Single-class genera of orthogonal groups
- Einklassige Geschlechter orthogonaler Gruppen
Lorch, David; Kirschmer, Markus (Thesis advisor); Nebe, Gabriele (Thesis advisor)
Aachen (2019, 2020)
Dissertation / PhD Thesis
Dissertation, RWTH Aachen University, 2019
This thesis presents a classification, up to similarity, of all single-class genera of totally definite lattices in quadratic spaces of rank m >= 3 over totally real algebraic number fields. The Hasse-Minkowski theorem shows a "local-global principle" for quadratic spaces over number fields K: two regular quadratic spaces over K are isometric, if and only if they are isometric over every completion of K. In general, this principle is false for lattices in quadratic spaces over number fields. Given some lattice L in a regular quadratic space V, the set of lattices in V which are isometric to L over every completion is finite. This set is called the "genus" of L. A lattice is called "single-class", if its genus consists of a single isometry class. So, single-class lattices fulfill a local-global principle for isometry, contrary to the general situation. For totally definite lattices L over totally real number fields, this case occurs very rarely. Indeed G. L. Watson has shown that in the special case K=Q, no single-class, definite lattices exist unless m <= 10, and has published an incomplete classification. For general totally real number fields, Pfeuffer has shown the finiteness of the classification problem, and proved that m <= 16.The present work begins by proving bounds to the invariants of single-class, square-free, totally definite lattices L (and their enveloping quadratic spaces V) over totally real number fields K. These are based upon the results of Pfeuffer, who built upon the basis of Siegel's mass formula. The finite set of possible base fields K can be found in existing tables, once sufficiently strong bounds to the root discriminant of K have been proved. Over a given base field K, an algorithm based on O'Meara's classification of lattices over local fields allows to construct representatives of all genera satisfying the bounds to their invariants. In the following, an algorithm is developed that allows to enumerate representatives of the isometry classes in a given genus. In the case of totally definite lattices, a prerequisite is being able to enumerate the classes in a given spinor genus, using Kneser's neighbours algorithm. We continue to describe how the different spinor genera in the genus are linked by neighbour calculation at suitable primes. Finally, the classification of single-class, totally definite lattices is completed by applying a descent algorithm to a set of single-class, totally definite and square-free lattices. This algorithm uses reduction methods developed by Watson and Gerstein.