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

摘要

Let F be a totally real number field and let be the ring of integers in F. A totally positive quadratic form f over 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 with at most k exceptions. This generalizes the corresponding finiteness result for positive definite ternary quadratic forms over by Watson (PhD Thesis, University College, London, 1953; Mathematika 1 (1954) 104-110).