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On the zeros of automorphic forms.

Jung, Junehyuk.

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On the zeros of automorphic forms.

Record Type:	Electronic resources : Monograph/item
Title/Author:	On the zeros of automorphic forms./
Author:	Jung, Junehyuk.
Description:	102 p.
Notes:	Source: Dissertation Abstracts International, Volume: 74-09(E), Section: B.
Contained By:	Dissertation Abstracts International74-09B(E).
Subject:	Mathematics. -
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3562204
ISBN:	9781303097515

On the zeros of automorphic forms.
Jung, Junehyuk.

On the zeros of automorphic forms. - 102 p.

Source: Dissertation Abstracts International, Volume: 74-09(E), Section: B.

Thesis (Ph.D.)--Princeton University, 2013.

The subject of this thesis is the zeros of automorphic forms. In the first part, we study the asymptotic behavior of nodal lines of Maass (cusp) forms on hyperbolic surfaces via taking intersection with various curves. The first result is the upper bounds for the number of intersection between nodal lines of Maass cusp forms &phis; and various fixed analytic curves. Let lambda&phis; is the Laplacian eigenvalue of &phis; and let Z&phis; be the set of nodal lines of &phis;. When Y is a compact hyperbolic surface and gamma a geodesic circle, or when Y is a non-compact hyperbolic surface with finite volume and gamma is a closed horocycle we prove that the number of intersections between Z&phis; and gamma is O( lf ).

ISBN: 9781303097515Subjects--Topical Terms:

515831
Mathematics.

On the zeros of automorphic forms.
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020 $a 9781303097515
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100 1 $a Jung, Junehyuk. $3 2049793
245 1 0 $a On the zeros of automorphic forms.
300 $a 102 p.
500 $a Source: Dissertation Abstracts International, Volume: 74-09(E), Section: B.
500 $a Adviser: Peter C. Sarnak.
502 $a Thesis (Ph.D.)--Princeton University, 2013.
520 $a The subject of this thesis is the zeros of automorphic forms. In the first part, we study the asymptotic behavior of nodal lines of Maass (cusp) forms on hyperbolic surfaces via taking intersection with various curves. The first result is the upper bounds for the number of intersection between nodal lines of Maass cusp forms &phis; and various fixed analytic curves. Let lambda&phis; is the Laplacian eigenvalue of &phis; and let Z&phis; be the set of nodal lines of &phis;. When Y is a compact hyperbolic surface and gamma a geodesic circle, or when Y is a non-compact hyperbolic surface with finite volume and gamma is a closed horocycle we prove that the number of intersections between Z&phis; and gamma is O( lf ).
520 $a The second result is a quantitative statement of the quantum ergodicity for Maass-Hecke cusp forms on X=SL2,Z \H . As an application we deduce that the number of nodal domains of &phis; which intersect a fixed geodesic segment in {iy | y > 0} ⊂ H increases with the eigenvalue, with a small number of exceptional &phis;'s.
520 $a In the second part of the thesis, we prove for various families of automorphic forms that the positive-definite automorphic forms are sparse. If pi is a self-dual cuspidal automorphic form on GLm/ Q , then we say pi is positive-definite if Lambda(1/2 + it, pi) is a positive-definite function in t ∈ R , where Lambda(s, pi) is the completed L-function attached to pi. For Maass cusp forms, the nodal line not meeting the y-axis and the positive-definiteness are the same. A holomorphic cusp form is positive-definite if and only if it has no zero on the y-axis.
520 $a In the proof we formulate an axiomatic criterion about sets of automorphic forms pi satisfying certain averages when suitably ordered, which ensures that almost all pi's are not positive-definite within such sets. We then apply the result to various well known families of automorphic forms.
590 $a School code: 0181.
650 4 $a Mathematics. $3 515831
650 4 $a Applied Mathematics. $3 1669109
690 $a 0405
690 $a 0364
710 2 $a Princeton University. $b Mathematics. $3 2049791
773 0 $t Dissertation Abstracts International $g 74-09B(E).
790 $a 0181
791 $a Ph.D.
792 $a 2013
793 $a English
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3562204