Zeta function factorisation, Dwork hypersurfaces, hypergeometric hypersurfaces

2009/12/09 by Goutet, Philippe
#11G25 (Secondary) #14G10 (Primary) #14G15 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.0912.1685

Abstract

Let \mathbbF_q be a finite field with q elements, ψ a non-zero element of \mathbbF_q, and n an integer ≥ 3 prime to q. The aim of this article is to show that the zeta function of the projective variety over \mathbbF_q defined by X_ψ\colon x₁ⁿ+...+x_nⁿ - n ψx₁... x_n=0 has, when n is prime and X_ψ is non singular (i.e. when ψⁿ ≠ 1), an explicit decomposition in factors coming from affine varieties of odd dimension ≤ n-4 which are of hypergeometric type. The method we use consists in counting separately the number of points of X_ψ and of some varieties of the preceding type and then compare them. This article answers, at least when n is prime, a question asked by D. Wan in his article "Mirror Symmetry for Zeta Functions".

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