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.
N
 A: 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 $\mathbb{Q}$.
A: Examples of non-rigid Calabi-Yau varieties can be found in the paper
  http://arxiv.org/abs/0812.4450
