Frobenius map and the $p$-adic Gamma function

TL;DR

This paper explores the connection between the Dwork exponential's power series and the Mahler expansion of the $p$-adic Gamma function, using this to analyze the Frobenius map in mirror symmetry.

Contribution

It establishes a relationship between two expansions and uses it to express Frobenius map quantities via derivatives of the $p$-adic Gamma function, proving a conjecture.

Findings

Expressed Frobenius map entries in terms of $p$-adic Gamma derivatives

Proved a conjecture about the Frobenius matrix of the mirror quintic

Linked Dwork exponential and Mahler expansion in $p$-adic analysis

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 previous computations of the Frobenius map can be expressed in terms of the derivatives of the $p$-adic Gamma function at 0. This is used to prove a conjecture about the non-trivial off-diagonal entry in the Frobenius matrix of the mirror quintic threefold.