I try to grab a (very) little understanding of what a Shimura variety is, and although I still don't understand the formal definition of that notion, a few vague ideas have come to my mind.
Hence the following questions:
1) can the degree of an L-function $F$ (that is, an element of the Selberg class which is automorphic) be viewed as the dimension of the tangent space of a Shimura variety $X_F$ "canonically" associated to $F$?
2) is there a natural notion of "direct sum" and "tensor product" of Shimura varieties such that $X_{F.G}=X_{F}\oplus X_{G}$ and $X_{F\otimes G}=X_{F}\otimes X_{G}$ where $\forall p\in\mathbb{P}$, the $p$-th Dirichlet coefficient $a_{p}(F\otimes G)=a_{p}(F).a_{p}(G)$? In other words, is the map $X:F\mapsto X_{F}$ some kind of homomorphism of monoids?
Thank you in advance for any insight.