Designing Poincare Series for Number Theoretic Applications

2014/01/08 by Amy T. DeCelles, DeCelles, Amy T.
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT) #math.NT #math.RT

paper · pdf · doi:10.48550/arxiv.1401.1780

17 pages, supercedes arxiv:1104.4313. arXiv admin note: substantial text overlap with arXiv:1104.5406

arxiv created 2014/01/08 · arxiv updated 2014/01/09

Abstract

The GL₂ Poincaré series giving the subconvexity results of Diaconu and Garrett is the solution to an automorphic partial differential equation, constructed by winding-up the solution to the corresponding differential equation on the free space. Generalizing this approach allows design of higher rank Poincaré series with specific number theoretic applications in mind: a Poincaré series for producing an explicit formula for the number of lattice points in an expanding region in a symmetric space, a Poincaré series producing moments of GL_n × GL_n L-functions, and a Poincaré series designed for applications involving pseudo-Laplacians.

Citations

• Fundamental solution for (Delta - lambda_z)ⁿ on a symmetric space G/K
• An exact formula relating lattice points in symmetric spaces to the automorphic spectrum

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