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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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  • $\begingroup$ I don’t think this answers the question, which is about abelian varieties over global fields, not over finite fields. Milne’s article does not discuss the Hasse-Weil L-function over global fields, and I do not think the fact that the higher cohomology groups are exterior powers of H^1 makes it straightforward to relate their global L-functions to that of H^1. $\endgroup$
    – Vik78
    Commented Nov 4 at 17:40
  • $\begingroup$ The L-function of an Abelian variety over a number field is the product of L-function over finite fields. So the exterior power calculation takes care of the Euler products at the good primes. The fun part of course is with the bad primes and additional work is needed, but for CM Abelian varieties this is standard (Abelian) Artin conductor calculation, at least when the base field contains the CM field. $\endgroup$
    – W Sao
    Commented Dec 5 at 16:52

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