---
abstract: 'We give an elementary approach to proving the birationality of multiple Berglund-Hübsch-Krawitz (BHK) mirrors by using Shioda maps. We do this by creating a birational picture of the BHK correspondence in general. Although a similar result has been obtained in recent months by Shoemaker, our proof is new in that it sidesteps using toric geometry and drops an unnecessary hypothesis. We give an explicit quotient of a Fermat variety to which the mirrors are birational.'
address: 'Department of Mathematics, University of Pennsylvania, 209 S. 33rd St, Philadelphia, PA 19103'
author:
- 'Tyler L. Kelly'
date: 'April 11, 2013'
title: 'Berglund-Hübsch-Krawitz Mirrors via Shioda Maps'
---

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Introduction
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The mirror symmetry conjecture predicts that for a Calabi-Yau variety, $M$, there exists another Calabi-Yau variety, $W$, so that various geometric and physical data is exchanged between $M$ and $W$. A classical relationship found between so-called mirror pairs is that on the level of cohomology $$H^{p,q}(M, {\mathbb{C}}) \cong H^{N-p,q}(W, {\mathbb{C}}),$$ provided that both Calabi-Yau varieties $M$ and $W$ are $N$-dimensional. In 1992, Berglund and Hübsch proposed such a mirror symmetry relationship between finite quotients of hypersurfaces in weighted projective $n$-space [@BH]. Suppose $F_A$ is a polynomial, $$F_A = \sum_{i=0}^n \prod_{j=0}^n 