Quadratic forms over p-adic integer rings

introduction

• Hilbert symbol
• Hasse invariant

\( \newcommand\Zp{\Z_p} \newcommand\Qp{\Q_p} \newcommand\Qpx{\Qp^\times} \newcommand\GL[2]{\operatorname{GL}_{#1}(#2)} \newcommand\GLnZ{\GL n\Z} \newcommand\GLnZp{\GL n{\Zp}} \newcommand\Znn{\Z_{\ge0}} \newcommand{\ord}[2]{{\rm ord}_{#1}(#2)} \newcommand\inv{^{-1}} \)

Hilbert symbol

• For \(a,b\in\Qpx\) the Hilbert symbol \((a,b)_p\) is \(1\) if \(aX^2+bY^2=Z^2\) has nontrivial

solutions in \(\Qp^3\) and \(-1\) if not.

Hasse invariant

• For \(u\in\GL m\Qp^{\rm sym}\) the Hasse invariant of \(u\) is \(h_p(u)=\prod_{i\le j}(a_i,a_j)_p\) where \(u\) is \(\GL m\Qp\)-equivalent to the diagonal matrix having entries \(a_1,\cdots,a_m\).

computational resource

• https://drive.google.com/file/d/0B8XXo8Tve1cxTERFcHVSQnpKUFU/view