# On the orthogonal group of a lattice on a quadratic space over dyadic local field

Let $$F$$ be a local field with valuation ring $$R$$. $$V$$ is a n dimensional non-singular quadratic space over $$F$$ with bilinear form $$B$$ and quadratic map $$Q$$. As usual, $$O(V)$$ denotes the orthogonal group of $$V$$. Given a lattice $$L$$ on $$V$$. The orthogonal group of $$L$$ is defined to be $$O(L)=:\{\sigma\in O(V):\ \sigma (L)=L\}.$$

Clearly, for a maximal anisotropic vector $$u\in V$$, the symmetry $$\tau_u\in O(L)$$ if and only if $$\frac{2B(u,L)}{Q(u)}\in R$$. When $$F$$ is non-dyadic, this is equivalent to saying that the sublattice $$Ru$$ splits in $$L$$. And by induction on the dimension $$n$$, it is easy to show that each element of $$O(L)$$ is a product of at most $$2n-1$$ symmetries in $$O(L)$$.

Along this line, when $$F$$ is dyadic, are there any results on $$O(L)$$ similar to the non-dyadic case?

I don't think that it is true for dyadic local fields. But I think in that case $$O(L)$$ is generated by symmetries and Eichler-transformations. Try the paper $$$$Generation of integral orthogonal groups over dyadic local fields" by Fei Xu, appeared in Pacific J, 1995.