An Explicit Factorisation of the Zeta Functions of Dwork Hypersurfaces
نویسندگان
چکیده
Let Fq be a finite field with q elements, ψ a non-zero element of Fq, 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 Fq defined by Xψ : x n 1 + · · · + x n n − nψx1 . . . xn = 0 has, when n is prime and Xψ is non singular (i.e. when ψ 6= 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 Xψ 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”.
منابع مشابه
Isotypic decomposition of the cohomology and factorization of the zeta functions of Dwork hypersurfaces
The aim of this article is to illustrate, on the Dwork hypersurfaces xn1+· · ·+x n n−nψx1 . . . xn = 0 (with n an integer ≥ 3 and ψ ∈ F∗ q a parameter satisfying ψ n 6= 1), how the study of the representation of a finite group of automorphisms of a hypersurface in its etale cohomology allows to factor its zeta function.
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