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Does anyone know if there's any reference on the $\ell$-adic cohomology of some simple moduli spaces/Shimura varieties, like Siegel moduli varieties $A_{g,N}$ of genus $g$ and level $N,$ for small $g$ and $N?$ Like dimensions of the cohomology spaces and the weights. Thanks.

Edit: I'm particularly interested in the weights of these $\ell$-adic cohomology of moduli varieties defined over finite field, or even the precise Frobenius eigenvalues, for the purpose of independence of $\ell$ and automorphy. Therefore I would like to know $H^i$ for all $i,$ in particular the middle cohomology (e.g. $H^3(A_{2,N})$).

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4 Answers

Let me tell you what I know about the cohomology of congruence subgroups of Sp_{2g}(\Z). As far as cohomology with rational coefficients goes, this was determined by Borel. In the limit as g->\infty, it is isomorphic to a polynomial algebra with generators in degrees 4k+2. See his paper

A. Borel, Stable real cohomology of arithmetic groups, Ann. Sci. ´Ecole Norm. Sup. (4) 7 (1974), 235–272 (1975).

I don't know of many integral calculations. I calculated H1 of the level L congruence subgroups for L odd and g at least 3 in my paper "The abelianization of the level L mapping class group", which is available on my webpage (click my name for a link). This was also determined independently by Perron (unpublished) and M. Sato. Sato's paper is "The abelianization of the level 2 mapping class group", and is available on the arXiv. He also works out H_1 for L even.

Another paper with information on H^2 is my paper "The Picard group of the moduli space of curves with level structures", which is also available on my webpage.

As a remark, both of the papers of myself mentioned above are really papers about the mapping class group and the moduli space of curves, but I ended up proving results about PPAV's and Sp_{2g}(\Z) along the way

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My apologies for forgetting to mention your paper, Andy! –  JSE Oct 17 '09 at 1:35
    
No problem, Jordan! As a side note, I was very surprised when I was working on these problems that I found so few papers on them. I have a feeling that more must be known, but it probably is written in a fancier language. If anyone knows, I'd be very interested to find out! –  Andy Putman Oct 17 '09 at 1:51
    
Thank you both, and I'll check out these papers. –  shenghao Oct 17 '09 at 3:53
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This question is quite old, but I just remembered another relevant paper. Namely, in his paper "The rational cohomology ring of the moduli space of abelian 3-folds" (available here), Hain determines the rational cohomology rings (including the weights) for both A_{g} (g=2 or 3) and their Satake compactifications.

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thanks, Andy! == –  shenghao Nov 11 '09 at 0:59
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For the Siegel modular varieties, you're just asking for the cohomology of the symplectic group Sp_{2g}(Z) and or some of its congruence subgroups; your lit search may work better for material on the cohomology of arithmetic groups than for cohomology of moduli spaces.

I it will be easier to find statements about H^i(A_{g,N}) where i is small relative to g; is that the sort of thing you need, or do you need to know the cohomology in all degrees?

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This is for Shimura varieties only; I've read only parts of the introduction so I don't know exactly what's done, but both seem to be related to $l$-adic cohomology of Shimura varieties :

  1. Taylor & Harris's paper "Some geometry and cohomology of simple Shimura varieties" - this might contain some relevant things, but is very lengthy; in the introduction it mentions "we are able to identify the action of the decomposition group at a prime of bad reduction on the l-adic cohomology of the “simple” Shimura varieties studied by Kottwitz", so I presume it is related somewhat. See http://people.math.jussieu.fr/~preprints/pdf/227.pdf .

2.Kottwitz - "$\lambda$-adic representations associated to some simple Shimura varieties"; this doesn't quite do $\ell$-adic cohomology, from what I've read in the introduction, but I think what it does ($\lambda$-adic representations) is related. It is cited as a main reference in Taylor & Harris's paper. (This one's on MathSciNet).

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Thanks rajamanikkam. Kottwitz's \lambda-adic rep is about alternating sums of compact supp cohomologies (resp. intersection cohomologies) of a Shimura variety as a virtual Galois rep. For intersection cohom one can even recover the individual IH^i from the Euler characteristic since the weights are different. In the comp supp cohom case one cannot recover the individual H^i_c in general, I guess...For indep of \ell, Gabber proved indep of \ell for intersection cohom in general (i.e. not just Shimura varieties but all compact varieties), and results in Kottwitz didn't give indep of \ell for H_c –  shenghao Dec 24 '09 at 17:38
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