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Since $\Gamma(N)$ is normal in $\mathrm{SL}(2,\mathbb{Z})$, the quotient group $\mathrm{SL}(2,\mathbb{Z}/N)$ acts on the spaces of cusp forms $S_k(\Gamma(N))$. How do these spaces decompose into irreducible representations?

I can do the case $N=2$. I'm mostly interested in the case of $N$ a prime.

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I wonder if this question of mine is related:… – Ilya Nikokoshev Nov 9 '09 at 20:08
up vote 4 down vote accepted

See Theorem 1.0.3 of Jared Weinstein's phd thesis (it uses equivariant Riemann Roch).

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As usual, once I spot a question on here I have anything useful to say about, somebody has already answered it.

I can sum up that part of my thesis this way: let M be the induced representation of the character (-I) --> (-1)^k of the center up to all of SL(2,Z/NZ). Then S_k(Gamma(N)) is roughly k/12 copies of M, plus some error term which can be given precisely, with some effort.

When you ask instead about Hilbert modular forms over a totally real field K, the "1/12" becomes the absolute value of zeta_K(-1).

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Thank you moonface and Jared, that's perfect. I wish I could give more than one accepted answer. :) – Dan Petersen Nov 10 '09 at 8:35

If you think about this question in terms of automorphic representations then it sort of becomes trivial. The space $Sk(\Gamma(N))$ can be re-interpreted as the direct sum of $\pi^{U(N)}$, where $\pi$ is running through the automorphic representations of $GL_2$ which are holomorphic of weight $k$. Each factor is $SL(2,Z/NZ)$-invariant and often irreducible but sometimes has small finite length. The representation of $SL(2,Z/NZ)$ that shows up on $\pi{^U(N)}$ is the "type" of $\pi$. For explicit $\pi$s one will be able to explicitly determine this representation.

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I'm not sure that they are irreducible (at least if I'm thinking about GL_2(F_p)) - e.g. can't I get things like the induction of the trivial character on the Borel showing up? – TSG Nov 9 '09 at 23:48

You can almost do this with nothing more than Riemann-Hurwitz; in particular, by R-H you can compute the action of SL_2(Z/NZ) on H_1(X(N),C), which is just the sum of the representation you want with its dual.

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The great thing about this approach is that it's completely different to the automorphic forms approach, and if the two are used together then they give information about the number of automorphic forms of a given "type". Sort of a generalisation of counting how many level 1 forms there are by computing the dimension of the space of level 1 forms by some global method. – Kevin Buzzard Nov 10 '09 at 7:38

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