Spinor norms of rotations of local integral quadratic forms
Let L be a regular \({\mathfrak o}\)-lattice with \({\mathfrak s}L\subseteq {\mathfrak o}\) and rank \(n\geq 3\), where \({\mathfrak o}\) is the ring of integers of a ramified quadratic extension of \({\mathbb{Q}}_ 2\). The author proves in the paper under review that if \(ord_{{\mathfrak p}} dL<n(n-2)\quad or\quad (n- 1)^ 2,\) according to whether n is even or odd, then \(U\subseteq \Theta (O^+(L))\), where U is the group of units in \({\mathfrak o}\) and \(\Theta\) is the spinor norm map. Using this result, combined with knowledge on spinor norm for non-dyadic local fields, the author gets a sufficient condition for \(g^+(L)\) dividing \(h_ F(S)\), where L is a lattice over a number field F and \(g^+(L)\) is the number of proper spinor genera in gen L and S is a set consisting of all but finitely many discrete spots in F and \(h_ F(S)\) is the order of the ideal class group of O(S), the ring of integers with respect to S in F.
scientific article; zbMATH DE number 4123812