Modularity of fibres in rigid local systems

Abstract

It is believed that any p-adic Galois representation which is potentially semistable arises from a modular form. The main theorem of Wiles establishes this modularity when the representation in question satisfies various technical restrictions, together with the key hypothesis that its reduction modulo p arises itself from a modular form. This article explains how a strong version of Wiles' "lifting theorem" implies the modularity of all hypergeometric abelian varieties - so-called because their periods are expressed in terms of values of classical hypergeometric functions. While this strong version remains unproved, various strengthenings of Wiles' original result, notably those arising from work of Wiles and Skinner, allow one to unconditionally establish the modularity of infinite collections of hypergeometric abelian varieties.