Norm Euclidean Quaternionic Orders

We determine the norm Euclidean orders in a positive definite quaternion algebra over Q. Lagrange (1770) proved the four square theorem via Euler’s four square identity and a descent argument. Hurwitz [4] gave a quaternionic proof using the order Λ(2) with Z-basis: 1, i, j, 1 2 (1 + i + j + k). Here i = j = −1 and ij = −ji = k, the standard basis of the quaternions. The key property of Λ(2) is that it is norm Euclidean, namely, given α, β ∈ Λ(2) with β 6= 0, there exist q, r ∈ Λ(2) such that α = βq + r and N(r) < N(β). Liouville (1856) showed there are exactly seven positive definite quaternion norm forms (that is, 2-fold Pfister forms) x+ay+ bz +abw, with a, b positive integers, that represent all positive integers. These are (1, a, b, ab) = (1, 1, 1, 1) and (1, 1, 2, 2) (1, 1, 3, 3) (1, 2, 2, 4) (1, 2, 3, 6) (1, 2, 4, 8) (1, 2, 5, 10). See volume III of Dickson [1] for more details. Recently Deutsch constructed norm Euclidean orders to prove the universality of all but the last. Here we show there are, up to equivalence, exactly three norm Euclidean orders in a positive definite quaternion algebra over Q. This includes one in (−2,−5 Q ). 1 Quaternionic orders Q will denote a positive definite quaternion algebra over Q. Write Q = ( Q ). Here a, b < 0 are rational and Q has a basis e1 = 1, e2, e3, e4 with e 2 2 = a,