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Bulletin of the London Mathematical Society Advance Access originally published online on September 2, 2008
Bulletin of the London Mathematical Society 2008 40(6):1025-1037; doi:10.1112/blms/bdn085
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© 2008 London Mathematical Society

Positive definite almost regular ternary quadratic forms over totally real number fields

Wai Kiu Chan

Department of Mathematics and Computer Science
Wesleyan University
Middletown, CT 06459
USA

Maria Ines Icaza

Instituto de Matematica y Fisica
Universidad de Talca
Avenida Liray s/n, Talca
Chile
icazap@inst-mat.utalca.cl

Received 4 December 2007. Revision received 17 June 2008.

Let F be a totally real number field and let o be the ring of integers in F. A totally positive quadratic form f over o is said to be almost regular with k exceptions if f represents all but k elements in F that are represented by f locally everywhere. In this paper, we show that for a fixed positive integer k, there are only finitely many similarity classes of positive definite almost regular ternary quadratic forms over o with at most k exceptions. This generalizes the corresponding finiteness result for positive definite ternary quadratic forms over Z by Watson (PhD Thesis, University College, London, 1953; Mathematika 1 (1954) 104–110).

2000 Mathematics Subject Classification 11E12, 11E20 (primary).

The research of the second author was supported by ACT05 and FONDECYT 1051004.


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