Abstract
We study the hypergeometric functions associated to five one-parameter deformations of Delsarte K3 quartic hypersurfaces in projective space. We compute all of their Picard–Fuchs differential equations; we count points using Gauss sums and rewrite this in terms of finite-field hypergeometric sums; then we match up each differential equation to a factor of the zeta function, and we write this in terms of global L-functions. This computation gives a complete, explicit description of the motives for these pencils in terms of hypergeometric motives.
Highlights
1.1 Motivation There is a rich history of explicit computation of hypergeometric functions associated to certain pencils of algebraic varieties
Our main theorem (Theorem 1.4.1 below) shows that hypergeometric functions are naturally associated to this collection of Delsarte hypersurface pencils in two ways: as Picard–Fuchs differential equations and as traces of Frobenius yielding point counts over finite fields
In previous work [18], we showed that these five pencils share a common factor in their zeta functions, a polynomial of degree 3 associated to the hypergeometric Picard– Fuchs differential equation satisfied by the holomorphic form—see recent work of Kloosterman [36]
Summary
1.1 Motivation There is a rich history of explicit computation of hypergeometric functions associated to certain pencils of algebraic varieties. The link between the study of Picard–Fuchs equations and point counts via hypergeometric functions has intrigued many mathematicians. 73], and Candelas–de la Ossa–Rodríguez-Villegas considered the factorization of the zeta function for the Dwork pencil of Calabi–Yau threefolds in [9,10], linking physical and mathematical approaches. Given a finite-field hypergeometric function defined over Q, Beukers– Cohen–Mellit [3] construct a variety whose trace of Frobenius is equal to the finite-field hypergeometric sum up to certain trivial factors. 1.2 Our context In this paper, we provide a complete factorization of the zeta function and more generally a factorization of the L-series for some pencils of Calabi–Yau varieties, namely families of K3 surfaces.
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