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Applications of spinor class fields: embeddings of orders and quaternionic lattices

Luis Arenas-Carmona (2003)

Annales de l'Institut Fourier

We extend the theory of spinor class fields and relative spinor class fields to study representation problems in several classical linear algebraic groups over number fields. We apply this theory to study the set of isomorphism classes of maximal orders of central simple algebras containing a given maximal Abelian suborder. We also study isometric embeddings of one skew-Hermitian Quaternionic lattice into another.

Hermitian and quadratic forms over local classical crossed product orders

Y. Hatzaras, Th. Theohari-Apostolidi (2000)

Colloquium Mathematicae

Let R be a complete discrete valuation ring with quotient field K, L/K be a Galois extension with Galois group G and S be the integral closure of R in L. If a is a factor set of G with values in the group of units of S, then (L/K,a) (resp. Λ =(S/R,a)) denotes the crossed product K-algebra (resp. crossed product R -order in A). In this paper hermitian and quadratic forms on Λ -lattices are studied and the existence of at most two irreducible non-singular quadratic Λ -lattices is proved (Theorem 3.5)....

Inertial law of quadratic forms on modules over plural algebra

Marek Jukl (1995)

Mathematica Bohemica

Quadratic forms on a free finite-dimensional module are investigated. It is shown that the inertial law can be suitably generalized provided the vector space is replaced by a free finite-dimensional module over a certain linear algebra over ( real plural algebra) introduced in [1].

Kronecker modules and reductions of a pair of bilinear forms

Giovanni Falcone, M. Alessandra Vaccaro (2004)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

We give a short overview on the subject of canonical reduction of a pair of bilinear forms, each being symmetric or alternating, making use of the classification of pairs of linear mappings between vector spaces given by J. Dieudonné.

La controverse de 1874 entre Camille Jordan et Leopold Kronecker

Frédéric Brechenmacher (2007)

Revue d'histoire des mathématiques

Une vive querelle oppose en 1874 Camille Jordan et Leopold Kronecker sur l’organisation de la théorie des formes bilinéaires, considérée comme permettant un traitement « général » et « homogène » de nombreuses questions développées dans des cadres théoriques variés au xixe siècle et dont le problème principal est reconnu comme susceptible d’être résolu par deux théorèmes énoncés indépendamment par Jordan et Weierstrass. Cette controverse, suscitée par la rencontre de deux théorèmes que nous considérerions...

Levels of rings - a survey

Detlev W. Hoffmann (2016)

Banach Center Publications

Let R be a ring with 1 ≠ 0. The level s(R) of R is the least integer n such that -1 is a sum of n squares in R provided such an integer exists, otherwise one defines the level to be infinite. In this survey, we give an overview on the history and the major results concerning the level of rings and some related questions on sums of squares in rings with finite level. The main focus will be on levels of fields, of simple noncommutative rings, in particular division rings, and of arbitrary commutative...

Matrix of ℤ-module1

Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama (2015)

Formalized Mathematics

In this article, we formalize a matrix of ℤ-module and its properties. Specially, we formalize a matrix of a linear transformation of ℤ-module, a bilinear form and a matrix of the bilinear form (Gramian matrix). We formally prove that for a finite-rank free ℤ-module V, determinant of its Gramian matrix is constant regardless of selection of its basis. ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm and cryptographic systems with lattices [22]...

Modular lattices from finite projective planes

Tathagata Basak (2014)

Journal de Théorie des Nombres de Bordeaux

Using the geometry of the projective plane over the finite field 𝔽 q , we construct a Hermitian Lorentzian lattice L q of dimension ( q 2 + q + 2 ) defined over a certain number ring 𝒪 that depends on q . We show that infinitely many of these lattices are p -modular, that is, p L q ' = L q , where p is some prime in 𝒪 such that | p | 2 = q .The Lorentzian lattices L q sometimes lead to construction of interesting positive definite lattices. In particular, if q 3 mod 4 is a rational prime such that ( q 2 + q + 1 ) is norm of some element in [ - q ] , then we find a 2 q ( q + 1 ) dimensional...

On the arithmetic of cross-ratios and generalised Mertens’ formulas

Jouni Parkkonen, Frédéric Paulin (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

We develop the relation between hyperbolic geometry and arithmetic equidistribution problems that arises from the action of arithmetic groups on real hyperbolic spaces, especially in dimension 5 . We prove generalisations of Mertens’ formula for quadratic imaginary number fields and definite quaternion algebras over , counting results of quadratic irrationals with respect to two different natural complexities, and counting results of representations of (algebraic) integers by binary quadratic, Hermitian...

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