Generation of integral orthogonal groups over dyadic local fields
Consider a dyadic field \(F\). Let \(V\) be a regular quadratic space over \(F\) and let \(L\) be a lattice on \(V\). The main theorem of the paper under review states that the orthogonal group \(O(L)\) of \(L\) is generated by symmetries and Eichler transformations unless the image of \(O(L)\) under the spinor norm coincides with the multiplicative group of \(F\). The same statement holds if \(L\) is modular or 2 is unramified. The latter result is due to O'Meara and Pollak. The author introduces the notion of `minimal norm Jordan splittings' and combines some of his earlier results with the techniques of O'Meara and Pollak for the proof of his main theorem. One possible application is to study the spinor genus theory over number fields. This will be done in a joint paper of the author, Hsia and Shao.
scientific article; zbMATH DE number 769667
scientific article