Hasse principleHassePrinciple2003-08-12 11:25:162003-08-14 18:16:51DefinitionHasse principleHasse conditionlocally solubleHasse principle% this is the default PlanetMath preamble. as your knowledge
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\newtheorem{cor}{Corollary}Let $V$ be an algebraic variety defined over a field $K$. By
$V(K)$ we denote the set of points on $V$ defined over $K$. Let
$\bar{K}$ be an algebraic closure of $K$. For a valuation $\nu$ of
$K$, we write $K_{\nu}$ for the completion of $K$ at $\nu$. In
this case, we can also consider $V$ defined over $K_{\nu}$ and
talk about $V(K_{\nu})$.
\begin{defn}\quad
\begin{enumerate}
\item If $V(K)$ is not empty we say that $V$ is \emph{soluble} in
$K$.
\item If $V(K_{\nu})$ is not empty then we say that $V$ is
\emph{locally soluble} at $\nu$.
\item If $V$ is locally soluble for all $\nu$ then we say that $V$
satisfies the \emph{Hasse condition}, or we say that $V/K$ is
\emph{everywhere locally soluble}.
\end{enumerate}
\end{defn}
The \emph{Hasse Principle} is the idea (or desire) that an
everywhere locally soluble variety $V$ must have a rational point,
i.e. a point defined over $K$. Unfortunately this is not true,
there are examples of varieties that satisfy the Hasse condition
but have no rational points.
{\bf Example}: A quadric (of any dimension) satisfies the Hasse
condition. This was proved by Minkowski for quadrics over
$\mathbb{Q}$ and by Hasse for quadrics over a number field.
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\bibitem{milne} Swinnerton-Dyer, {\em Diophantine Equations: Progress and Problems}, \PMlinkexternal{online notes}{http://swc.math.arizona.edu/notes/files/DLSSw-Dyer1.pdf}.
\end{thebibliography}