"Quadratic forms over p-adic integer rings"의 두 판 사이의 차이

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imported>Pythagoras0
잔글 (Pythagoras0 사용자가 Diagonalization over p-adic integers 문서를 Quadratic forms over p-adic integer rings 문서로 옮겼습니다)
imported>Pythagoras0
1번째 줄: 1번째 줄:
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==introduction==
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* Hilbert symbol
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* Hasse invariant
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$
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\newcommand\Zp{\Z_p}
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\newcommand\Qp{\Q_p}
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\newcommand\Qpx{\Qp^\times}
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\newcommand\GL[2]{\operatorname{GL}_{#1}(#2)}
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\newcommand\GLnZ{\GL n\Z}
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\newcommand\GLnZp{\GL n{\Zp}}
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\newcommand\Znn{\Z_{\ge0}}
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\newcommand{\ord}[2]{{\rm ord}_{#1}(#2)}
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\newcommand\inv{^{-1}}
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$
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==Hilbert symbol==
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* For $a,b\in\Qpx$ the Hilbert symbol $(a,b)_p$ is $1$ if $aX^2+bY^2=Z^2$ has nontrivial
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solutions in $\Qp^3$ and $-1$ if not.
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==Hasse invariant==
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* 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
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$u$ 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) 19:40 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 $a_1,\cdots,a_m$.


computational resource

원본 주소 "https://wiki.mathnt.net/index.php?title=Quadratic_forms_over_p-adic_integer_rings&oldid=40859"