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The number of $\mathbb{F}_q$-points on diagonal hypersurfaces with monomial deformation

Dermot McCarthy

Vol. 328 (2024), No. 2, 339–359
Abstract

We consider the family of diagonal hypersurfaces with monomial deformation

Dd,λ,h : x1d + x 2d + + x nd dλx 1h1 x2h2 xnhn = 0,

where d = h1 + h2 + + hn with gcd (h1,h2, ,hn) = 1. We first provide a formula for the number of 𝔽q-points on Dd,λ,h in terms of Gauss and Jacobi sums. This generalizes a result of Koblitz, which holds in the special case dq 1. We then express the number of 𝔽q-points on Dd,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 = = hn = 1 and d = n, i.e., the Dwork hypersurface, we also generalize a previous result of the author which holds when q is prime.

Keywords
diagonal hypersurface, Gauss sum, Jacobi sum, finite field hypergeometric function, $p$-adic hypergeometric function, counting points
Mathematical Subject Classification
Primary: 11G25, 33E50
Secondary: 11S80, 11T24, 33C99
Milestones
Received: 9 December 2023
Accepted: 4 April 2024
Published: 30 April 2024
Authors
Dermot McCarthy
Department of Mathematics & Statistics
Texas Tech University
Lubbock, TX
United States

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