L-functions

TL;DR

This paper investigates the p-adic variation of unit root L-functions derived from toric exponential sums, revealing that their p-adic unit roots behave like ratios of A-hypergeometric functions, extending understanding of p-adic properties in geometric L-functions.

Contribution

It demonstrates that the p-adic unit root in a family of unit root L-functions can be expressed as a ratio of A-hypergeometric functions, linking p-adic variation to classical hypergeometric functions.

Findings

Each unit root L-function has a unique p-adic unit root.

The unit root varies similarly to classical exponential sum families.

The unit root can be expressed as a ratio of A-hypergeometric functions.

Abstract

Dwork's conjecture, now proven by Wan, states that unit root L-functions "coming from geometry" are p-adic meromorphic. In this paper we study the p-adic variation of a family of unit root L-functions coming from a suitable family of toric exponential sums. In this setting, we find that the unit root L-functions each have a unique p-adic unit root. We then study the variation of this unit root over the family of unit root L-functions. Surprisingly, we find that this unit root behaves similarly to the classical case of families of exponential sums. That is, the unit root is essentially a ratio of A-hypergeometric functions.