2000/06/08 by Stephen D. Miller, Miller, Stephen D.
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.math/0006058
47 pages, + 7 page appendix chart available at http://www.math.yale.edu/users/steve/sl3
arxiv created 2000/06/08 · openalex publication_date 2000/06/08 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We develop a partial trace formula which circumvents some technical difficulties in computing the Selberg trace formula for the quotient SL3(\Z)\backslash 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)\backslash SL3(\R)/SO3(\R) as well as non-self-dual ones. A self-contained description of our proof for SL2(\Z)\backslash \U is included to convey the main new ideas. Heavy use is made of truncation and the Maass-Selberg relations.