2021/03/16 by Wen-Ching Winnie Li, Линг Лонг, Li, Wen-Ching Winnie +3 · 2 citations
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Advanced Mathematical Identities
paper · pdf · doi:10.48550/arxiv.2103.08858
A well-known formula of Whipple relates certain hypergeometric values 7F6(1) and 4F3(1). In this paper we revisit this relation from the viewpoint of the underlying hypergeometric data HD, to which there are also associated hypergeometric character sums and Galois representations. We explain a special structure behind Whipple's formula when the hypergeometric data HD are primitive and self-dual. If the data are also defined over \mathbb Q, by the work of Katz, Beukers, Cohen, and Mellit, there are compatible families of ℓ-adic representations of the absolute Galois group of \mathbb Q attached to HD. For specialized choices of HD, these Galois representations are shown to be decomposable and automorphic. As a consequence, the values of the corresponding hypergeometric character sums can be explicitly expressed in terms of Fourier coefficients of certain modular forms. We further relate the hypergeometric values 7F6(1) in Whipple's formula to the periods of these modular forms.