Primitive minima of positive definite quadratic forms

Aloys Krieg

Acta Arithmetica (1993)

  • Volume: 63, Issue: 1, page 91-96
  • ISSN: 0065-1036

Abstract

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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]).

How to cite

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Aloys Krieg. "Primitive minima of positive definite quadratic forms." Acta Arithmetica 63.1 (1993): 91-96. <http://eudml.org/doc/206509>.

@article{AloysKrieg1993,
abstract = { 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]). },
author = {Aloys Krieg},
journal = {Acta Arithmetica},
keywords = {reduction theory; generalization of Hermite's inequality; finiteness of class number; relations with Rankin's minima; Venkov's reduction; primitive minima; positive definite quadratic form; successive minima},
language = {eng},
number = {1},
pages = {91-96},
title = {Primitive minima of positive definite quadratic forms},
url = {http://eudml.org/doc/206509},
volume = {63},
year = {1993},
}

TY - JOUR
AU - Aloys Krieg
TI - Primitive minima of positive definite quadratic forms
JO - Acta Arithmetica
PY - 1993
VL - 63
IS - 1
SP - 91
EP - 96
AB - 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]).
LA - eng
KW - reduction theory; generalization of Hermite's inequality; finiteness of class number; relations with Rankin's minima; Venkov's reduction; primitive minima; positive definite quadratic form; successive minima
UR - http://eudml.org/doc/206509
ER -

References

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  10. [10] S. S. Ryshkov and E. P. Baranovskiĭ, Classical methods in the theory of lattice packings, Russian Math. Surveys 34 (4) (1979), 1-68. Zbl0441.52008
  11. [11] A. Terras, Harmonic Analysis on Symmetric Spaces and Applications II, Springer, New York 1988. 
  12. [12] A. B. Venkov, On the reduction of positive quadratic forms, Izv. Akad. Nauk SSSR Ser. Mat. 4 (1940), 37-52 (in Russian). 
  13. [13] G. Voronoï, Sur quelques propriétés des formes quadratiques positives parfaites, J. Reine Angew. Math. 133 (1907), 97-178. Zbl38.0261.01
  14. [14] B. L. van der Waerden, Die Reduktionstheorie der positiven quadratischen Formen, Acta Math. 96 (1956), 263-309. Zbl0072.03601
  15. [15] H. Weyl, Theory of reduction for arithmetical equivalence, Trans. Amer. Math. Soc. 48 (1940), 126-164. Zbl0024.14802

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