A Whipple $_7F_6$ formula revisited

Abstract

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 defined over Q. In this case, by the work of Katz, Beukers, Cohen, and Mellit, there are compatible families of -adic representations of the absolute Galois group of 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 modular forms occurred.

On Some Hypergeometric Supercongruence Conjectures of Long

“…To fully prove Conjecture 1.2 from Theorem 1.4 it would be necessary to generalize Dwork's work for these cases and to establish the expected modularity of the Galois representations associated to our hypergeometric data by Katz [11,12] and Beukers, Cohen, and Mellit [2]. Li, Long, and Tu [14] have recently proven this modularity for the hypergeometric data corresponding to the first three tuples (r 1 , r 2 , q) listed in Conjecture 1.2. The Dirichlet characters and modular forms corresponding to these hypergeometric data are listed below in Figure 1.…”