Spinor norms of local integral rotations. I

Abstract

The spinor norms of integral rotations on a modular quadratic form over a local field are determined. Whenever possible, these results are expressed in convenient closed forms.

Spinor norms of local integral rotations. II

“…[3], [4]). These computations have been performed whenever the local form is modular (see [2]). In the case of an arbitrary form, the Jordan splitting can be used to decompose the given form as an orthogonal sum of modular forms.…”

Representations by spinor genera

“…On the other hand, when d = 2, the theory is a good deal more intricate. We show that here too in most cases K is representable by every proper spinor genus in the genus of L; the exceptional cases will be pointed out, and there one needs to know the precise results for the local computations of the spinor norms of local integral rotations on L p ] the known facts about these are found in [3] for nondyadic p, in [1] for unramified dyadic p, and in [2] …”

Almost universal ternary sums of polygonal numbers

“…Therefore the only isometries of L 2 + ν are those fixing ν, and hence are precisely the isometries of K 2 described above. In particular, it follows from [11,Lemma 1] (L 2 + ν)). When p = 3, then we consider the generalized lattice M/L, as defined in [25], which has the orthogonal group (L 3 + ν)) is generated by pairs of symmetries coming…”