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On representations of spinor genera. (English) Zbl 1041.11020

The authors address the problem of determining when one quadratic form over the ring of integers of an algebraic number field is represented by the spinor genus of another such form. Explicit criteria for such a representation to exist are determined for the case when the rank of the representing form is three or four. The results extend previous work by J. S. Hsia, Y. Y. Shao and F. Xu [J. Reine Angew. Math. 494, 129–140 (1998; Zbl 0883.11016)].
The authors also point out connections with recent results in the literature. First, it is shown that an embedding theorem for quaternion algebras proved by T. Chinburg and E. Friedman [J. Lond. Math. Soc. (2) 60, 33–44 (1999; Zbl 0940.11053)] can be deduced from the representation theory of ternary quadratic forms. To illustrate the value of this point of view, a theorem from that previous paper is generalized to the case of Eichler orders. It is further shown how the results of F. Xu [Duke Math. J. 110, 279–307 (2001; Zbl 1021.11008)] on representation masses of spinor genera of quadratic forms are related to results of M. Borovoi and Z. Rudnick [Invent. Math. 111, 37–66 (1995; Zbl 0917.11025)] on Hardy-Littlewood varieties.

MSC:

11E12 Quadratic forms over global rings and fields
11E25 Sums of squares and representations by other particular quadratic forms
11G35 Varieties over global fields
11R52 Quaternion and other division algebras: arithmetic, zeta functions
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