Einklassige Geschlechter von Gittern über Maximalordnungen in definiten schiefhermiteschen Räumen über indefiniten rationalen Quaternionenalgebren ungerader Diskriminante

• Single-class genera of lattices over maximal orders in definite skew-hermitian spaces over indefinite rational quaternion algebras of odd discriminant

Pawelzik, Nils Malte; Nebe, Gabriele (Thesis advisor); Kirschmer, Markus (Thesis advisor)

Aachen : RWTH Aachen University (2023)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2023

Abstract

The subject of this dissertation is the classification of single-class genera of lattices, i. e. of the lattices for which the isometry of all the localisations implies global isometry, over maximal orders in definite skew-Hermitian spaces over rational quaternion algebras of odd discriminant. This work stems from the classification of single-class genera of rank ≥ 3 for definite quadratic forms over totally-real number fields by Markus Kirschmer and David Lorch resp. of single- and double-class genera over definite Hermitian forms by Markus Kirschmer. This dissertation is the first to consider the last open case of single-class genera over skew-Hermitian quaternionic forms. First, the unary case is reduced to the description of orders in imaginary quadratic fields for whose discriminant the main genus of quadratic binary forms consists of one class. These are known under assumption of the Generalised Riemann Hypothesis. Next, bounds for the mass and the local factors, using the formula by Wee Teck Gan und Jiu-Kang Yu, are developed to construct the possible skew-Hermitian spaces and the single-class genera therein. In the end, the results are represented by genus symbols. It is shown that single-class genera in at least binary definite skew-Hermitian spaces over indefinite rational quaternion algebras of odd discriminant exists if and only if the dimension is 2 and the ramified places form a two-element subset of {3, 5, 7, 13}. In total, there are 206 similarity classes of such genera resp. lattices.

Institutions

• Department of Mathematics [110000]
• Mathematics (Algebra) Teaching and Research Area [114820]