The number of \mathbbF_q-points on diagonal hypersurfaces with monomial deformation

2023/08/03 by Dermot McCarthy, McCarthy, Dermot
Mathematics · #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2308.01522

openalex publication_date 2023/08/03 · openalex created_date 2023/08/18 · openalex updated_date 2026/07/28

Abstract

We consider the family of diagonal hypersurfaces with monomial deformation D_{d, λ, h}: x₁^d + x₂^d … + x_n^d - d λ x₁^h₁ x₂^h₂ … x_n^h_n=0 where d = h₁+h₂ +… + h_n with gcd(h₁, h₂, … h_n)=1. We first provide a formula for the number of \mathbbF_q-points on D_{d, λ, h} in terms of Gauss and Jacobi sums. This generalizes a result of Koblitz, which holds in the special case d | q-1. We then express the number of \mathbbF_q-points on D_{d, λ, h} in terms of a p-adic hypergeometric function previously defined by the author. The parameters in this hypergeometric function mirror exactly those described by Koblitz when drawing an analogy between his result and classical hypergeometric functions. This generalizes a result by Sulakashna and Barman, which holds in the case gcd(d,q-1)=1. In the special case h₁ = h₂ = … =h_n = 1 and d=n, i.e., the Dwork hypersurface, we also generalize a previous result of the author which holds when q is prime.

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