#### Vol. 288, No. 1, 2017

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

### C. Douglas Haessig and Steven Sperber

Vol. 288 (2017), No. 1, 129–156
##### 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
Primary: 11T23