Hello,
Are there any examples of varieties which are not Shimura varieties or abelian varieties and whose Lfunctions have been shown to be a product of automorphic Lfunctions?
Thanks.
N
Hello, Are there any examples of varieties which are not Shimura varieties or abelian varieties and whose Lfunctions have been shown to be a product of automorphic Lfunctions? Thanks. N 


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 2dimensional orthogonal Galois representations. Rigid CalabiYau 3folds 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 BarnetLamb 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 $\mathbb{Q}$. 


Examples of nonrigid CalabiYau varieties can be found in the paper http://arxiv.org/abs/0812.4450 

