Universität zu Köln

Cesana, Giulia (2023). On the Asymptotic Behavior of Modular Forms and Related Objects. PhD thesis, Universität zu Köln.

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Abstract

This thesis consists of research articles on the asymptotic behavior of modular forms and various related objects. First we determine the bivariate asymptotic behavior of Fourier coefficients for a wide class of eta-theta quotients with simple poles in the upper half plane by employing a variant of Wright’s Circle Method. These kind of quotients show up in many different areas not only in mathematics. For example they show up in investigations into Vafa– Witten invariants or the counting of so-called BPS-states via wall-crossing, but also in Watson’s quintuple product formula which has many applications in number theory and combinatorics. Further, we offer a general framework to prove asymptotic equidistribution, convexity, and log-concavity of coefficients of generating functions in arithmetic progressions. We do this by using a variant of Wright’s Circle Method and give a selection of different examples of such results for various (modular typed) objects. We end the thesis by employing the Circle Method to prove exact formulae for Fourier coefficients of an infinite family of weight zero mixed false modular forms showing up as characters of modules of rational vertex operator algebras.

Item Type:	Thesis (PhD thesis)
Translated title:	Title Language Über das asymptotische Verhalten von Modulformen und verwandten Objekten German
Creators:	Creators Email ORCID ORCID Put Code Cesana, Giulia gcesana@math.uni-koeln.de UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-704050
Date:	11 July 2023
Place of Publication:	Köln
Language:	English
Faculty:	Faculty of Mathematics and Natural Sciences
Divisions:	Faculty of Mathematics and Natural Sciences > Department of Mathematics and Computer Science > Mathematical Institute
Subjects:	Mathematics
Uncontrolled Keywords:	Keywords Language Circle Method English Modular Objects English Fourier coefficients English Asymptotic Analysis English
Date of oral exam:	11 July 2023
Referee:	Name Academic Title Bringmann, Kathrin Prof. Dr. Zwegers, Sander Prof. Dr.
Refereed:	Yes
URI:	http://kups.ub.uni-koeln.de/id/eprint/70405