Hypergeometric functions and relations to Dwork hypersurfaces

Abstract

Abstract. We give an expression for number of points for the family of Dwork K3 surfacesover finite fields of order q ≡ 1 (mod 4) in terms of Greene's finite field hypergeometric functions. We also develop hypergeometric point count formulas for all odd primes using McCarthy's p-adic hypergeometric function. Furthermore, we investigate the relationship between certain period integrals of these surfaces and the trace of Frobenius over finite fields. We extend this work to higher dimensional Dwork hypersurfaces.

The number of -points on Dwork hypersurfaces and hypergeometric functions

“…1 for the case n is prime and p ≡ 1 (mod n), which was conjectured by Goodson [13] and proven by Barman et al [2].…”

“…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.…”

“…We also proved that the values of this particular hypergeometric function were linearly related to the Fourier coefficients of a certain modular form. In [13], Goodson considers the n = 4 case and gives formulas for the number of F q -points in terms of finite field hypergeometric functions when q ≡ 1 (mod 4), and extends to all odd primes using this author's p-adic hypergeometric function. She also conjectures a formula in the special case that n is prime and that p ≡ 1 (mod n), which was proven by Barman et al [2].…”

Hypergeometric properties of genus 3 generalized Legendre curves

“…In [1], Ahlgren and Ono gave a formula for the number of F p points on a modular Calabi-Yau threefold. We extended this work in [9,10] by showing that the number of points on Dwork hypersurfaces over finite fields can be expressed in terms of Greene's finite field hypergeometric functions.…”