Chapter 17. Projections (Models) of Hyperbolic Planes

AbstractLetKbe a number field, and letFbe a symmetric bilinear form in 2Nvariables overK. LetZbe a subspace ofKN. A classical theorem of Witt states that the bilinear space (Z,F) can be decomposed into an orthogonal sum of hyperbolic planes and singular and anisotropic components. We prove the existence of such a decomposition of small height, where all bounds on height are explicit in terms of heights ofFandZ. We also prove a special version of Siegel's lemma for a bilinear space, which provides a small-height orthogonal decomposition into one-dimensional subspaces. Finally, we prove an effective version of the Cartan–Dieudonné theorem. Namely, we show that every isometry σ of a regular bilinear space (Z,F) can be represented as a product of reflections of bounded heights with an explicit bound on heights in terms of heights ofF,Z, and σ.

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Quadratic Forms

10.23943/princeton/9780691166902.003.0002 ◽

2017 ◽

Author(s):

Bernhard M¨uhlherr ◽

Holger P. Petersson ◽

Richard M. Weiss

Keyword(s):

Quadratic Forms ◽

Splitting Field ◽

Quadratic Space ◽

Polar Spaces ◽

Quadratic Module ◽

Quadratic Modules ◽

Hyperbolic Planes ◽

Scalar Extensions

This chapter presents a few standard definitions and results about quadratic forms and polar spaces. It begins by defining a quadratic module and a quadratic space and proceeds by discussing a hyperbolic quadratic module and a hyperbolic quadratic space. A quadratic module is hyperbolic if it can be written as the orthogonal sum of finitely many hyperbolic planes. Hyperbolic quadratic modules are strictly non-singular and free of even rank and they remain hyperbolic under arbitrary scalar extensions. A hyperbolic quadratic space is a quadratic space that is hyperbolic as a quadratic module. The chapter also considers a split quadratic space and a round quadratic space, along with the splitting extension and splitting field of of a quadratic space.

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New discretization of complex analysis: The Euclidean and hyperbolic planes

Proceedings of the Steklov Institute of Mathematics ◽

10.1134/s0081543811040122 ◽

2011 ◽

Vol 273 (1) ◽

pp. 238-251 ◽

Cited By ~ 4

Author(s):

S. P. Novikov

Keyword(s):

Complex Analysis ◽

Hyperbolic Planes

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On Fundamental Domains for Subgroups of Isometries Acting in

ISRN Geometry ◽

10.5402/2012/685103 ◽

2012 ◽

Vol 2012 ◽

pp. 1-27

Author(s):

Antonio Lascurain Orive ◽

Rubén Molina Hernández

Keyword(s):

Alternative Proof ◽

Index Subgroup ◽

Transitive Action ◽

Space Model ◽

Fundamental Domains ◽

Möbius Group ◽

Fundamental Polyhedron ◽

Hyperbolic Planes ◽

Generating Theorem ◽

Finite Index Subgroup

Given a fundamental polyhedron for the action of , a classical kleinian group, acting in -dimensional hyperbolic space, and , a finite index subgroup of , one obtains a fundamental domain for pasting copies of by a Schreier process. It also generalizes the side pairing generating theorem for exact or inexact polyhedra. It is proved as well that the general Möbius group acting in is transitive on “-spheres”. Hence, describing the hyperbolic -planes in the upper half space model intrinsically, and providing also an alternative proof of the transitive action on them. Some examples are given in detail, derived from the classical modular group and the Picard group.

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