We prove that if two Calabi-Yau invertible pencils have the same dual weights, then they share a common factor in their zeta functions. By using Dwork cohomology, we demonstrate that this common factor is related to a hypergeometric Picard-Fuchs differential equation. The factor in the zeta function is defined over the rationals and has degree at least the order of the Picard-Fuchs equation. As an application, we relate several pencils of K3 surfaces to the Dwork pencil, obtaining new cases of arithmetic mirror symmetry.

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Abstract. It is a well known result that the number of points over a finite field on the Legendre family of elliptic curves can be written in terms of a hypergeometric function modulo p. In this paper, we extend this result, due to Igusa, to a family of monomial deformations of a diagonal hypersurface. We find explicit relationships between the number of points and generalized hypergeometric functions as well as their finite field analogues.

The real multiple zeta values ζ(k 1 , . . . , k r ) are known to form a Q-algebra; they satisfy a pair of well-known families of algebraic relations called the double shuffle relations. In order to study the algebraic properties of multiple zeta values, one can replace them by formal symbols Z(k 1 , . . . , k r ) subject only to the double shuffle relations. These form a graded Hopf algebra over Q, and quotienting this algebra by products, one obtains a vector space. A difficult theorem due to G. Racinet proves that this vector space carries the structure of a Lie coalgebra; in fact Racinet proved that the dual of this space is a Lie algebra, known as the double shuffle Lie algebra ds.J. Ecalle developed a deep theory to explore combinatorial and algebraic properties of the formal multiple zeta values. His theory is sketched out in some publications (essentially [E1] and [E2]). However, because of the depth and complexity of the theory, Ecalle did not include proofs of many of the most important assertions, and indeed, even some interesting results are not always stated explicitly. The purpose of the present paper is to show how Racinet's theorem follows in a simple and natural way from Ecalle's theory.

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.