This is my plenary talk presented at the workshop “String-Math 2015” at Sanya.
We consider Calabi–Yau varieties of dimension d 3 defined over Q, and address the modular-
ity/automorphy question of such Calabi–Yau varieties. When the dimension of the associated Galois rep-
resentations are large, e.g., > 2, the problem poses a serious challenge and is out of reach in the general
situations.
In this talk, I will concentrate on Calabi–Yau varieties of CM type, and establish their (motivic) mod-
ularity/automorphy. The presentation is focused on the two examples: K3 surfaces with non-symplectic
automorphisms, and Calabi–Yau threefolds of Borcea–Voisin type. If time permits, we will discuss arith-
metic mirror symmetry for mirror pairs of Calabi–Yau varieties at CM points.