In this paper we consider the class of K3 surfaces defined as hypersurfaces in weighted projective space, that admit a non-symplectic automorphism of non-prime order, excluding the orders 4, 8, and 12. We show that on these surfaces the Berglund-Hübsch-Krawitz mirror construction and mirror symmetry for lattice polarized K3 surfaces constructed by Dolgachev agree; that is, both versions of mirror symmetry define the same mirror K3 surface. IntroductionSince its discovery by physicists nearly 30 years ago, mirror symmetry has been the focus of much interest for both physicists and mathematicians. Although mirror symmetry has been "proven" physically, we have much to learn about the phenomenon mathematically. When we speak of mirror symmetry mathematically, there are many different constructions or rules for determining when a Calabi-Yau manifold is "mirror" to another. The constructions are often formulated in terms of families of Calabi-Yau manifolds. A natural question is whether, in a situation where more than one version can apply, they produce the same mirror (or mirror family). In this article, we consider two versions of mirror symmetry for K3 surfaces, and show that in this case the answer is affirmative, as we might expect.The first version of mirror symmetry of interest to us is known as BHK mirror symmetry. This was formulated by Berglund-Hübsch [10], Berglund-Henningson [9] and Krawitz [22] for Landau-Ginzburg models. Using the ideas of the Landau-Ginzburg/Calabi-Yau correspondence, BHK mirror symmetry also produces a version of mirror symmetry for certain Calabi-Yau manifolds (see Section 2).In the BHK construction, one starts with a quasihomogeneous and invertible polynomial W and a group G of symmetries of W satisfying certain conditions (see Section 2.2 for more details). From this data, we obtain the Calabi-Yau (orbifold) defined as the hypersurface Y W,G = {W = 0} /G. Given an LG pair (W, G), BHK mirror symmetry allows to obtain another LG pair (W T , G T ) satisfying the same conditions, and therefore another Calabi-Yau (orbifold) Y W T ,G T . We say that Y W,G and Y W T ,G T form a BHK mirror pair. In our case, we resolve singularities to obtain K3 surfaces X W,G and X W T ,G T , which we call a BHK mirror pair. When no confusion arise, we will denote these mirror K3 surfaces simply by X and X T , respectively.Another form of mirror symmetry for K3 surfaces, which we will call LPK3 mirror symmetry, is described by Dolgachev in [17]. LPK3 mirror symmetry says that the mirror family of a given K3 surface admitting a polarization by a lattice M is the family of K3 surfaces polarized by the mirror lattice M ∨ . We say that the two K3 surfaces are LPK3 mirror when they are lattice polarized and they belong to LPK3 mirror families (see details in Section 2.1).Returning to the question posed earlier, one can ask whether the BHK mirror symmetry and LPK3 mirror symmetry produce the same mirror. A similar question was considered by Belcastro in [8]. She considers a family of K3 surfaces that arise as (th...

We prove a big monodromy result for a smooth family of complex algebraic surfaces of general type, with invariants pg = q = 1 and K 2 = 3, that has been introduced by Catanese and Ciliberto. This is accomplished via a careful study of degenerations. As corollaries, when a surface in this family is defined over a finitely generated extension of Q, we verify the semisimplicity and Tate conjectures for the Galois representation on the middle ℓ-adic cohomology of the surface.

Following an idea of Ishida, we develop polynomial equations for certain unramified double covers of surfaces with pg = q = 1 and K 2 = 2. Our first main result provides an explicit surface X with these invariants defined over Q that has Picard number ρ(X) = 2, which is the smallest possible for these surfaces. This is done by giving equations for the double coverX of X, calculating the zeta function of the reduction ofX to F 3 , and extracting from this the zeta function of the reduction of X to F 3 ; the basic idea used in this process may be of independent interest.Our second main result is a big monodromy theorem for a family that contains all surfaces with pg = q = 1, K 2 = 2, and K ample. It follows from this that a certain Hodge correspondence of Kuga and Satake, between such a surface and an abelian variety, is motivated (and hence absolute Hodge). This allows us to deduce our third main result, which is that the Tate Conjecture in characteristic zero holds for all surfaces with pg = q = 1, K 2 = 2, and K ample.

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