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Update on the modularity of Calabi-Yau varieties (With an appendix by Helena Verill). (English) Zbl 1092.11030
Yui, Noriko (ed.) et al., Calabi-Yau varieties and mirror symmetry. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3355-3/hbk). Fields Inst. Commun. 38, 307-362 (2003).
This is an update article on recent results concerning the modularity of Calabi-Yau varieties in dimension and is a sequel to two survey articles by the same author, who also provides many conjectures, questions and related research problems around the modularity [see M.-H. Saito and N. Yui, J. Math. Kyoto Univ. 41, No. 2, 403–419 (2001; Zbl 1077.14546); R. Livné and N. Yui, ibid. 45, No. 4, 645–665 (2005; Zbl 1106.14025)]. The article concludes with an appendix by H. Verill.
The introductory Section 1 is devoted to the definition of Calabi-Yau varieties and their L-functions. Section 2 introduces the modularity of Calabi-Yau varieties and related conjectures. It is known that 1-dimensional Calabi-Yau varieties (elliptic curves) over $$\mathbb Q$$ are modular. It follows from Livné’s proof of modularity of rank 2 orthogonal Galois representations of Gal$$(\overline{\mathbb Q}/ \mathbb Q)$$ that the extremal (singular) $$K3$$ surfaces over $$\mathbb Q$$ are also modular. (The author provides two proofs of this in section 4). It is conjectured that the rigid Calabi-Yau threefolds over $$\mathbb Q$$ are modular. There are, up-to-date, 35 rigid Calabi-Yau threefolds over $$\mathbb Q$$ (some of them perhaps being birational over $$\mathbb Q$$) for which this conjecture is known to be true. These threefolds are discussed in Section 5.
The modularity of non-extremal $$K3$$ surfaces and non-rigid Calabi-Yau threefolds is more subtle, since the associated Galois representations are of dimension $$>2$$. Some strategies for establishing the modularity conjecture of these varieties are given in Section 3. Section 6 contains some examples of non-rigid modular Calabi-Yau varieties. In Sections 7-8, the author discusses the Calabi-Yau varieties of CM type as a class of non-rigid Calabi-Yau varieties for which the modularity conjecture might be more accessible.
In the Appendix, (Section 9) the L-series of the rigid Calabi-Yau threefolds arising as the self-fibre products of elliptic curves are determined by Helena Verill.
For the entire collection see [Zbl 1022.00014].

MSC:
 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 14G35 Modular and Shimura varieties 14J30 $$3$$-folds 14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties 14J28 $$K3$$ surfaces and Enriques surfaces 14J32 Calabi-Yau manifolds (algebro-geometric aspects)