Bon Durant, William R. Spinor norms of rotations of local integral quadratic forms. (English) Zbl 0686.10015 J. Number Theory 33, No. 1, 83-94 (1989). 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. Reviewer: Li Delang MSC: 11E08 Quadratic forms over local rings and fields 11E41 Class numbers of quadratic and Hermitian forms 11R23 Iwasawa theory Keywords:regular n-ary quadratic lattice; class number; indefinite quadratic form; quadratic number field; spinor norm map; spinor genera PDFBibTeX XMLCite \textit{W. R. Bon Durant}, J. Number Theory 33, No. 1, 83--94 (1989; Zbl 0686.10015) Full Text: DOI References: [1] Earnest, A. G.; Hsia, J. S., Spinor norms of local integral rotations, II, Pacific J. Math., 61, 71-86 (1975), errata, 115 (1984), 493-494 · Zbl 0334.10012 [2] A. G. Earnest and J. S. Hsia; A. G. Earnest and J. S. Hsia · Zbl 0393.10022 [3] Gerstein, L. J., Splitting quadratic forms over integers of global fields, Amer. J. Math., 91, 106-134 (1969) · Zbl 0179.08001 [4] Hsia, J. S., Spinor norms of local integral rotations, I, Pacific J. Math., 57, 199-206 (1975) · Zbl 0283.10009 [5] Kneser, M., Klassenzahlen indefiniter quadratischer Formen in drei oder mehr Veränderlichen, Arch. Math. (Basel), VII, 323-332 (1956) · Zbl 0071.27205 [6] O’Meara, O. T., (Introduction to Qaudratic Forms (1963), Springer: Springer New York) · Zbl 0107.03301 [7] O’Meara, O. T.; Pollak, Barth, Generation of local integral orthogonal groups, Math. Z., 87, 385-400 (1965) · Zbl 0134.26404 [8] O’Meara, O. T.; Pollak, Barth, Generation of local integral orthogonal groups, II, Math. Z., 93, 171-188 (1966) · Zbl 0161.02303 [9] Riehm, C., On the integral representations of quadratic forms over local fields, Amer. J. Math., 86, 25-62 (1964) · Zbl 0135.08702 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.