title: Quaternionic Drinfeld modular forms
creator: Butenuth, Ralf
subject: ddc-510
subject: 510 Mathematics
description: Drinfeld modular forms were introduced by D. Goss in 1980 for congruence subgroups of ${\rm GL}_2(\mathbb{F}_q[T])$. They are a counterpart of classical modular forms in the function field world. In this thesis I study Drinfeld modular forms for inner forms of ${\rm GL}_2$ that correspond to unit groups $\Lambda^\star$ of quaternion division algebras over $\mathbb{F}_q(T)$ split at the place $\infty = 1/T$. I show, following work of Teitelbaum for ${\rm GL}_2(\mathbb{F}_q[T])$, that these forms have a combinatorial interpretation as certain maps from the edges of the Bruhat-Tits tree $\mathcal{T}$ associated to ${\rm PGL}_2(K_\infty)$. Here $K_\infty$ denotes the completion of $K$ at $\infty$. A major focus of this thesis is on computational aspects: I present an algorithm for computing a fundamental domain for the action of $\Lambda^\star$ on $\mathcal{T}$ with an edge pairing, and describe how to obtain a basis of the space of these forms out of this fundamental domain. On this basis one can compute the Hecke action.  
date: 2012-10-24
type: Dissertation
type: info:eu-repo/semantics/doctoralThesis
type: NonPeerReviewed
format: application/pdf
identifier: https://archiv.ub.uni-heidelberg.de/volltextserverhttps://archiv.ub.uni-heidelberg.de/volltextserver/13885/1/Dissertation_Ralf-Butenuth.pdf
identifier: DOI:10.11588/heidok.00013885
identifier: urn:nbn:de:bsz:16-heidok-138856
identifier:   Butenuth, Ralf  (2012) Quaternionic Drinfeld modular forms.  [Dissertation]     
relation: https://archiv.ub.uni-heidelberg.de/volltextserver/13885/
rights: info:eu-repo/semantics/openAccess
rights: http://archiv.ub.uni-heidelberg.de/volltextserver/help/license_urhg.html
language: eng