Hypergeometric type identities in the $p$-adic setting and modular forms

Abstract

We prove hypergeometric type identities for a function defined in terms of quotients of the p-adic gamma function. We use these identities to prove a supercongruence conjecture of Rodriguez-Villegas between a truncated 4F3 hypergeometric series and the Fourier coefficients of certain weight four modular form.

A finite field hypergeometric function associated to eigenvalues of a Siegel eigenform

“…One of the more interesting applications of hypergeometric functions over finite fields is their links to elliptic modular forms and in particular Hecke eigenforms [1,3,8,11,12,13,21,22,24,27,28]. It is anticipated that these links represent a deeper connection that also encompasses Siegel modular forms of higher degree, and the purpose of this paper is to provide new evidence in this direction.…”

Summation identities and special values of hypergeometric series in the p-adic setting

“…The first and second authors [5,6] provide transformations for 2 G 2 [· · · ] q by counting points on various families of elliptic curves over F q . Recently, the third author and Fuselier [10] provide two more transformations for n G n [· · · ] p when n = 3 and n = 4, respectively. They also provide two transformations for n G n [· · · ] p for any n. However, these transformations are over F p .…”

Supercongruences Occurred to Rigid Hypergeometric Type Calabi–Yau Threefolds

“…a supercongruence originally conjectured by van Hamme in [54]. The later development of the method towards proving some other instances of (1) was undertaken in [21,40]. Quite remarkably, a somewhat simpler companion identity associated to (6) exists,…”

“…The cases (r 1 , r 2 ) = ( 1 2 , 1 2 ) and ( 1 5 , 2 5 ) have been obtained earlier by Kilbourn [29] and McCarthy [40], respectively. Furthermore, the reduction of case (r 1 , r 2 ) = ( 1 2 , 1 4 ) to Kilbourn's result in [29] has been performed by McCarthy and Fuselier [21]. The remaining eleven cases are new.…”