In Shimura's paper "On the Zeta Function of an Abelian Variety With Complex Multiplication", in his terminology, the `one-dimensional part' of the zeta function is identified with a Hecke $L$-function in a wider context than previously. By one-dimensional part, it is meant the Hasse-Weil $L$-function for first etale cohomology. Of course, this is quite a reasonable restriction to make. For one thing, if the abelian variety is the jacobian of a curve, then this really is the $L$-function of the curve.

I would like to think that the Hasse-Weil $L$-functions for the other cohomology groups can be deduced from this identification. It does not seem too unreasonable, given that, at least over a separably closed field, the cup product allows us to identify the kth etale cohomology group with the kth exterior power of $H^1$ and the real substance of the above paragraph is an identification of representations. Thus, one should be able to write down the full zeta function of (a finite type over $\mathbb{Z}$ model of) an abelian variety. Either this is harder than I think, or this is simply not interesting. Or both.

Allowing myself to run with the idea that it is not interesting, I am lead to ask, is there a geometric interpretation of the L-functions of the other cohomology groups of an abelian variety? For example, for an elliptic curve these L-functions are those of the base number field (shifted a bit for H^2).


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The r-th etale cohomology of an Abelian variety -- CM or otherwise -- is the r-th exterior power of the first etale cohomology of the Abelian variety, from there you can then work out the Hasse-Weil zeta function. See e.g. Milne's article "Abelian varieties" in Cornell-Silverman, "Arithmetic Geometry".


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