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

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

2000-06-08

arxiv.org/abs/math/0006058v1

Mathematics

Number Theory

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

Scientific paper

We develop a partial trace formula which circumvents some technical difficulties in computing the Selberg trace formula for the quotient $SL_3({\Z})\backslash SL_3({\R})/SO_3({\R})$. 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 $SL_3({\Z})\backslash SL_3({\R})/SO_3({\R})$ 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.

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