On the Existence and Temperedness of Cusp Forms for SL3(Z)

We develop a partial trace formula which circumvents some technical difficulties in computing the Selberg trace formula for the quotient SL3(Z)\SL3(R)/SO3(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 SL3(Z)\SL3(R)/SO3(R) as well as non-self-dual ones. A self-contained description of our proof for SL2(Z)\H is included to convey the main new ideas. Heavy use is made of truncation and the Maass-Selberg relations.