"Quadratic forms over p-adic integer rings"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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==Hasse invariant== | ==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 | + | * 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 $a_1,\cdots,a_m$. |
| − | $u$ is $\GL m\Qp$-equivalent to the diagonal matrix having entries $a_1,\cdots,a_m$. | ||
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==computational resource== | ==computational resource== | ||
* https://drive.google.com/file/d/0B8XXo8Tve1cxTERFcHVSQnpKUFU/view | * https://drive.google.com/file/d/0B8XXo8Tve1cxTERFcHVSQnpKUFU/view | ||
2018년 2월 14일 (수) 20:08 판
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 $a_1,\cdots,a_m$.