×

zbMATH — the first resource for mathematics

Finiteness theorems for positive definite \(n\)-regular quadratic forms. (English) Zbl 1026.11046
A positive definite integral quadratic form is said to be regular if it represents all integers that are represented by its genus. It was first proved by G. L. Watson [Ph.D. thesis, Univ. London (1953)] that there are only finitely many equivalence classes of primitive integral positive definite ternary quadratic forms that are regular in this sense. The results in the present paper represent a far-reaching generalization of this finiteness result to the context of representations of one positive definite quadratic lattice by another of greater or equal rank.
All lattices under consideration in this paper are integral \(\mathbb Z\)-lattices on positive definite quadratic spaces over \(\mathbb Q\). Such a lattice is said to be \(n\)-regular if it represents all lattices of rank \(n\) that are represented by its genus. The study of these higher-dimensional analogues of regular quadratic forms was initiated by the reviewer in [Trans. Am. Math. Soc. 345, 853-863 (1994; Zbl 0810.11019)], where it was proved that there exist only finitely many inequivalent primitive positive definite quaternary lattices that are 2-regular.
The main result of the paper under review is that for any \(n\geq 2\) there exist at most finitely many inequivalent primitive \(n\)-regular positive definite lattices of rank \(n+3\). This result is proved by establishing bounds on the successive minima of the lattices in question and then using a fundamental inequality from reduction theory to obtain a bound for their discriminants. In contrast to the analytic techniques used initially by G. L. Watson [Mathematika 1, 104-110 (1954; Zbl 0056.27201)] and later by the reviewer [op. cit.] to tackle such problems, the techniques used here are arithmetic, centered around a generalized version of the regularity-preserving transformations that play a key role in the arguments in Watson’s thesis [op. cit.].
The authors also prove related finiteness results for “almost \(n\)-regular” lattices (those that represent all but at most finitely many of the equivalence classes of lattices of rank \(n\) represented by their genus) and “spinor \(n\)-regular” lattices (those that represent all lattices of rank \(n\) represented by their spinor genus).

MSC:
11E12 Quadratic forms over global rings and fields
11E20 General ternary and quaternary quadratic forms; forms of more than two variables
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] J. W. Benham, A. G. Earnest, J. S. Hsia, and D. C. Hung, Spinor regular positive ternary quadratic forms, J. London Math. Soc. (2) 42 (1990), no. 1, 1 – 10. · Zbl 0686.10014
[2] W. K. Chan and A. G. Earnest, Discriminant bounds for spinor regular ternary quadratic lattices, submitted. · Zbl 1084.11011
[3] Y. C. Chung, On 2-regular forms, Ph.D. Thesis, National Seoul University, 2001.
[4] L. E. Dickson, Ternary quadratic forms and congruences, Ann. of Math. 28 (1927), 331-341. · JFM 53.0133.03
[5] William Duke and Rainer Schulze-Pillot, Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids, Invent. Math. 99 (1990), no. 1, 49 – 57. · Zbl 0692.10020
[6] A. G. Earnest, The representation of binary quadratic forms by positive definite quaternary quadratic forms, Trans. Amer. Math. Soc. 345 (1994), no. 2, 853 – 863. · Zbl 0810.11019
[7] A. G. Earnest, An application of character sum inequalities to quadratic forms, Number theory (Halifax, NS, 1994) CMS Conf. Proc., vol. 15, Amer. Math. Soc., Providence, RI, 1995, pp. 155 – 158. · Zbl 0833.11012
[8] A. G. Earnest, Universal and regular positive quadratic lattices over totally real number fields, Integral quadratic forms and lattices (Seoul, 1998) Contemp. Math., vol. 249, Amer. Math. Soc., Providence, RI, 1999, pp. 17 – 27. · Zbl 0955.11008
[9] A. G. Earnest and J. S. Hsia, One-class spinor genera of positive quadratic forms, Acta Arith. 58 (1991), no. 2, 133 – 139. · Zbl 0689.10028
[10] J. S. Hsia, Representations by spinor genera, Pacific J. Math. 63 (1976), no. 1, 147 – 152. · Zbl 0328.10018
[11] John S. Hsia, Yoshiyuki Kitaoka, and Martin Kneser, Representations of positive definite quadratic forms, J. Reine Angew. Math. 301 (1978), 132 – 141. · Zbl 0374.10013
[12] William C. Jagy, Irving Kaplansky, and Alexander Schiemann, There are 913 regular ternary forms, Mathematika 44 (1997), no. 2, 332 – 341. · Zbl 0923.11060
[13] B. M. Kim, Complete determination of regular positive diagonal quaternary integral quadratic forms, preprint.
[14] Yoshiyuki Kitaoka, Arithmetic of quadratic forms, Cambridge Tracts in Mathematics, vol. 106, Cambridge University Press, Cambridge, 1993. · Zbl 0785.11021
[15] O. T. O’Meara, The integral representations of quadratic forms over local fields, Amer. J. Math. 80 (1958), 843 – 878. · Zbl 0085.02801
[16] O. T. O’Meara, Introduction to quadratic forms, Die Grundlehren der mathematischen Wissenschaften, Bd. 117, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. · Zbl 0107.03301
[17] G. L. Watson, Some problems in the theory of numbers, Ph.D. Thesis, University of London, 1953.
[18] G. L. Watson, The representation of integers by positive ternary quadratic forms, Mathematika 1 (1954), 104-110. · Zbl 0056.27201
[19] G. L. Watson, The class-number of a positive quadratic form, Proc. London Math. Soc. (3) 13 (1963), 549 – 576. · Zbl 0122.05802
[20] G. L. Watson, Regular positive ternary quadratic forms, J. London Math. Soc. (2) 13 (1976), no. 1, 97 – 102. · Zbl 0319.10024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.