Representation by spinor genera of ternary quadratic forms
Rainer Schulze‐Pillot, Xu, Fei · Publications of the UdS (Saarland University) · 2003
over the ring of integers Z, where A and B are non-degenerate and symmetric matrices of size m × m and n × n over Z respectively, and A is indefinite with m ≥ 3. It is a necessary condition for solubility of equation (1.1) that it is solvable over Zp for all primes p and the real numbers R. This necessary condition is already sufficient if m − n ≥ 3 [Kn1,Hs]. However the equation (1.1) is no longer a purely local problem when m − n ≤ 2. By the Hasse principle, the necessary condition implies there is a rational solution of (1.1). In the previous papers [CX] and [X1], one of us has given conditions that allow to decide for m−n ≤ 2 whether the equation (1.1) is solvable over Z by looking at a given rational solution whose denominator is prime to the determinant of A. Can one also determine the solubility of (1.1) if the denominator of the rational solution is not prime to the determinant of A? In this note, we try to give such a condition. Notation and terminology are standard if not explained, or adopted from [CX] and [X1]. Let V be a quadratic space over a number field F with a non-degenerate symmetric bilinear form 〈x, y〉, Q(x) = 〈x, x〉 be the quadratic map on V and SO(V ) be the special orthogonal group of V . A lattice in V means a finitely generated oF module in V such that it generates a non-degenerate quadratic subspace of V . A full lattice means a lattice which generates the whole space. For a full lattice L, L denotes the dual lattice of L. For two lattices K and L in V , 〈K, L〉 denotes the fractional ideal generated by 〈x, y〉 for x ∈ K and y ∈ L. We use τz for the reflection if Q(z) 6= 0. We also denote n(L) and s(L) as norm and scale of a lattice L in the sense of [O] respectively. For any prime p of F , Vp (resp. Fp, etc.) denotes the local completion of V (resp. F , etc.). Let oF be the ring of integers of F . If p is a finite prime, the group of units of oFp is denoted by up, and πp is a uniformizer of Fp. We use θp to denote the spinor norm map of SO(Vp). For a lattice Kp and a full lattice Lp in Vp, let