Hermitesche Gitter und der Nachbarschaftsoperator

Henn, Andreas (Author); Krieg, Aloys (Thesis advisor)

Aachen / Publikationsserver der RWTH Aachen University (2013, 2014) [Dissertation / PhD Thesis]

Page(s): 122 S.

Abstract

This thesis examines unimodular even lattices in Euclidean vector spaces, called theta lattices in the following. We consider lattices with an additional Hermitian structure, i.e. a structure as a module over the ring of integers of an imaginary quadratic number field or a maximal order in a definite quaternion algebra over the rationals. Theta lattices are of importance for the theory of modular forms since their theta series of degree n are Siegel modular forms for the full Siegel modular group. In the case of lattices with a Hermitian structure, one can also define theta series which are Hermitian or quaternionic modular forms respectively. Of central significance in the theory of modular forms are the so-called Hecke operators, certain endomorphisms of the space of modular forms. For the classification of lattices frequently Kneser's neighbor method, a global construction which changes the completion of a lattice only at one prime, has been applied. In connection with the neighbour method, the neighbourhood operator, an endomorphism of the space of formal linear combinations of theta lattices, is defined. This operator can also be regarded as an operator on the space spanned by the theta series of degree n of the considered theta lattices. From formulas describing the effect of Hecke operators on theta series it follows that this operator is a Hecke operator. Here corresponding results in the Hermitian and quaternionic case shall be derived, but only in the case of the full modular group. We now describe the contents of this thesis in more detail. In the first chapter, the foundations of the theory of lattices with a Hermitian structure are compiled. Then we investigate the three Euclidean maximal orders $mathcal M_{infty,2}$, $mathcal M_{infty,3}$ and $mathcal M_{infty,5}$ in the quaternion algebras $mathbb Q_{infty,2}$, $mathbb Q_{infty,3}$ and $mathbb Q_{infty,5}$ respectively, which will be of interest in the following. In regard to the classification of theta lattices with a Hermitian structure over these orders we derive a mass formula for these lattices, but generally for maximal orders in definite quaternion algebras over the rationals (without restriction to the class number). This mass formula suffices for demonstrating that up to isometry there are exactly one theta lattice of rank 2 and two theta lattices of rank 4 over $mathcal M_{infty,3}$. The theta lattices of rank 6 over $mathcal M_{infty,3}$ can be determined by investigation of the automorphism groups of the known Niemeier lattices. In this case, there are six theta lattices up to isometry. In the case of theta lattices of rank 8 over $mathcal M_{infty,3}$, we have carried out the classification with Kneser's neighbor method. First we treat the theoretical foundations of this method, then we describe the corresponding algorithm which we have implemented in Magma. We have found 83 isometry classes of theta lattices. With the same method, we have classified the theta lattices of rank 2, 4, and 6 over $mathcal M_{infty,5}$. In this case there is one theta lattic of rank 2, two of rank 4, and 21 of rank 6; the classification in rank 8 is out of reach. Subsequently we discuss the definition and the computation of the neighborhood operator. The next chapter addresses Hermitian and quaternionic modular forms; we restrict ourselves to the case of class number 1. After an introduction into the foundations of the respective theories, we determine the effect of some concrete Hecke operators on theta series of theta lattices and thereby show that the neighbourhood operator corresponds to a Hecke operator on the space spanned by theta series. For that, we combine methods of A. Krieg and E. Freitag from the Siegel case. Furthermore, we make use of the law of commutation of Hecke operators with the Siegel $Phi$-operator. As Hecke operators are normal with regard to the Peterson scalar product, there is a basis of eigenforms of the neighborhood operator. In the third and last chapter, in some cases we describe the filtration of the affiliated linear combinations of theta series (and thereby the dimensions of the spaces spanned by theta series) by computing Fourier coefficients with Magma. Two appendices follow this chapter. In the first one, Gram matrices are given for the lattices we have found; the second one contains the Magma code for our implementation of the neighborhood method.

Identifier

  • URN: urn:nbn:de:hbz:82-opus-48817
  • REPORT NUMBER: RWTH-CONV-143817