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Represented value sets for integral binary quadratic forms and lattices

Author(s): A. G. Earnest; Robert W. Fitzgerald
Journal: Proc. Amer. Math. Soc. 135 (2007), 3765-3770.
MSC (2000): Primary 11E16; Secondary 11E12, 11E25, 11R29
Posted: August 30, 2007
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Abstract: A characterization is given for the integral binary quadratic forms for which the set of represented values is closed under products. It is also proved that for an integral binary quadratic lattice over a Dedekind domain, the product of three values represented by the form is again a value represented by the form. This generalizes the trigroup property observed by V. Arnold in the case of integral binary quadratic forms.

References:

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F. Aicardi and V. Timorin, On binary quadratic forms with semigroup property, preprint.

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V. I. Arnold, Arithmetics of binary quadratic forms, symmetry of their continued fractions and geometry of their de Sitter world, Bull. Braz. Math. Soc. 34 (2003), 1-41. MR 1991436 (2004h:11030)

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D. A. Cox, Primes of the form $ x^{2} + ny\sp{2}$. Fermat, class field theory and complex multiplication, John Wiley & Sons, New York, 1989. MR 1028322 (90m:11016)

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A. G. Earnest and D. R. Estes, Class groups in the genus and spinor genus of binary quadratic lattices, Proc. London Math. Soc. 40 (1980), 40-52. MR 560994 (81i:10025)

[5]
O. T. O'Meara, Introduction to Quadratic Forms, Springer-Verlag, Berlin, 1963.

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H. Weber, Beweis des Satzes, dass jede eigentlich primitive quadratische Form unendlich viele Primzahlen darzustellen fähig ist, Math. Ann. 20 (1882), 301-329. MR 1510171

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Additional Information:

A. G. Earnest
Affiliation: Department of Mathematics, Southern Illinois University, Carbondale, Illinois 62901
Email: aearnest@math.siu.edu

Robert W. Fitzgerald
Affiliation: Department of Mathematics, Southern Illinois University, Carbondale, Illinois 62901
Email: rfitzg@math.siu.edu

DOI: 10.1090/S0002-9939-07-08895-8
PII: S 0002-9939(07)08895-8
Received by editor(s): June 14, 2006
Received by editor(s) in revised form: September 5, 2006
Posted: August 30, 2007
Communicated by: Ken Ono
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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