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arXiv · 1212.4308

Modularity of Calabi--Yau varieties: 2011 and beyond

Noriko Yui
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Abstract

This paper presents the current status on modularity of Calabi-Yau varieties since the last update in 2003. We will focus on Calabi-Yau varieties of dimension at most three. Here modularity refers to at least two different types: arithmetic modularity and geometric modularity. These will include: (1) the modularity (automorphy) of Galois representations of Calabi-Yau varieties (or motives) defined over Q or number fields, (2) the modularity of solutions of Picard--Fuchs differential equations of families of Calabi-Yau varieties, and mirror maps (mirror moonshine), (3) the modularity of generating functions of invariants counting certain quantities on Calabi-Yau varieties, and (4) the modularity of moduli for families of Calabi-Yau varieties.

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Noriko Yui. 2012-12-18. Modularity of Calabi--Yau varieties: 2011 and beyond. https://arxiv.org/abs/1212.4308

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