A useful approach to finding dimensions of spaces of Siegel cusp forms is to investigate the representation theory of GSp(2n). We can translate results in representation theory to results on spaces of cusp forms and vice-versa. As observed by Harish-Chandra, cuspidal representations are the building blocks for the representation theory of certain groups in a way analogous to the construction of Eisenstein series from cusp forms. More precisely, a cusp form f\in S_k(\Gamma(N)) gives rise to a cuspidal automorphic representation (\pi,V) of {\rm GSp}(2n,\mathbb{A}) and vice-versa. These cuspidal automorphic representations can be written in terms of local components. The local components of the automorphic representation in turn give rise to local components of the cusp form. The dimensions of these spaces tell us essentially how many choices we have for the local factors of the representation and therefore the number of choices of local cusp forms.

Let F be a non-archimedean local field of characteristic zero with ring of integers \mathfrak{o} and maximal ideal \mathfrak{p} such that \mathfrak{o}/\mathfrak{p} is isomorphic to \mathbb{F}_q, the finite field of order q=p^n for p an odd prime. We consider the group {\rm GSp}(4,F) and hence Siegel modular forms of degree 2. By the properties of our field F we have {\rm GSp}(4,\mathfrak{o}/\mathfrak{p})\cong {\rm GSp}(4,\mathbb{F}_q). We define the congruence subgroup of level \mathfrak{p}^n, denoted by \Gamma(\mathfrak{p}^n), by

\Gamma(\mathfrak{p}^n)=\{g\in {\rm GSp}(4, \mathfrak{o})\, :\, g\equiv I\, ({\rm mod}\, \mathfrak{p}^n)\}

For the maximal compact subgroup K = {\rm GSp}(4,\mathfrak{o}) and an admissible representation (\pi, V) of {\rm GSp}(4, F), K acts on the space V^{\Gamma(\mathfrak{p})} of vectors in V fixed by the action of the congruence subgroup \Gamma(\mathfrak{p}). This space is finite dimensional by the admissibility of the representation. By definition, \Gamma(\mathfrak{p}) acts trivially on this space and so we have a more interesting action of the group K/\Gamma(\mathfrak{p})\cong{\rm GSp}(4,\mathfrak{o}/\mathfrak{p})\cong{\rm GSp}(4,\mathbb{F}_q). We can then determine the dimension of V^{\Gamma(\mathfrak{p})} by looking at the finite group analogue of \pi.

An investigation of this finite group analogue yields information that is then translated to the language of modular forms. We can then obtain results such as the dimension of a space of cusp forms. We have these results for all such local fields F that satisfy the conditions above. Our local method can then be used for global results on cusp forms, such as dimension formulas for the space S_k(\Gamma(N)) of Siegel cusp forms on the principal congruence subgroup of odd square-free level N.

I thought the following video would be a good start to this blog. It is of the announcement of John Tate receiving the 2010 Abel Prize at the Norwegian Academy of Science and Letters on March 24, 2010

John Tate wins the Abel Prize 2010

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