Frobenius map and the p -adic Gamma function

Abstract

In this note we study the relationship between the power series expansion of the Dwork exponential and the Mahler expansion of the p -adic Gamma function. We exploit this relationship to prove that certain quantities that appeared in our computations in Shapiro (2009) [12] can be expressed in terms of the derivatives of the p -adic Gamma function at 0. We then use this to prove the claim made in Shapiro (2009) [12] about the non-trivial off-diagonal entry in the Frobenius matrix of the mirror quintic threefold.