Réduction de formes quadratiques dans un corps algébrique fini

Humbert, Pierre

doi:10.1007/bf02565591

Cited by 21 publications

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“…Hence, From (3) it follows that M has the reciprocal (4) M1= (% -.D I Two matrices A(n) and B(n) of R are called a hermitian pair if AB = Bit, and (AB) is of rank n. The hermitian pair is called integral if the elements of A and B are integral. For a given integral hermitian pair A, B of 9 we may consider the rows xA, xB, where x = x(1 n) is a variable row of R. The pair A, B is called coprime if integral xA and xB imply that x itself is integral.…”

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confidence: 98%

“…In what follows we have to make use of some of Humbert's [3] results concerning the reduce domain t which has been defined in Section 5. We formulate Ilumbert's results only for SZ.…”

mentioning

confidence: 99%

“…It has been proved by Pierre Humbert [3] that the Minkowski reduction theory of quadratic forms can be generalized. In the case of an imaginary quadratic number field R Humbert's results say that there exists a finite number h of non-singular, n-rowed, integral matrices Al, A2, ** *, Ah of f, depending only upon n and 9, which have the following property:…”

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confidence: 99%

“…The result of the following sections will be that the group .& has the same properties as the modular group of degree n. In order to prove this, we apply in general the methods developed by Siegel for the modular group of degree n [1,2]1. However, there has to be applied the Humbert reduction theory [3] instead of the Minkowski reduction theory. Therefore, the result stated above is not evident from the beginning.…”

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confidence: 99%

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Hermitian Modular Functions. III: The Hermitian Modular Group

Braun¹

1951

The Annals of Mathematics

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JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Annals of Mathematics.

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confidence: 98%

“…In what follows we have to make use of some of Humbert's [3] results concerning the reduce domain t which has been defined in Section 5. We formulate Ilumbert's results only for SZ.…”

mentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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Hermitian Modular Functions. III: The Hermitian Modular Group

Braun¹

1951

The Annals of Mathematics

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“…Write M = aa + a 2 # 2 with ñá = 0.Since ô á>ñ is on Jf we have (ñÌ) a^ M, hence (ñ M) ea ^ M, hence ô 6á>ñ is also on M.And the transformation ó defined by óá = (l -å~é) a, ax 2 = x 27 ) See the generation theorems of Hurwitz on pp. The second inclusion relation is trivial.…”

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confidence: 99%

On the finite generation of linear groups over Hasse domains.

O’Meara¹

1965

CRLL

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By 0. T. O'Meara at Notre Dame (Indiana, U. S. A.)The purpose of this paper is to investigate the finite and non-finite generation of certain linear groups over the Hasse domains of global fields. A global field F is a field whieh is either an algebraic number field or an algebraic function field in one variable over a finite constant field. A Hasse domain o on F is a Dedekind domain which can be obtained äs the intersection of almost all valuation rings of jP; these domains are an accepted generalization of the classical concept of the ring of iptegers of a number field.Consider, for the moment, the general linear group GL n and the special linear group SL n defined äs matrix groups over the Hasse domain o. Suppose we are in the number theoretic ease, i. e. in charaeteristic 0. Then 1) the group of units of o, i. e. the group GL 17 is finitely generated. 2) Using the Euclidean algorithm it is easy to show that SL n is finitely generated when o is the ring of rational integers Z. 3) Using a weak form of the Euclidean algorithm Hurwitz [8] extended the last result to the ring of integers of an algebraic number field 2 ). 4) Several authors have established the finite generation of the other classical groups by analytic, arithrnetic and other methods. 5) These results have culminated in the very general work of Borel and Harish-Chandra [4].These theories still have to be developed in charaeteristic p, i. e. in the function field case. However, it is known that 1) the group of units of o, i. e. the group GL 1? is finitely generated. 2) If o is the polynomial ring k[x] of a rational global field k(x), then it again follows from the Euclidean algorithm that SL n is finitely generated, but only if n > 3. If n = 2, it is not. This means that exceptional behavior must be expected in charaeteristic p.Our main task here is to investigate the finite generation of GL n and SL n in the function theoretic case. This will be done using the elementary arithmetic methods of Artin and Whaples which do not distinguish between charaeteristic 0 and charaeteristic p. So in fact we shall give a unified account that includes the classical case of charaeteristic 0.The linear groups under discussion can be interpreted äs groups of automorphisms of free modules over o. It is more appropriate and more general to consider groups of automorphisms of a lattice M on an rc-dimensional vector space over F = o -f-o. For such an M we shall introduce the general linear group GL«, the special linear group SL n , the group TL n generated by transvections on Af, and the group EL n generated by x )

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The First Years

Narkiewicz

2012

Springer Monographs in Mathematics

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Réduction de formes quadratiques dans un corps algébrique fini

Cited by 21 publications

References 1 publication

Hermitian Modular Functions. III: The Hermitian Modular Group

Hermitian Modular Functions. III: The Hermitian Modular Group

On the finite generation of linear groups over Hasse domains.

The First Years

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