Are there cusp forms for the full modular group Sp(2,Z) and representations det^3 \otimes Sym^2j(\rho_standard) - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T09:02:24Z http://mathoverflow.net/feeds/question/102575 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/102575/are-there-cusp-forms-for-the-full-modular-group-sp2-z-and-representations-det3 Are there cusp forms for the full modular group Sp(2,Z) and representations det^3 \otimes Sym^2j(\rho_standard) Tom 2012-07-18T20:11:12Z 2012-07-19T02:48:58Z <h1>What are modular forms or cusps forms, resp. ?</h1> <p>We start with defining their common domains $\mathbb{H}_g$ as the set of symmetric $g \times g$ matrices with positive definite imaginary parts. The symplectic group $Sp(g,\mathbb{Z})$ is the subgroup of $SL(2g,\mathbb{Z})$ such that all elements satisfy $M=J_g^t M J_g$ with $J_g$ being the canonical almost complex structure.</p> <p>$Sp(g,\mathbb{Z})$ acts on $\mathbb{H}_g$ by $M(Z)=(AZ+B)(CZ+D)^{-1}$ where A,B,C and D are the block matrix entries of M.</p> <p>Let $\rho : GL(g,\mathbb{C}) \to GL(V)$ be a rational representation on a finite dimensional $\mathbb{C}$-vector space then the associated modular forms are the holomorphic functions $f : \mathbb{H}_g \to V$ satisfying $f(M(Z))=\rho(CZ+D)f(Z)$ for all $M \in Sp(g,\mathbb{Z})$.</p> <p>Cusps forms can be easily characterized as the elements of Siegel's $\Phi$ operator's kernel.</p> <h1>Modular forms in genus 2</h1> <p>If g equals 2 then the observed representations are the ones of $GL(2,\mathbb{C})$. We know from representation theory that all irreducible representations are isomorphic to a rep of the type $det^k \otimes Sym^{2j}(\rho_{standard})$. </p> <p>We denote by $\rho_{standard}$ the standard representation $X \mapsto X$. $Sym^{2j}(\rho_{standard})$ is the associated symmetric product $GL(2,\mathbb{C}) \to Sym^{2j}(\mathbb{C}^2)$. $det$ is just the 1 dimensional determinant representation $GL(2,\mathbb{C}) \to \mathbb{C}$.</p> <p>For $k\geq 4$ Tshushima has given a dimension formula for the vector space of cusps forms in An explicit dimension formula for the spaces of generalized automorphic forms with respect to Sp(2, Z). Proceedings of the Japan Academy, Ser. A, Mathematical Sciences, 59:139–142, 1983.</p> <p>Satoh and Ibukiyama gave (but partly didn't publish AFAIK) generators for the modules of vector valued modular forms to the representations $det^k \otimes Sym^{2j}(\rho_{standard})$ with running k and fixed j in ${1,2,3}$. </p> <h1>The actual question</h1> <p>So the next question for me was are there cusps forms to $det^3 \otimes Sym^{2j}(\rho_{standard})$ and can they ( at least a single one) be given explicitly, in particular for j=4 ?</p> <p>cheers Tom</p> <p>p.s. please excuse all mistakes I made but it was the first time for me publishing on such a plattform.</p> http://mathoverflow.net/questions/102575/are-there-cusp-forms-for-the-full-modular-group-sp2-z-and-representations-det3/102613#102613 Answer by ncr for Are there cusp forms for the full modular group Sp(2,Z) and representations det^3 \otimes Sym^2j(\rho_standard) ncr 2012-07-19T02:48:58Z 2012-07-19T02:48:58Z <p>Though I haven't seen it, I've heard that Tomoya Kiyuna, a student at Kyushu University, has done this for the case $j=4$ as part of his master's thesis. In particular, he finds eighteen explicit generators of the module of vector-valued Siegel modular forms of the symmetric tensor 8. </p>