Dwork crystals III: from excellent Frobenius lifts towards supercongruences

TL;DR

This paper advances the understanding of supercongruences by leveraging Cartier operations and excellent Frobenius lifts, especially in the context of Calabi-Yau varieties, extending Dwork's foundational work.

Contribution

It introduces a novel approach connecting Frobenius lifts with supercongruences for rational function coefficients, expanding their application beyond elliptic and abelian varieties.

Findings

Supercongruences proven for expansion coefficients of rational functions.

Excellent Frobenius lifts identified as key to supercongruences.

Examples provided for Calabi-Yau families.

Abstract

This paper is a continuation of our Dwork crystals series. Here we exploit the Cartier operation to prove supercongruences for expansion coefficients of rational functions. In the process it appears that excellent Frobenius lifts are a driving force behind supercongruences. Originally introduced by Dwork, these excellent lifts have occurred rather infrequently in the literature, and only in the context of families of elliptic curves and abelian varieties. In the final sections of this paper we present a list of examples that occur in the case of families of Calabi-Yau varieties.