THE NUMBER OF Fq-POINTS ON DIAGONAL HYPERSURFACES WITH MONOMIAL DEFORMATION

Date

2024

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Abstract

We consider the family of diagonal hypersurfaces with monomial deformation Dd,λ,h: xd1 + xd2 +· · ·+ xdn − dλ xh21 xh32… xhn n = 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 d | q −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.

Description

© 2024 The Author, under license to MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). Open Access made possible by subscribing institutions via Subscribe to Open. cc-by

Keywords

counting points, diagonal hypersurface, finite field hypergeometric function, Gauss sum, Jacobi sum, p-adic hypergeometric function

Citation

Mccarthy, D.. 2024. THE NUMBER OF Fq-POINTS ON DIAGONAL HYPERSURFACES WITH MONOMIAL DEFORMATION. Pacific Journal of Mathematics, 328(2). https://doi.org/10.2140/pjm.2024.328.339

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