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Transactions of the American Mathematical Society

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

Author: Jeffrey Stopple
Journal: Trans. Amer. Math. Soc. 316 (1989), 281-293
MSC: Primary 11F72
MathSciNet review: 939806
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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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Keywords: Scattering matrix, Roelcke-Selberg conjecture, Maass wave form
Article copyright: © Copyright 1989 American Mathematical Society