Aloys Krieg

The Minnesota notes on Jordan algebras and their applications

Domains of Positivity.- Omega Domains.- Jordan Algebras.- Real and Complex Jordan Algebras.- Complex Jordan Algebras.- Jordan Algebras and Omega Domains.- Half-Spaces.- Appendix: The Bergman kernel function.

The Maaß Space and Hecke Operators.

We give a new proof of the fact that the Maaß space is invariant under all Hecke operators. It is based on the characterization of the Maaß space by a symmetry relation and certain commutation relations of the Hecke algebra for the Jacobi group. 1 Jonas Gallenkämper, Lehrstuhl A für Mathematik, RWTH Aachen, D-52056 Aachen, jonas.gallenkaemper@rwth-aachen.de 2 Bernhard Heim, GUtech, Way No. 36, Building No. 331, North Ghubrah, Muscat, Sultanate of Oman, bernhard.heim@gutech.edu.om 3 Aloys Krieg, Lehrstuhl A für Mathematik, RWTH Aachen, D-52056 Aachen, krieg@rwth-aachen.de

The Hecke-algebras related to the unimodular and modular group over the Hurwitz order of integral quaternions

In the present paper the elementary divisor theory over the Hurwitz order of integral quaternions is applied in order to determine the structure of the Hecke-algebras related to the attached unimodular and modular group of degreen. In the casen = 1 the Hecke-algebras fail to be commutative. Ifn > 1 the Hecke-algebras prove to be commutative and coincide with the tensor product of their primary components. Each primary component turns out to be a polynomial ring inn resp.n + 1 resp. 2n resp. 2n+1 algebraically independent elements. In the case of the modular group of degreen, the law of interchange with the Siegel ϕ-operator is described. The induced homomorphism of the Hecke-algebras is surjective except for the weightsr = 4n-4 andr = 4n-2.

Eisenstein-Series on the Four-Dimensional Hyperbolic Space

Abstract The paper deals with Eisenstein-series on the four-dimensional hyperbolic space in the realization of a quaternionic upper half-plane. The Rankin-Selberg method leads to a duality theorem saying that these Eisenstein-series arise from those on the complex upper half-plane by the Petersson inner product with certain theta-series and vice versa. Moreover the meromorphic continuation and a Kronecker limit formula are proved and Hecke-operators are introduced.

The elementary divisor theory over the Hurwitz order of integral quaternions

There exists a diagonal form with certain divisibility conditions for matrices over the Hurwitz order of integral quaternions under unimodular equivalence. The diagonal entries are uniquely determined up to similarity. Given two such diagonal forms, where the diagonal entries are similar by pairs, the matrices prove to be ummodularly equivalent, whenever the rank of the matrices is creater than one.

On the classification of even unimodular lattices with a complex structure

This paper classifies the even unimodular lattices that have a structure as a Hermitian -lattice of rank r ≤ 12 for rings of integers in imaginary quadratic number fields K of class number 1. The Hermitian theta series of such a lattice is a Hermitian modular form of weight r for the full modular group, therefore we call them theta lattices. For arbitrary imaginary quadratic fields we derive a mass formula for the principal genus of theta lattices which is applied to show completeness of the classifications.

The singular modular forms on the 27-dimensional exceptional domain

SummaryDue to a result of Resnikoff the non-constant singular modular forms on the 27-dimensional exceptional domain are exactly those of weight 4 and weight 8. They were constructed by Kim using analytic continuation of non-holomorphic Eisenstein series. In this paper a simpler construction is described. These modular forms arise from theta series on the Cayley half-plane of degree two, which is a 10-dimensional boundary component, by means of the Fourier-Jacobi expansion.

The theory of Jacobi forms over the Cayley numbers

M. EIE AND A. KRIEGAbstract. As a generalization of the classical theory of Jacobi forms we discussJacobi forms on /xC8 , which are related with integral Cayley numbers. Usingthe Selberg trace formula we give a simple explicit formula for the dimensionof the space of Jacobi forms. The orthogonal complement of the space of cuspforms is shown to be spanned by certain types of Eisenstein series.

The Maaß space and Hecke operators

We give a new proof of the fact that the Maas space is invariant under all Hecke operators. It is based on the characterization of the Maas space by a symmetry relation and certain commutation relations of the Hecke algebra for the Jacobi group.

THETA SERIES OVER THE HURWITZ QUATERNIONS

There are six theta constants over the Hurwitz quaternions on the quaternion half-space of degree 2. The paper describes the behavior of these theta constants under the transpose mapping, which can be derived from the Fourier expansions. The results are applied to the theta series of the first and second kind.