Some explicit cases of the Selberg trace formula for vector valued functions

Stopple, Jeffrey

doi:10.1090/S0002-9947-1989-0939806-9

Some explicit cases of the Selberg trace formula for vector valued functions
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Trans. Amer. Math. Soc. 316 (1989), 281-293 Request permission

Abstract:

The trace formula for $SL(2,{\mathbf {Z}})$ can be developed for vector-valued functions which satisfy an automorphic condition involving a group representation $\pi$. This paper makes this version explicit for the class of representations which can be realized as representations of the finite group $PSL(2,{\mathbf {Z}}/q)$ for some prime $q$. The body of the paper is devoted to computing, for the singular representations $\pi$, the determinant of the scattering matrix $\Phi (s,\pi )$ on which the applications depend. The first application is a version of the Roelcke-Selberg conjecture. This follows from known results once the scattering matrix is given. The study of representations of $SL(2,{\mathbf {Z}})$ in finite-dimensional vector spaces of (scalar-valued) holomorphic forms dates back to Hecke. Similar problems can be studied for vector spaces of Maass wave forms, with fixed level $q$ and eigenvalue $\lambda$. One would like to decompose the natural representation of $SL(2,{\mathbf {Z}})$ in this space, and count the multiplicities of its irreducible components. The eigenvalue estimate obtained for vector-valued forms is equivalent to an asymptotic count, as $\lambda \to \infty$, of these multiplicities.

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