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 provide a sufficient condition for a general hypersurface in a Q-Fano toric variety to be a Calabi-Yau variety in terms of its Newton polytope. Moreover, we define a generalization of the Berglund-Hübsch-Krawitz construction in case the ambient is a Q-Fano toric variety with torsion free class group and the defining polynomial is not necessarily of Delsarte type. Finally, we introduce a duality between families of Calabi-Yau hypersurfaces which includes both Batyrev and Berglund-Hübsch-Krawitz mirror constructions. This is given in terms of a polar duality between pairs of polytopes ∆ 1 ⊆ ∆ 2 , where ∆ 1 and ∆ * 2 are canonical.{W = 0} ⊂ P(w)/G ←→ {W * = 0} ⊂ P(w * )/G * ,

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In this paper we classify the elliptic fibrations on K3 surfaces which are the double cover of a blow up of P 2 branched along rational curves and we give equations for many of these elliptic fibrations. Thus we obtain a classification of the van Geemen-Sarti involutions (which are symplectic involutions induced by a translation by a 2-torsion section on an elliptic fibration) on such a surface. Each van Geemen-Sarti involution induces a 2-isogeny between two K3 surfaces, which is described in this paper.