SL(2,Z/N)-decomposition of space of cusp forms for Gamma(N) - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T18:22:14Z http://mathoverflow.net/feeds/question/4763 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/4763/sl2-z-n-decomposition-of-space-of-cusp-forms-for-gamman SL(2,Z/N)-decomposition of space of cusp forms for Gamma(N) Dan Petersen 2009-11-09T20:06:23Z 2009-11-10T07:36:47Z <p>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? </p> <p>I can do the case $N=2$. I'm mostly interested in the case of $N$ a prime.</p> http://mathoverflow.net/questions/4763/sl2-z-n-decomposition-of-space-of-cusp-forms-for-gamman/4768#4768 Answer by Kevin Buzzard for SL(2,Z/N)-decomposition of space of cusp forms for Gamma(N) Kevin Buzzard 2009-11-09T20:20:54Z 2009-11-10T07:36:47Z <p>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.</p> http://mathoverflow.net/questions/4763/sl2-z-n-decomposition-of-space-of-cusp-forms-for-gamman/4791#4791 Answer by moonface for SL(2,Z/N)-decomposition of space of cusp forms for Gamma(N) moonface 2009-11-10T00:15:39Z 2009-11-10T00:20:12Z <p>See Theorem 1.0.3 of Jared Weinstein's <a href="http://www.math.ucla.edu/~jared/jswthesis.pdf" rel="nofollow">phd thesis</a> (it uses equivariant Riemann Roch). </p> http://mathoverflow.net/questions/4763/sl2-z-n-decomposition-of-space-of-cusp-forms-for-gamman/4800#4800 Answer by JSE for SL(2,Z/N)-decomposition of space of cusp forms for Gamma(N) JSE 2009-11-10T01:32:50Z 2009-11-10T01:32:50Z <p>You can <em>almost</em> 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.</p> http://mathoverflow.net/questions/4763/sl2-z-n-decomposition-of-space-of-cusp-forms-for-gamman/4821#4821 Answer by Jared Weinstein for SL(2,Z/N)-decomposition of space of cusp forms for Gamma(N) Jared Weinstein 2009-11-10T06:54:49Z 2009-11-10T06:54:49Z <p>As usual, once I spot a question on here I have anything useful to say about, somebody has already answered it. </p> <p>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. </p> <p>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). </p>