cohomology of moduli spaces - MathOverflow

cohomology of moduli spaces

2009-10-16T21:59:52Z

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. $H^3(A_{2,N})$).

Answer by JSE for cohomology of moduli spaces

2009-10-17T00:55:42Z

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?

Answer by Andy Putman for cohomology of moduli spaces

2009-10-17T01:11:29Z

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 This was also determined independently by Perron (unpublished) and M. Sato.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

Answer by Andy Putman for cohomology of moduli spaces

2009-11-06T04:31:53Z

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.

Answer by Vinoth for cohomology of moduli spaces

2009-12-24T10:34:03Z

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 :

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).