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