The modularity/automorphy of Calabi-Yau varieties of CM type

Yui, Noriko

doi:10.1090/pspum/096/01659

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The modularity/automorphy of Calabi-Yau varieties of CM type

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Yui, Noriko
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.1090/pspum/096
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引用

Yui, N. (2017). The modularity/automorphy of Calabi-Yau varieties of CM type. In String-Math 2015: December 31, 2015-January 4, 2016, Tsinghua Sanya International Mathematics Forum, Sanya, China (pp. 265-297). Providence, RI: American Mathematical Society.

引用: https://hdl.handle.net/21.11116/0000-0004-6A0E-7

要旨

We consider Calabi–Yau varieties of dimension
d ≤ 3 defined over Q, and address the modularity/automorphy of such Calabi–Yau varieties. When the dimension of the associated Galois representations are large, e.g., >2, the problem poses a serious challenge and is out of reach in the general situations.
In this paper, I will concentrate on Calabi–Yau varieties of CM type, and establish their (motivic) modularity/automorphy. The expositions are focused on two examples: K3 surfaces with non-symplectic automorphisms, and Calabi-Yau threefolds of Borcea–Voisin type. We will briefly discuss arithmetic mirror symmetry for quite specific examples of K3 surfaces and Calabi–Yau threefolds of Borcea–Voisin type.