Number of \mathbbF_q-points on Diagonal hypersurfaces and hypergeometric function

2022/10/21 by Sulakashna, Barman, Rupam
#11G25 #11S80 #11T24 #33E50 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2210.11732

Abstract

Let D_λ^d denote the family of monomial deformations of diagonal hypersurface over a finite field \mathbbF_q given by D_λ^d: X₁^d+X₂^d+⋯+X_n^d=λd X₁^h₁X₂^h₂⋯ X_n^h_n, where d,n≥2, h_i≥1, ∑_i=1ⁿ h_i=d, and gcd(d,h₁,h₂,…, h_n)=1. The Dwork hypersurface is the case when d=n, that is, h₁=h₂=⋯ =h_n=1. Formulas for the number of \mathbbF_q-points on the Dwork hypersurfaces in terms of McCarthy's p-adic hypergeometric functions are known. In this article we provide a formula for the number of \mathbbF_q-points on D_λ^d in terms of McCarthy's p-adic hypergeometric function which holds for d≥ n.

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