Local densities of 2-adic quadratic forms
Let \((L,q)\) be a non-degenerate even integral quadratic lattice over \({\mathbb{Z}}_p\) of rank \(m\). Let \(T\) be an \(n\times n\) symmetric matrix over \({\mathbb{Z}}_p\). We define the quadratic local density of \(L\) representing \(T\) as \[ W(L,T)=\int_{\text{Sym}_n({\mathbb{Q}}_p)}\int_{L^n}\psi(\text{tr}\, bq(x))\,dx\;\psi (-\text{tr}\, Tb)\,db; \] where the additive character \(\psi:{\mathbb{Q}}_p\rightarrow {\mathbb{C}}^1\) is given by \(\psi(x)=e^{-2\pi i\lambda(x)}\) with \(\lambda:{\mathbb{Q}}_p/{\mathbb{Z}}_p\hookrightarrow {\mathbb{Q}}/{\mathbb{Z}}.\) The quadratic local density polynomial \(W(X;T,L)=W(T,L_s)\) is defined via \(W(p^s;T,L)=W(T,L_s)\), where \(L_s=L\oplus H^s\). Here \(H={\mathbb{Z}}_p^2\) is the standard hyperplane. In the paper under review the author deals with the case \(n=p=2\) and obtains explicit formulae for the local density polynomial \(W(X;T,L)\). In general, (sections 2-4) we get complicate formulae according to the shape of the Jordan form of \(T\). These results are obtained with a computer program. However, in the case where \(B\) is a quaternion algebra over \({\mathbb{Q}}_2\) with maximal order \({\mathcal{O}}_B\), and \(L\) is the lattice of trace zero elements in \({\mathcal{O}}_B\) with the reduced norm as its quadratic form, the formula for \(W(X;T,L)\) turns out to be simple in terms of the Gross-Keating invariants and to be independent of the shape of the Jordan form of \(T\).
scientific article; zbMATH DE number 2162041