"Quadratic forms over p-adic integer rings"의 두 판 사이의 차이
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imported>Pythagoras0 |
Pythagoras0 (토론 | 기여) |
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| (다른 사용자 한 명의 중간 판 하나는 보이지 않습니다) | |||
| 2번째 줄: | 2번째 줄: | ||
* Hilbert symbol | * Hilbert symbol | ||
* Hasse invariant | * Hasse invariant | ||
| − | + | <math> | |
\newcommand\Zp{\Z_p} | \newcommand\Zp{\Z_p} | ||
\newcommand\Qp{\Q_p} | \newcommand\Qp{\Q_p} | ||
| 12번째 줄: | 12번째 줄: | ||
\newcommand{\ord}[2]{{\rm ord}_{#1}(#2)} | \newcommand{\ord}[2]{{\rm ord}_{#1}(#2)} | ||
\newcommand\inv{^{-1}} | \newcommand\inv{^{-1}} | ||
| − | + | </math> | |
==Hilbert symbol== | ==Hilbert symbol== | ||
| − | * For | + | * For <math>a,b\in\Qpx</math> the Hilbert symbol <math>(a,b)_p</math> is <math>1</math> if <math>aX^2+bY^2=Z^2</math> has nontrivial |
| − | solutions in | + | solutions in <math>\Qp^3</math> and <math>-1</math> if not. |
==Hasse invariant== | ==Hasse invariant== | ||
| − | * For | + | * For <math>u\in\GL m\Qp^{\rm sym}</math> the Hasse invariant of <math>u</math> is <math>h_p(u)=\prod_{i\le j}(a_i,a_j)_p</math> where <math>u</math> is <math>\GL m\Qp</math>-equivalent to the diagonal matrix having entries <math>a_1,\cdots,a_m</math>. |
==computational resource== | ==computational resource== | ||
| 30번째 줄: | 30번째 줄: | ||
[[분류:quadratic forms]] | [[분류:quadratic forms]] | ||
| + | [[분류:migrate]] | ||
2020년 11월 16일 (월) 04:31 기준 최신판
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 \(a_1,\cdots,a_m\).