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Universal positive quaternary quadratic lattices over totally real number fields

Part of: Forms and linear algebraic groups

Published online by Cambridge University Press:  26 February 2010

A. G. Earnest  and
Azar Khosravani
A. G. Earnest
Affiliation:
Department of Mathematics, Southern Illinois University, Carbondale, IL 62901-4408, U.S.A.
Azar Khosravani
Affiliation:
Department of Mathematics, University of Wisconsin, Oshkosh, Oshkosh, WI 54901-8631, U.S.A.
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Extract

In [CKR], Chan, Kim and Raghavan determine all universal positive ternary integral quadratic forms over real quadratic number fields. In this context, universal means that the form represents all totally positive elements of the ring of integers of the underlying field. This generalizes the usage of the term introduced by Dickson for the case of the ring of rational integers [D]. In the present paper, we will continue the investigation of quadratic forms with this property, considering positive quaternary forms over totally real number fields. The main goal of the paper is to prove that if E is a totally real number field of odd degree over the field of rational numbers, then there are at most finitely many inequivalent universal positive quaternary quadratic forms over the ring of integers of E. In fact, the stronger result will be proved that this finiteness holds for those forms which represent all totally positive multiples of any fixed totally positive integer. The necessity of the assumption of oddness of the degree of the extension for a general result of this type can be seen from the existence of universal ternary forms over certain real quadratic fields (for example, the sum of three squares over the field ℚ(√5), as first shown by Maass [M]).

MSC classification

Secondary: 11E12: Quadratic forms over global rings and fields
Type
Research Article
Information
Mathematika , Volume 44 , Issue 2 , December 1997 , pp. 342 - 347
Copyright
Copyright © University College London 1997

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References

CKR.Chan, W.-K.Kim, M.-H. and Raghavan, S.. Ternary universal integral quadratic forms over real quadratic fields. Japanese J. Math., 22 (1996), 263–273.CrossRefGoogle Scholar
D.Dickson, L. E.. Quaternary quadratic forms representing all integers. Amer. J. Math., 49 (1937), 39–56.CrossRefGoogle Scholar
E.Earnest, A. G.. The representation of binary quadratic forms by positive definite quaternary quadratic forms. Trans. Amer. Math. Soc., 345 (1994), 853–863.CrossRefGoogle Scholar
K.Kaplansky, I.. Ternary positive quadratic forms that represent all odd positive integers. Ada Arith., 70 (1995), 209–214.CrossRefGoogle Scholar
M.Maass, H.. Über die Darstellung total positiver Zahlen des Körpers R(√5) als Summe von drei Quadraten. Abh. Math. Sem. Hamburg, 14 (1941), 185–191.CrossRefGoogle Scholar
O.O'Meara, O. T.. Introduction to Quadratic Forms, (Springer-Verlag, Berlin, 1963).CrossRefGoogle Scholar
R.Ross, A. E.. On a problem of Ramanujan. Amer. J. Math., 68 (1946), 29–46.CrossRefGoogle Scholar
SP.Schulze-Pillot, R.. Darstellung durch definite ternäre quadratische Formen. J. Number Theory, 14 (1982), 237–250.CrossRefGoogle Scholar
V.der Waerden, B. L. van. Die Reduktionstheorie der positiven quadratischen Formen. Ada Math., 96 (1956), 265–309.Google Scholar
W.Willerding, M. F.. Determination of all classes of positive quaternary quadratic forms which represent all (positive) integers. Bull. Amer. Math. Soc., 54 (1948), 334–337.CrossRefGoogle Scholar