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 A-hypergeometric functions.
Bibliographical noteFunding Information:
We would like to thank the referee for their careful reading. Haessig was partially supported by a grant from the Simons Foundation (#314961). MSC2010: 11T23.
- Unit root