Automorphic distributions, L-functions, and Voronoi summation for GL(3)

By Stephen D. Miller, W. Schmid. 2004

208 citations recorded by the source corpus.

This paper is third in a series of three, following "Summation Formulas, from Poisson and Voronoi to the Present" (math.NT/0304187) and "Distributions and Analytic Continuation of Dirichlet Series" (math.FA/0403030). The first is primarily an expository paper explaining the present one, whereas the second contains some distributional machinery used here as well. These papers concern the boundary distributions of automorphic forms, and how they can be applied to study questions about cusp forms on semisimple Lie groups. The main result of this paper is a Voronoi-style summation formula for the Fourier coefficients of a cusp form on GL(3,Z)\GL(3,R). We also give a treatment of the standard L-function on GL(3), focusing on the archimedean analysis as performed using distributions. Finally a new proof is given of the GL(3)xGL(1) converse theorem of Jacquet, Piatetski-Shapiro, and Shalika. This paper is also related to the later papers math.NT/0402382 and math.NT/0404521.

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

View Paper: https://www.semanticscholar.org/paper/6db3e7a7dc1276b1369bb3bcf6d40f57e9af2d85

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In Algebra and Number Theory

1. R. Munshi
2. Xiaoqing Li
3. Nicolas Templier
4. Atsushi Ichino
5. V. Blomer
6. Djordje Milićević

In Geometry and Topology

1. Xiaoqing Li
2. R. Munshi

In Mathematical Physics

1. R. Munshi
2. Xiaoqing Li
3. Nicolas Templier
4. Atsushi Ichino
5. V. Blomer
6. Djordje Milićević

Citations in the corpus

• The Central value of the Rankin–Selberg L-Functions (2008, methodology): “Proposition 2.1. ([ MS1 ], [GL]) Let �(x) ∈ C ∞ c (0, ∞). Let A(n,m) denote the (n,m)-th Fourier coefficient of a Maass form forSL(3, Z) as in (2.1).”
• Riemann's zeta function and beyond (2003, unknown): “Using very clever and intricate arguments, Hecke was able to express his L-series in terms of generalized “θ-functions”, and to then derive their analytic continuation, functional equation, and boundedness in vertical strips, a la Riemann.”
• The circle method and bounds for L-functions---II: Subconvexity for twists of GL(3) L-functions (2012, methodology): “The Voronoi summation formula (see [7], [8]) will play a crucial role in our analysis.”
• Bounds for GL(3)×GL(2) L -functions and GL(3) L -functions (2008, methodology): “The Voronoi formula for GL(3) was first derived by Miller and Schmidt [MS] (see [GL] for a simple proof).”
• Cancellation in additively twisted sums on GL(n) (2004, unknown): “Our assumption in Theorem 1.3 is that the L -function L ( s ) = ∞ a n − s , assumed to not be the Riemann ζ -function, is the standard L -function of a cusp form GL ( m ) over Q .”