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Spinor regular positive ternary quadratic forms. (English) Zbl 0686.10014

In terminology introduced by L. E. Dickson [Ann. Math. 28, 333-341 (1927; JFM 53.0133.03)], a positive definite integral ternary quadratic form is said to be regular if it represents all integers represented by its genus. A historical survey of the search for such regular forms can be found in [J. S. Hsia, Mathematika 28, 231-238 (1981; Zbl 0469.10009)]. In the present paper, a refinement of the notion of regularity, called spinor regularity, is introduced; a form is said to be spinor regular if it represents all integers represented by its spinor genus.
It is proven that there exist only finitely many integral equivalence classes of positive definite primitive integral ternary quadratic forms which have this property, and all such forms are determined which lie in genera containing multiple spinor genera and have discriminant less than 2000. Among these forms appears one previously unknown regular form (of discriminant 864) and eleven spinor regular forms which are not regular.
Reviewer: A.G.Earnest

MSC:

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