On the Existence and Temperedness of Cusp Forms for SL(3,Z)

Abstract

We develop a partial trace formula which circumvents some technical difficulties in computing the Selberg trace formula for the quotient $S{L}_3(\mathbb{Z})\backslash S{L}_3(\mathbb{R})/S{O}_3(\mathbb{R})$. As applications, we establish the Weyl asymptotic law for the discrete Laplace spectrum and prove that almost all of its cusp forms are tempered at infinity. The technique shows there are non-lifted cusp forms on $S{L}_3(\mathbb{Z})\backslash S{L}_3(\mathbb{R})/S{O}_3(\mathbb{R})$ as well as non-self-dual ones. A self-contained description of our proof for $SL_2({\Z})\backslash \U$ is included to convey the main new ideas. Heavy use is made of truncation and the Maass-Selberg relations.

47 pages, + 7 page appendix chart available at http://www.math.yale.edu/users/steve/sl3