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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by Jeffrey Stopple

Trans. Amer. Math. Soc. 316 (1989), 281-293

DOI: https://doi.org/10.1090/S0002-9947-1989-0939806-9

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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.

References

Erich Hecke, Mathematische Werke, 3rd ed., Vandenhoeck & Ruprecht, Göttingen, 1983 (German). With introductory material by B. Schoeneberg, C. L. Siegel and J. Nielsen. MR 749754
Dennis A. Hejhal, The Selberg trace formula for $\textrm {PSL}(2,\,\textbf {R})$. Vol. 2, Lecture Notes in Mathematics, vol. 1001, Springer-Verlag, Berlin, 1983. MR 711197, DOI 10.1007/BFb0061302
Heinz Huber, Über die Darstellungen der Automorphismengruppe einer Riemannschen Fläche in den Eigenräumen des Laplace-Operators, Comment. Math. Helv. 52 (1977), no. 2, 177–184 (German). MR 492233, DOI 10.1007/BF02567363
H. D. Kloosterman, The behaviour of general theta functions under the modular group and the characters of binary modular congruence groups. I, Ann. of Math. (2) 47 (1946), 317–375. MR 21032, DOI 10.2307/1969082
Hans Maass, Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 121 (1949), 141–183 (German). MR 31519, DOI 10.1007/BF01329622

Complex representations of

for finite fields

Karl Prachar, Primzahlverteilung, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1957 (German). MR 0087685
Burton Randol, A remark on the multiplicity of the discrete spectrum of congruence groups, Proc. Amer. Math. Soc. 81 (1981), no. 2, 339–340. MR 593486, DOI 10.1090/S0002-9939-1981-0593486-1
A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.) 20 (1956), 47–87. MR 88511
Jean-Pierre Serre, Linear representations of finite groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott. MR 0450380

A functional equation for some Selberg zeta functions

Jeffrey Stopple, Selberg zeta functions with virtual characters and the class number, Acta Arith. 54 (1989), no. 1, 37–42. MR 1024415, DOI 10.4064/aa-54-1-37-42
Shun’ichi Tanaka, Construction and classification of irreducible representations of special linear group of the second order over a finite field, Osaka Math. J. 4 (1967), 65–84. MR 219635