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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].

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)
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