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CM values and Fourier coefficients of harmonic Maass forms

Alfes, Claudia (2015):
CM values and Fourier coefficients of harmonic Maass forms.
TU Darmstadt, [Online-Edition: http://tuprints.ulb.tu-darmstadt.de/4458],
[Ph.D. Thesis]

Abstract

In this thesis, we show that the Fourier coefficients of certain half-integral weight harmonic Maass forms are given as ``twisted traces'' of CM values of integral weight harmonic Maass forms. These results generalize work of Zagier, Bruinier, Funke, and Ono on traces of CM values of harmonic Maass forms of weight 0 and -2.

We utilize two theta lifts: one of them is a generalization of the Kudla-Millson theta lift considered by Bruinier, Funke, and Ono and the other one is defined using a theta kernel recently studied by Hövel.

Both of the lifts have interesting applications. For instance, we show that the vanishing of the central derivative of the Hasse-Weil zeta function of an elliptic curve over the rational numbers is encoded by the Fourier coefficients of a harmonic Maass form arising from the Weierstrass zeta-function of the elliptic curve.

Item Type: Ph.D. Thesis
Erschienen: 2015
Creators: Alfes, Claudia
Title: CM values and Fourier coefficients of harmonic Maass forms
Language: English
Abstract:

In this thesis, we show that the Fourier coefficients of certain half-integral weight harmonic Maass forms are given as ``twisted traces'' of CM values of integral weight harmonic Maass forms. These results generalize work of Zagier, Bruinier, Funke, and Ono on traces of CM values of harmonic Maass forms of weight 0 and -2.

We utilize two theta lifts: one of them is a generalization of the Kudla-Millson theta lift considered by Bruinier, Funke, and Ono and the other one is defined using a theta kernel recently studied by Hövel.

Both of the lifts have interesting applications. For instance, we show that the vanishing of the central derivative of the Hasse-Weil zeta function of an elliptic curve over the rational numbers is encoded by the Fourier coefficients of a harmonic Maass form arising from the Weierstrass zeta-function of the elliptic curve.

Divisions: 04 Department of Mathematics
04 Department of Mathematics > Algebra
Date Deposited: 22 Mar 2015 20:55
Official URL: http://tuprints.ulb.tu-darmstadt.de/4458
URN: urn:nbn:de:tuda-tuprints-44587
Referees: Bruinier, Prof. Dr. Jan Hendrik and Ono, Prof PhD Ken and Scheithauer, Pro.f Dr. Nils
Refereed / Verteidigung / mdl. Prüfung: 5 February 2015
Alternative Abstract:
Alternative abstract Language
In der vorliegenden Dissertation wird gezeigt, dass die Fourier-Koeffizienten gewisser harmonischer Maaß Formen halb-ganzen Gewichts die getwisteten Spuren von CM-Werten von harmonischen Maaß Formen ganzen Gewichts sind. Diese Ergebnisse verallgemeinern Arbeiten von Zagier, Bruinier, Funke und Ono über die Spuren von CM-Werten von harmonischen Maaß Formen von Gewicht 0 und -2. Wir betrachten zwei Thetaliftungen, den sogenannten Kudla-Millson und den Bruinier-Funke Thetalift, um diese Resultate zu erhalten. Beide Liftungen haben interessante Anwendungen. Insbesondere kann mit Hilfe des Bruinier-Funke Lifts gezeigt werden, dass das Verschwinden der zentralen Ableitung der Hasse-Weil Zeta-Funktion einer elliptischen Kurve über den rationalen Zahlen mit der Algebraizität der Spur von CM-Werten einer zu der elliptischen Kurve assoziierten harmonischen Maaß Form zusammenhängt.UNSPECIFIED
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