On a supercongruence conjecture of Rodriguez-Villegas

Abstract

Abstract. In examining the relationship between the number of points over Fp on certain Calabi-Yau manifolds and hypergeometric series which correspond to a particular period of the manifold, Rodriguez-Villegas identified numerically 22 possible supercongruences. We prove one of the outstanding supercongruence conjectures between a special value of a truncated generalized hypergeometric series and the p-th Fourier coefficient of a modular form.

Hypergeometric evaluation identities and supercongruences

“…McCarthy [2009] proved another supercongruence of this type and his approach provides a general combinatorial framework for all these congruences. We will establish a few supercongruences mainly via hypergeometric evaluation identities and combinatorics.…”

The number of -points on Dwork hypersurfaces and hypergeometric functions

“…This technique is also often used to establish results involving finite field hypergeometric functions [2,3,11,13,19,21]. We define the Teichmüller character to be the primitive character ω :…”

“…Formulas for counting the number of points on algebraic varieties over finite fields using hypergeometric functions are of special interest. The hypergeometric functions involved often display interesting properties, in particular, their links to Fourier coefficients of modular forms [1,[10][11][12]20,21] and to the periods of the variety [5,6,13,24]. To date, these formulas have focused on single varieties and have been developed on an ad hoc basis.…”

“…He then highlights the analogy between this formula and the Barnes integral for classical hypergeometric series. In [21], this author provided a simple formula for the number of F p -points on the Dwork hypersurface, in the case n = 5 and λ = 1, in terms of a finite field hypergeometric function, when p ≡ 1 (mod 5). We then extended this result to all odd primes using a hypergeometric type function defined in terms of the p-adic gamma function (see Definition 2.1 below).…”

On the divisibility of the truncated hypergeometric function F23

“…For more congruences on truncated hypergeometric function, the readers may refer to [4,9,11,5,6,14,12]. Motivated by the above results, in this short note, we shall consider the truncated hypergeometric function 3 F 2…”