Abstract/Details

Automorphic partial differential equations and spectral theory with applications to number theory


2011 2011

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Abstract (summary)

While proofs of the Riemann hypothesis and the Lindelöf hypothesis remain elusive, for some number-theoretic applications any bound that surpasses the “trivial” or “convex” bound for the growth of an L-function, i.e. any subconvex bound, suffices. In this paper, we construct a Poincaré series suitable for proving a subconvex bound for Rankin-Selberg convolutions for GL n × GLn over totally complex number fields. The Poincaré series, with transparent spectral expansion, is obtained by winding-up a free space fundamental solution for the operator (Delta - lambdaz)ν on the free space G/K. As a sample application, not obviously related to subconvexity, a Perron transform extracts, from the Poincaré series, information about the number of lattice points in an expanding region in G/K, and from the spectral expansion, terms corresponding to the automorphic spectrum of the Laplacian. The result is an explicit formula relating the automorphic spectrum to the number of lattice points in an expanding region. A global automorphic Sobolev theory as well as a zonal spherical Sobolev theory legitimize derivations and manipulations of spectral expansions. This line of inquiry is relevant not only to the hoped-for subconvexity result but also to the development of techniques applicable to harmonic analysis of automorphic forms on higher rank groups.

Indexing (details)


Subject
Applied Mathematics;
Mathematics
Classification
0364: Applied Mathematics
0405: Mathematics
Identifier / keyword
Applied sciences; Pure sciences; Asymptotics of l-functions; Automorphic spectral theory; Fundamental solutions; Higher rank groups; Lattice-point counting; Partial differential equations; Poincare series
Title
Automorphic partial differential equations and spectral theory with applications to number theory
Author
DeCelles, Amy Therese
Number of pages
111
Publication year
2011
Degree date
2011
School code
0130
Source
DAI-B 72/08, Dissertation Abstracts International
Place of publication
Ann Arbor
Country of publication
United States
ISBN
9781124670690
Advisor
Garrett, Paul B.
Committee member
Diaconu, Calin A.; Lawson, Tyler D.; Odlyzko, Andrew M.
University/institution
University of Minnesota
Department
Mathematics
University location
United States -- Minnesota
Degree
Ph.D.
Source type
Dissertations & Theses
Language
English
Document type
Dissertation/Thesis
Dissertation/thesis number
3457055
ProQuest document ID
873564958
Copyright
Database copyright ProQuest LLC; ProQuest does not claim copyright in the individual underlying works.
Document URL
http://search.proquest.com/docview/873564958
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