Vol. 288, No. 1, 2017

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ISSN: 0030-8730
$p$-adic variation of unit root $L$-functions

C. Douglas Haessig and Steven Sperber

Vol. 288 (2017), No. 1, 129–156

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 A̧-hypergeometric functions.

L-function, unit root, hypergeometric
Mathematical Subject Classification 2010
Primary: 11T23
Received: 22 April 2016
Revised: 28 August 2016
Accepted: 7 September 2016
Published: 8 April 2017
C. Douglas Haessig
Department of Mathematics
University of Rochester
Rochester, NY 14627
United States
Steven Sperber
School of Mathematics
University of Minnesota
Minneapolis, MN 55455
%-0436 United States