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



modularity of algebraic varieties
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 $\mathbb{Q}$.


