A Modular Non-Rigid Calabi-Yau Threefold

Abstract

We construct an algebraic variety by resolving singularities of a quintic Calabi-Yau threefold. The middle cohomology of the threefold is shown to contain a piece coming from a pair of elliptic surfaces. The resulting quotient is a two-dimensional Galois representation. By using the Lefschetz fixed-point theorem in \'etale cohomology and counting points on the variety over finite fields, this Galois representation is shown to be modular.