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Calcul du nombre de classes d'un corps quadratique imaginaire ou réel, d'après Shanks, Williams, McCurley, A. K. Lenstra et Schnorr

Henri Cohen (1989)

Journal de théorie des nombres de Bordeaux

Dans cette note nous décrivons différentes méthodes utilisées en pratique pour calculer le nombre de classes d'un corps quadratique imaginaire ou réel ainsi que pour calculer le régulateur d'un corps quadratique réel. En particulier nous décrivons l'infrastructure de Shanks ainsi que la méthode sous-exponentielle de McCurley.

Caractères quadratiques de groupes abéliens finis et sommes de Gauss

T.A. Springer (1976)

Mémoires de la Société Mathématique de France

Cayley orders

Arjeh M. Cohen, Gabriele Nebe, Wilhelm Plesken (1996)

Compositio Mathematica

Certain linear combinations of two Pfister forms and the isotropy problem. (Certaines combinaisons linéaires de deux formes de Pfister et le problème d'isotropie.)

Laghribi, Ahmed (2001)

Documenta Mathematica

Characteristic vectors of unimodular lattices which represent two

Mark Gaulter (2007)

Journal de Théorie des Nombres de Bordeaux

We improve the known upper bound of the dimension $n$ of an indecomposable unimodular lattice whose shadow has the third largest possible length, $n - 16$ .

Charakterisierung von Drehungen und Bewegungen des Rn durch lineare Invarianten.

W.Fr. Meyer (1911)

Jahresbericht der Deutschen Mathematiker-Vereinigung

Chebyshev polynomials in several variables.

Rudolf Lidl, Ch. Wells (1972)

Journal für die reine und angewandte Mathematik

Class number formulas for quaternary quadratic forms

Paul Ponomarev (1981)

Acta Arithmetica

Class numbers of binary quadratic forms over algebraic number fields.

Jih-Min Shyr (1979)

Journal für die reine und angewandte Mathematik

Class numbers of binary quadratic lattices over algebraic number fields

O. Körner (1981)

Acta Arithmetica

Class numbers of pairs of symmetric matrices

Jin Nakagawa (2002)

Acta Arithmetica

Class numbers of positive definite quaternary quadratic forms.

Arnold K. Pizer (1976)

Journal für die reine und angewandte Mathematik

Classes de Stiefel-Whitney de Formes quadratiques et de représentations galoisiennes réelles.

Bruno Kahn (1984)

Inventiones mathematicae

Classes de Stiefel-Whitney en cohomologie étale

O. Laborde (1976)

Mémoires de la Société Mathématique de France

Classification of hermitian forms and semisimple gorups over fields of virtual cohomological dimension one.

Claus Scheiderer (1996)

Manuscripta mathematica

Classification of two genera of 32-dimensional lattices of rank 8 over the Hurwitz order.

Bachoc, Christine, Nebe, Gabriele (1997)

Experimental Mathematics

Classification Theorems for Quadratic Forms over Fields

Richard Elman, T.Y. Lam (1974)

Commentarii mathematici Helvetici

Clifford algebras, Möbius transformations, Vahlen matrices, and $B$ -loops

Jimmie Lawson (2010)

Commentationes Mathematicae Universitatis Carolinae

In this paper we show that well-known relationships connecting the Clifford algebra on negative euclidean space, Vahlen matrices, and Möbius transformations extend to connections with the Möbius loop or gyrogroup on the open unit ball $B$ in $n$ -dimensional euclidean space $ℝ^{n}$ . One notable achievement is a compact, convenient formula for the Möbius loop operation $a * b = (a + b) {(1 - a b)}^{- 1}$ , where the operations on the right are those arising from the Clifford algebra (a formula comparable to $(w + z) {(1 + \bar{w} z)}^{- 1}$ for the Möbius loop multiplication...

Clifford k-algebras and k*-groups.

Luzius Grunenfelder (1984)

Mathematische Zeitschrift

CM liftings of supersingular elliptic curves

Ben Kane (2009)

Journal de Théorie des Nombres de Bordeaux

Assuming GRH, we present an algorithm which inputs a prime $p$ and outputs the set of fundamental discriminants $D < 0$ such that the reduction map modulo a prime above $p$ from elliptic curves with CM by $𝒪_{D}$ to supersingular elliptic curves in characteristic $p$ is surjective. In the algorithm we first determine an explicit constant $D_{p}$ so that $| D | > D_{p}$ implies that the map is necessarily surjective and then we compute explicitly the cases $| D | < D_{p}$ .

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