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## Acta Arithmetica

1993 | 63 | 1 | 91-96
Tytuł artykułu

### Primitive minima of positive definite quadratic forms

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EN
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EN
The main purpose of the reduction theory is to construct a fundamental domain of the unimodular group acting discontinuously on the space of positive definite quadratic forms. This fundamental domain is for example used in the theory of automorphic forms for GLₙ (cf. [11]) or in the theory of Siegel modular forms (cf. [1], [4]). There are several ways of reduction, which are usually based on various minima of the quadratic form, e.g. the Korkin-Zolotarev method (cf. [10], [3]), Venkov's method (cf. [12]) or Voronoï's approach (cf. [13]), which also works in the general setting of positivity domains (cf. [5]). The most popular method is Minkowski's reduction theory [6] and its generalizations (cf. [9], [15]).
Minkowski's reduction theory is based on attaining certain minima, which can be characterized as the successive primitive minima of the quadratic form. Besides these we have successive minima, but a reduction according to successive minima only works for n ≤ 4 (cf. [14]). In this paper we introduce so-called primitive minima}, which lie between successive and successive primitive minima (cf. Theorem 2). Using primitive minima we obtain a straightforward generalization of Hermite's inequality in Theorem 1. As an application we get a simple proof for the finiteness of the class number. Finally we describe relations with Rankin's minima (cf. [8]) and with Venkov's reduction (cf. [12]).
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
91-96
Opis fizyczny
Daty
wydano
1993
otrzymano
1992-04-30
Twórcy
autor
• Mathematisches Institut, Westfälische Wilhelms-Universität, Einsteinstr. 62, W-4400 Münster, Federal Republic of Germany
Bibliografia
• [1] A. N. Andrianov, Quadratic Forms and Hecke Operators, Grundlehren Math. Wiss. 286, Springer, Berlin 1987.
• [2] E. S. Barnes and M. J. Cohn, On the inner product of positive quadratic forms, J. London Math. Soc. (2) 12 (1975), 32-36.
• [3] D. Grenier, Fundamental domains for the general linear group, Pacific J. Math. 132 (1988), 293-317.
• [4] H. Klingen, Introductory Lectures on Siegel Modular Forms, Cambridge University Press, Cambridge 1990.
• [5] M. Koecher, Beiträge zu einer Reduktionstheorie in Positivitätsbereichen I, Math. Ann. 141 (1960), 384-432.
• [6] H. Minkowski, Diskontinuitätsbereich für arithmetische Äquivalenz, J. Reine Angew. Math. 129 (1905), 220-274.
• [7] M. Newman, Integral Matrices, Academic Press, New York 1972.
• [8] R. A. Rankin, On positive definite quadratic forms, J. London Math. Soc. 28 (1953), 309-319.
• [9] S. S. Ryshkov, On the Hermite-Minkowski reduction theory for positive quadratic forms, J. Soviet Math. 6 (1976), 651-671.
• [10] S. S. Ryshkov and E. P. Baranovskiĭ, Classical methods in the theory of lattice packings, Russian Math. Surveys 34 (4) (1979), 1-68.
• [11] A. Terras, Harmonic Analysis on Symmetric Spaces and Applications II, Springer, New York 1988.
• [12] A. B. Venkov, On the reduction of positive quadratic forms, Izv. Akad. Nauk SSSR Ser. Mat. 4 (1940), 37-52 (in Russian).
• [13] G. Voronoï, Sur quelques propriétés des formes quadratiques positives parfaites, J. Reine Angew. Math. 133 (1907), 97-178.
• [14] B. L. van der Waerden, Die Reduktionstheorie der positiven quadratischen Formen, Acta Math. 96 (1956), 263-309.
• [15] H. Weyl, Theory of reduction for arithmetical equivalence, Trans. Amer. Math. Soc. 48 (1940), 126-164.
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