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The zeta function of monomial deformations of Fermat hypersurfaces. (English) Zbl 1166.14016
Let \(d\in\mathbb{N}\). The main object of study in this paper are families of Fermat hypersurfaces \[ X_\lambda: \sum^n_{i=0} x^{d_i}_i+ \lambda\prod^n_{i=0} x^{a_i}_i,\quad\lambda\in \mathbb{F}_q, \] defined over the finite field \(\mathbb{F}_q\) of \(q\) elements. Here \(d_i\) is a natural number and \(a_i\) is a nonnegative integer, satisfying the conditions: \(d_iw_i= d\) for \(0\leq i\leq n\) and \(\sum^n_{i=0} w_i a_i= d\), so that \(X_\lambda\) may be regarded as a subvariety of the weighted projective space \(\mathbb{P}(w_0,\dots, w_n)\); moreover, it is assumed that h.c.f. \((q, d)= 1\).
The main technical result of the paper implies that the \(p\)-adic Picard-Fuchs equation, associated with the family \(X_\lambda\), is a generalised hypergeometric differential equation. The author discusses the Frobenius action on the cohomology of Fermat hypersurfaces and proves some results on the structure of the zeta function of a monomial deformation of such a hypersurface. As a by-product of his investigations, the author extends and improves on some results of S. Kadir and N. Yui [Motives and mirror symmetry for Calabi-Yau orbifolds. Modular forms and string duality, Fields Institute Communications 54, 3–46 (2008; Zbl 1167.14023)].
Reviewer: B. Z. Moroz (Bonn)

14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14G15 Finite ground fields in algebraic geometry
11G25 Varieties over finite and local fields
14J70 Hypersurfaces and algebraic geometry
14G22 Rigid analytic geometry
14D10 Arithmetic ground fields (finite, local, global) and families or fibrations
Zbl 1167.14023
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