Publikationsserver der RWTH Aachen University
On Hermitian theta series and modular forms
Abstract
dc:descriptionIn their paper glqq Hermitian quadratic forms and Hermitian modular formsgrqq ~(1978) the authors D.M. Cohen und H.L. Resnikoff introduce the construction of Hermitian modular forms using theta series. One starts with lattices over imaginary quadratic fields, which are, when considered as $mathbb{Z}$-modules, unimodular with respect to the trace form and will be denoted by $vartheta$-lattices within the thesis. We start with an introduction into the theory of lattices, followed by an introduction into Hermitian modular forms. Equipped with basic knowledge on lattices and modular forms we cite the central results of Cohen and Resnikoff and add some facts. Therefore one then is interested in finding such $vartheta$-lattices. Furthermore one is interested in the modular forms constructed using these lattices. We start with the Eisenstein field $mathbb{Q}(sqrt{-3})$ with discriminant $D=-3$ and classify the $vartheta$-lattices in rank $n=4$, $n=8$ and $n=12$ using the fact that a primitive third root of unity induces an automorphism of the $vartheta$-lattices which then can be identified in the automorphism groups of the associated $mathbb{Z}$-modules. Using dimension formulas and fourier coefficients we compute the filtration of the associated cusp forms, but cannot decide whether a certain form vanishes of degree $3$ or $2$. Furthermore we develop a mass formula in case of prime discriminants which can be used to verify if a set of lattices yields a system of representatives of isometry classes of a genus of $vartheta$-lattices. The method does not work if the underlying ring of integers of the imaginary quadratic field just contains the trivial units. So we introduce the neighbour-method. This explicit construction of lattices in a given genus was originally invented by Kneser. Schiemann adopted it in his paper glqq Classification of Hermitian Forms with the Neighbour Methodgrqq{} to field extensions $E/F$ of degree $2$. As a first application we try to extend the classification of $vartheta$-lattices over the Eisenstein field onto rank $16$, because there exist over a billion even and unimodular $mathbb{Z}$-lattices of rank $32$ and testing all the automorphism groups therefore is impossible. Furthermore we construct the $vartheta$-lattices of rank $32$ with root systems of full rank using codes and describe further methods of constructing those lattices. Then we turn to other imaginary quadratic fields of class number $1$ and classify some genera of $vartheta$-lattices using the mass formula and the neighbour method. In the other cases we give estimations of the number of isometry classes. In summary we clear the situation for imaginary quadratic fields except a few cases, whereat the situation was already known in the case of the Gaussian number field. In some cases we are able to compute the filtration of cusp forms. Finally we have a short look at the cases of imaginary quadratic fields of class number different from $1$. In these cases the $vartheta$-lattices are not necessarily free modules. First we show that the theta series construction applied to $vartheta$-lattices again yields Hermitian modular forms. In a short remark we describe problems appearing when one tries to classify $vartheta$-lattices, when the class number is different from $1$. This also gives a possibility for a natural continuation of this work.
Degree
thesis:*- Grantor dc:publisher
- Publikationsserver der RWTH Aachen University
- Year dc:date
- 2009
Author and committee
dc:creator, dc:contributor.*- Author dc:creator
-
- Hentschel, Michael
- Contributors dc:contributor
-
- Krieg, Aloys
Subjects
dc:subject × 9Rights
dc:rights- Statement dc:rights
-
- info:eu-repo/semantics/openAccess
- Language dc:language
- eng