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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).

11E12 Quadratic forms over global rings and fields
11E20 General ternary and quaternary quadratic forms; forms of more than two variables
Full Text: DOI
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