Counting Points on Dwork Hypersurfaces and -Adic Hypergeometric Functions

Abstract

We express the number of points on the Dwork hypersurfaceover a finite field of order q ≡ 1 (mod d) in terms of McCarthy's p-adic hypergeometric function for any odd prime d.

Hypergeometric functions and relations to Dwork hypersurfaces

“…After the initial submission of this paper to the Arxiv, Barman, Rahman, and Saikia have demonstrated the validity of this conjecture in [2].…”

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].…”

“…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].…”

“…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 :…”

Distribution of values of Gaussian hypergeometric functions

“…These properties include transformation laws, explicit evaluations, and contiguous relations. These functions have played central roles in the study of combinatorial supercongruences [1,3,36,43,46,47,51,54,55,56,57,58], Dwork hypersurfaces [9,45], Galois representations [40,41], L-functions of elliptic curves [6,10,11,25,39,44,52,60,63], hyperelliptic curves [7,8], K3 surfaces [4,19,52], Calabi-Yau threefolds [2,3,64], the Eichler-Selberg trace formula [24,25,26,27,38,48,58,59], among other topics.…”