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Modularity of Calabi–Yau Varieties: 2011 and Beyond

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Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds

Part of the book series: Fields Institute Communications ((FIC,volume 67))

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 \(\mathbb{Q}\) 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. The topic (4) is commonly known as geometric modularity.

Discussions in this paper are centered around arithmetic modularity, namely on (1), and (2), with a brief excursion to (3).

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Acknowledgements

I would like to thank Matthias Schütt for carefully reading the preliminary version of this paper and suggesting numerous improvements. I am indebted to Ron Livné for answering my questions about Galois representations.

I would also like to thank a number of colleagues for their comments and suggestions. This includes Jeng-Daw Yu, Ling Long, Ken-Ichiro Kimura, and Bert van Geemen.

We are grateful to V. Nikulin for allowing us to use the template of Nikulin’s pryamid in Fig. 1.

Last but not least, my sincere thanks is to the referee for reading through the earlier versions of this article and for very helpful constructive criticism and suggestions for the improvement of the article. I would also like to thank Arther Greenspoon of Mathematical Reviews for copy-editing the article.

The article was completed while the author held visiting professorship at various institutions in Japan: Tsuda College, Kavli Institute for Physics and Mathematics of the Universe, and Nagoya University. I thank the hospitality of these instutions.

The author was supported in part by NSERC Discovery Grant.

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  1. Department of Mathematics and Statistics, Queen’s University, Kingston, ON, Canada, K7L 3N6

    Noriko Yui

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  1. Noriko Yui
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Correspondence to Noriko Yui .

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  1. , Mathematics Department, Stony Brook University, N/A, Stony Brook, 11794-3651, New York, USA

    Radu Laza

  2. , Institut für Algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Matthias Schütt

  3. , Department of Math & Stats, Queen's University, University Ave., Kingston, K7L 3N6, Ontario, Canada

    Noriko Yui

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Yui, N. (2013). Modularity of Calabi–Yau Varieties: 2011 and Beyond. In: Laza, R., Schütt, M., Yui, N. (eds) Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds. Fields Institute Communications, vol 67. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6403-7_4

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