Abstract

Let X be a Calabi-Yau variety of dimension d. We will confine ourselves to Calabi-Yauvarieties of small dimensions, e.g., d ≤ 3. Dimension one Calabi–Yaus are elliptic curves, those of dimensiontwo are K3 surfaces, and dimension three ones are Calabi-Yau threefolds. Geometry and physics are bothvery much in evidence in Calabi-Yau varieties over the field of complex numbers. Today I will focus on Calabi-Yau varieties defined over the field Q of rational numbers (or number fields),and will discuss the modularity/automorphy of Calabi-Yau varieties in the framework of the LanglandsPhilosophy. In the last thirty years or so, we have witnessed tremendous advances on the modularityquestions for Calabi-Yau varieties. iMost of these results rest on the modularity of the two-dimensionalGalois representations associated to them. In the first part of the lecture, I will present these fascinating modularity results for Calabi–Yau varietiesover Q of dimension ≤ 3, as well as conjectures. Then I will discuss the current research in progess on the rank 4 weight 3 Calabi–Yau motives over Qand their Siegel modularity.