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

Stephen D. Miller
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Abstract

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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Stephen D. Miller. 2000-06-08. On the Existence and Temperedness of Cusp Forms for SL(3,Z). https://arxiv.org/abs/math/0006058

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