Zeta functions of projective toric hypersurfaces over finite fields

In this paper, we will investigate the zeta functions of projective toric hypersurfaces defined by Laurent polynomials. We can decompose the projective toric hypersurface into disjoint union of affine toric hypersurfaces in the algebraic tori corresponding to the nonempty faces of the polytope Delta( f). This zeta function is determined by the product of the L-functions of affine toric hypersurfaces. By the p-adic theory of Dwork, each L-function is a rational function and is determined by Dwork cohomology. When f is Delta-regular and Delta(f) is a simplex, we have some cancellations in the product of L-functions. The zeta function of the projective toric hypersurface is analogous to the zeta function of the hypersurface in Pn . We will see that the Newton polygon of the product of the L-functions is bounded below by its Hodge polygon.;Mirror Symmetry is an active area of research between Mathematics and Physics. As an application we will compute the zeta functions of the mirror pair of Calabi-Yau hypersurfaces in Mirror Symmetry.