# modularity of algebraic varieties

Hello,

Are there any examples of varieties which are not Shimura varieties or abelian varieties and whose L-functions have been shown to be a product of automorphic L-functions?

Thanks.

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Too long for a comment:

Yes. One family of examples is the singular K3 surfaces - a recent paper generalizing this is

http://arxiv.org/pdf/0904.1922

This is a consequence of a result of Livné about the modularity of 2-dimensional orthogonal Galois representations. Rigid Calabi-Yau 3-folds also give examples, after Serre's conjecture, cf. the following paper:

http://arxiv.org/pdf/0902.1466

(Although this implication was already in Serre's original paper: you can deduce a similar result for any motive with the right Hodge numbers).

These examples are however "close" to abelian varieties in some sense, so you might not find them very satisfying. I don't know of any others though.

Edit: I want also to mention information that potential automorphy theorems can give you. For example, in his thesis Barnet-Lamb showed that the zeta function of the Dwork hypersurface in $\mathbb{P}^4$ has meromorphic continuation, by showing that the cohomology is automorphic after possibly restricting to a totally real field extension of $\mathbb{Q}$.

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...and good enough for an answer: +1. I would say that these K3 surfaces are "not Shimura varieties" in the sense that they are not constructed from a "Shimura datum" via the Shimura-Deligne construction. As to whether they are isomorphic to Shimura varieties: probably some of them are but "most" of them are not. But it may not be easy to point to a specific example. –  Pete L. Clark May 10 '10 at 23:10
Thanks! There are two reasons why the K3 surfaces above might not be "satisfactory": first, the motive of a K3 surface embeds always in the motive of an abelian variety (possibly of much higher dimension). Second, the only interesting part of the cohomology of the singular K3 surfaces is the transcendental cycles, and the theorem of Livné I mentioned above says that this comes from CM eigenforms - these are less "interesting" than other Galois representations you might find out there. –  user1594 May 11 '10 at 0:54

Examples of non-rigid Calabi-Yau varieties can be found in the paper http://arxiv.org/abs/0812.4450

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