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Computation of Eisenstein series associated with discriminant forms

Opitz, Sebastian (2018):
Computation of Eisenstein series associated with discriminant forms.
Darmstadt, Technische Universität,
[Ph.D. Thesis]

Abstract

In this thesis, we describe methods to compute the Fourier coefficients of Eisenstein series for the Weil representation associated to an even lattice. The known formulas depend on an even lattice and use the "local" data derived from this lattice. A python program for use within sage was written to evaluate these formulas. The Eisenstein series itself only depends on the discriminant form of the lattice, and hence depends only on the "local" data. We examine the "global" formulas to see how they can be computed purely from "local" data, which can be encoded by a genus symbol or a Jordan decomposition. A comparison of two different approaches to the computation of the Fourier coefficients leads to formulas for the Igusa local zeta function. At last we use the implemented programs to classify all Borcherds products coming from a certain class of lattices.

Item Type: Ph.D. Thesis
Erschienen: 2018
Creators: Opitz, Sebastian
Title: Computation of Eisenstein series associated with discriminant forms
Language: English
Abstract:

In this thesis, we describe methods to compute the Fourier coefficients of Eisenstein series for the Weil representation associated to an even lattice. The known formulas depend on an even lattice and use the "local" data derived from this lattice. A python program for use within sage was written to evaluate these formulas. The Eisenstein series itself only depends on the discriminant form of the lattice, and hence depends only on the "local" data. We examine the "global" formulas to see how they can be computed purely from "local" data, which can be encoded by a genus symbol or a Jordan decomposition. A comparison of two different approaches to the computation of the Fourier coefficients leads to formulas for the Igusa local zeta function. At last we use the implemented programs to classify all Borcherds products coming from a certain class of lattices.

Place of Publication: Darmstadt
Divisions: 04 Department of Mathematics
04 Department of Mathematics > Algebra
04 Department of Mathematics > Algebra > Automorphic Forms, Number Theory, Algebraic Geometry
Date Deposited: 09 Dec 2018 20:55
URL / URN: https://tuprints.ulb.tu-darmstadt.de/8261
URN: urn:nbn:de:tuda-tuprints-82611
Additional Information:

https://zenodo.org/record/1464927 https://github.com/s-opitz/eisenstein_series

PPN:
Referees: Bruinier, Prof. Dr. Jan Hendrik ; Scheithauer, Prof. Dr. Nils
Refereed / Verteidigung / mdl. Prüfung: 27 November 2018
Alternative Abstract:
Alternative abstract Language

In der vorliegenden Dissertation werden Methoden entwickelt, um die Fourierkoeffizienten spezieller Reihen, namentlich vektorwertige Eisensteinreihen zur Weildarstellung eines geraden Gitters, zu berechnen. Die bisher bekannten Formeln gehen immer von einem geraden Gitter aus und leiten von diesem die „lokalen“ Daten des Gitters ab. Zur Berechnung dieser Formeln wurde ein Programm in der Sprache python zur Benutzung mit sage geschrieben. Die Eisensteinreihe selbst hängt nur von der Diskriminantenform des Gitters ab. Vor diesem Hintergrund untersuchen wir die „globalen“ Formeln, um zu verstehen, wie sie aus den „lokalen“ Daten des Gitters, wie zum Beispiel dem Geschlechtssymbol oder der Zerlegung in Jordankomponenten, berechnet werden können. Aus dem Vergleich verschiedener Ansätze zur Berechnung der Fourierkoeffizienten der Eisensteinreihen können wir Formeln für die lokale Igusazetafunktion ableiten. Zuletzt benutzen wir die geschriebenen Programme, um alle Borcherdsprodukte, die von einer gewissen Klasse von Gittern kommen, zu klassifizieren.

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