$p$-adic variation of unit root $L$-functions

C. Douglas Haessig and Steven Sperber

Vol. 288 (2017), No. 1, 129–156

DOI: 10.2140/pjm.2017.288.129

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, as studied by Adolphson and Sperber (2012). That is, the unit root is essentially a ratio of $\underset{̧}{A}$-hypergeometric functions.

Keywords

L-function, unit root, hypergeometric

Mathematical Subject Classification 2010

Primary: 11T23

Milestones

Received: 22 April 2016

Revised: 28 August 2016

Accepted: 7 September 2016

Published: 8 April 2017