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Automorphic partial differential equ...

DeCelles, Amy Therese.

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

Record Type:	Electronic resources : Monograph/item
Title/Author:	Automorphic partial differential equations and spectral theory with applications to number theory.
Author:	DeCelles, Amy Therese.
Description:	111 p.
Notes:	Source: Dissertation Abstracts International, Volume: 72-08, Section: B, page: 4697.
Notes:	Adviser: Paul B. Garrett.
Contained By:	Dissertation Abstracts International72-08B.
Subject:	Applied Mathematics.
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3457055
ISBN:	9781124670690

Automorphic partial differential equations and spectral theory with applications to number theory.
DeCelles, Amy Therese.

Automorphic partial differential equations and spectral theory with applications to number theory. - 111 p.

Source: Dissertation Abstracts International, Volume: 72-08, Section: B, page: 4697.

Thesis (Ph.D.)--University of Minnesota, 2011.

While proofs of the Riemann hypothesis and the Lindelof 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 Poincare series suitable for proving a subconvex bound for Rankin-Selberg convolutions for GL n x GLn over totally complex number fields. The Poincare series, with transparent spectral expansion, is obtained by winding-up a free space fundamental solution for the operator (Delta - lambdaz)nu on the free space G/K. As a sample application, not obviously related to subconvexity, a Perron transform extracts, from the Poincare 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.

ISBN: 9781124670690Subjects--Topical Terms:

530992
Applied Mathematics.

Automorphic partial differential equations and spectral theory with applications to number theory.
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020 $a 9781124670690
035 $a (UMI)AAI3457055
035 $a AAI3457055
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100 1 $a DeCelles, Amy Therese. $3 603144
245 1 0 $a Automorphic partial differential equations and spectral theory with applications to number theory.
300 $a 111 p.
500 $a Source: Dissertation Abstracts International, Volume: 72-08, Section: B, page: 4697.
500 $a Adviser: Paul B. Garrett.
502 $a Thesis (Ph.D.)--University of Minnesota, 2011.
520 $a While proofs of the Riemann hypothesis and the Lindelof 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 Poincare series suitable for proving a subconvex bound for Rankin-Selberg convolutions for GL n x GLn over totally complex number fields. The Poincare series, with transparent spectral expansion, is obtained by winding-up a free space fundamental solution for the operator (Delta - lambdaz)nu on the free space G/K. As a sample application, not obviously related to subconvexity, a Perron transform extracts, from the Poincare 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.
590 $a School code: 0130.
650 4 $a Applied Mathematics. $3 530992
650 4 $a Mathematics. $3 184409
690 $a 0364
690 $a 0405
710 2 $a University of Minnesota. $b Mathematics. $3 603145
773 0 $t Dissertation Abstracts International $g 72-08B.
790 1 0 $a Garrett, Paul B., $e advisor
790 1 0 $a Odlyzko, Andrew M. $e committee member
790 1 0 $a Diaconu, Calin A. $e committee member
790 1 0 $a Lawson, Tyler D. $e committee member
790 $a 0130
791 $a Ph.D.
792 $a 2011
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3457055