Hypergeometric Functions and Relations to Dwork Hypersurfaces

The number of $\mathbb{F}_p$-points on Dwork hypersurfaces and hypergeometric functions

“…Corollary 2.3 is a slight generalization of the result we mentioned in Section 1 for the case n is prime and p ≡ 1 (mod n), which was conjectured by Goodson [8] and proven by Barman et al in [1].…”

“…Theorem 4.1 can also be proved directly using the point counting technique in [17]. This technique is also often used to establish results involving finite field hypergeometric functions [1,2,7,8,13,14].…”

“…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). In [8], 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 in [1].…”

Counting Points on Dwork Hypersurfaces and -Adic Hypergeometric Functions

“…Koblitz [11] developed a formula for the number of points on diagonal hypersurfaces in the Dwork family in terms of Gauss sums. In [6], H. Goodson specializes Koblitz's formula to the family of Dwork K3 surfaces. She gives an expression for the number of points on the family of Dwork K3 surfaces X 4 λ :…”