Quaternion Orders and Ternary Quadratic Forms Orders of Class Number One and Representations of Algebraic Integers by Quadratic Forms

Doktorsavhandling, 2000

Let R be the ring of integers in a totally real quadratic field K. The purpose of the thesis is to study totally definite quaternion R-orders and representations of elements in R by totally positive definite integral ternary quadratic forms. The thesis consists of three papers. In the first paper, we prove that R-orders of class number one in totally definite quaternion algebras over K only exist for K=Q(Sqrt(d)) with d=2,5,13,17. We show that there are twenty-eight isomorphism classes of such orders for d=2, twenty-five for d=5, nine for d=13 and thirteen for d=17. We describe one order L from each of these classes by finding a ternary quadratic form f such that the even part of the Clifford algebra C0(f) is isomorphic to L. In the second paper, we study representations of totally positive numbers N .epsilon. R, where R is a principal ideal domain in a totally real number field K, by totally positive definite ternary quadratic forms over R. We prove a quantitative formula relating the number of representations of N by different classes in the genus of a form f to the class number of R[Sqrt({-cfN})], where cf .epsilon. R is a constant only depending on f. We give an algebraic proof of a classical result of H. Maass on representations by sums of three squares over the integers R=Z[(1+Sqrt(5))/2]. We also obtain an explicit formula for the number of primitive representations of N .epsilon. R by the sum of three squares, relating the number of representations to the class number of R[Sqrt({-N})]. In the third paper, we study simultaneous embeddings of two maximal orders in totally imaginary quadratic extensions of K=Q(Sqrt(d)), for d=2,5,13,17, into totally definite quaternion algebras A over K. We give necessary and sufficient conditions under which the images of two embeddings generate an R-order in A. We find class numbers relations between the embedded orders and also some applications to representations of totally positive numbers N .epsilon. R by certain quadratic forms.