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

By Stephen D. Miller. 2000

54 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0c98ea16e1d6ee942de3548341d17f9a6f75c32b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Erez Lapid
2. Akshay Venkatesh
3. Elon Lindenstrauss
4. Werner Mueller
5. Peter Sarnak
6. V. Blomer
7. Gergely Harcos
8. P. Michel

In Geometry and Topology

1. Akshay Venkatesh
2. Peter Sarnak
3. Elon Lindenstrauss
4. Werner Mueller
5. Erez Lapid
6. P. Michel
7. Manfred Einsiedler
8. Kannan Soundararajan

In Mathematical Physics

1. Akshay Venkatesh
2. Erez Lapid
3. Peter Sarnak
4. Elon Lindenstrauss
5. Werner Mueller
6. P. Michel
7. V. Blomer
8. Kannan Soundararajan

Citations in the corpus

• Spectra of hyperbolic surfaces (2003, methodology): “For X = SL(3;Z)nSL(3;R)=SO(3) this is done in [ Mi ] and for the general congruence quotient of SL(n;R)=SO(n )i n [Mu].”
• Existence and Weyl’s law for spherical cusp forms (2005, methodology): “/Kwhere Gis a semisimple algebraic group defined over Q, K⊂ G(R) a maximal compact subgroup, and Γ ⊂ G(Q) a congruence subgroup. 1 2 ELON LINDENSTRAUSS AND AKSHAY VENKATESH In his thesis, S. D. Miller [16] established this conjecture for G = PGL(3). More recently, W. Mu¨ller [18] has established this”
• SPECTRAL ASYMPTOTICS FOR ARITHMETIC QUOTIENTS OF SL(n,R)/SO(n) (2007, methodology): “ by Avakumovi´c [Ava56] and this was generalized to any elliptic pseudo-differential operator by H¨ormander [H¨or68].) The first example of a non-uniform lattice in higher rank was treated by S. Miller [Mil01] who proved that for SL(3,Z), the cuspidal spectrum satisfies Weyl’s law and the tempered cusp”