View source for Hasse principle

One of the central principles of Diophantine geometry, which reduces
the problem of the existence of rational points on an algebraic
variety over a global field to the analogous problem over local
fields.

Let $M$ be a class of algebraic varieties over a global field $K$. The
Hasse principle holds in $M$ if for any $X$ in $M$ such that for all
non-trivial absolute valuations $\nu$ on $K$ the set of $K_\nu$-rational
points $X(K_\nu)$ of $X$ is non-empty, the set of $K$-rational points $X(K)$ is
also not empty (where $K_\nu$ is the completion of $K$ relative to
$\nu$). In particular, if $K$ is the field $\Q$ of rational numbers, then
if the set of real points $X(\R)$ and the set of $p$-adic points $\Q_p$, for
all primes $p$, are not empty, it follows that the set of rational
points $X(\Q)$ is also not empty. The Hasse principle holds for quadrics
[[#References|[2]]], and so it is valid for algebraic curves of genus
0 (see
[[#References|[3]]]). For quadrics over a number field the Hasse
principle was stated and proved by H. Hasse in
[[#References|[1]]]. For cubic hypersurfaces the Hasse principle is
not true, in general (see
[[#References|[3]]],
[[#References|[4]]]); a counterexample (over $\Q$) is the projective
curve $3x^3+4y^3+5z^3 = 0$ or the projective surface $5x^3+12y^3+9z^3+10t^3=0$.

Let $G$ be an algebraic group over $K$ and let $M(G)$ be the class of
algebraic varieties consisting of all principal homogeneous spaces
over $G$ (see
[[Galois cohomology|Galois cohomology]];
[[Weil–Châtelet group|Weil–Châtelet group]], and also
[[#References|[2]]],
[[#References|[3]]],
[[#References|[5]]]). One says that the Hasse principle holds for $G$
if it holds for $M(G)$. The Hasse principle holds for simply-connected
and adjoint semi-simple algebraic groups over number fields
([[#References|[5]]],
[[#References|[6]]]). If $G$ is an Abelian variety, then the Hasse
principle holds for $G$ if and only if the Shafarevich–Tate group (cf.
[[Galois cohomology|Galois cohomology]]) of $G$ vanishes (see the
examples in
[[#References|[7]]],
[[#References|[8]]]).

====References====
<table><TR><TD valign="top">[1]</TD>
 <TD valign="top"> H. Hasse, "Darstellbarkeit von Zahlen durch quadratische Formen in einem beliebigen algebraischen Zahlkörper" ''J. Reine Angew. Math.'' , '''153''' (1924) pp. 113–130</TD>
</TR><TR><TD valign="top">[2]</TD>
 <TD valign="top"> J.W.S. Cassels (ed.) A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press (1967)</TD>
</TR><TR><TD valign="top">[3]</TD>
 <TD valign="top"> J.W.S. Cassels, "Diophantine equations with special reference to elliptic curves" ''J. London Math. Soc.'' , '''41''' (1966) pp. 193–291</TD>
</TR><TR><TD valign="top">[4]</TD>
 <TD valign="top"> Yu.I. Manin, "Cubic forms. Algebra, geometry, arithmetic" , North-Holland (1974) (Translated from Russian)</TD>
</TR><TR><TD valign="top">[5]</TD>
 <TD valign="top"> J.-P. Serre, "Cohomologie Galoisienne" , Springer (1964)</TD>
</TR><TR><TD valign="top">[6]</TD>
 <TD valign="top"> V. Chernusov, "The Hasse principle for groups of type $E_8$" , Minsk (1988) (In Russian)</TD>
</TR><TR><TD valign="top">[7]</TD>
 <TD valign="top"> K. Rubin, "Tate–Shafarevich groups and $L$-functions of elliptic curves with complex multiplication" ''Invent. Math.'' , '''89''' (1987) pp. 527–560</TD>
</TR><TR><TD valign="top">[8]</TD>
 <TD valign="top"> V. Kolyvagin, "The Mordell–Weil groups and the Shafarevich–Tate groups of Weil's elliptic curves" ''Izv. Akad. Nauk. SSSR Ser. Mat.'' , '''52''' : 6 (1988)</TD>
</TR></table>