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One of the central principles of Diophantine geometry, which reduces
+
{{MSC|11Dxx|11Gxx}}
the problem of the existence of rational points on an algebraic
+
 
variety over a global field to the analogous problem over local
+
The Hasse principle is one of the central principles of Diophantine
fields.
+
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
 
Let $M$ be a class of algebraic varieties over a global field $K$. The
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all primes $p$, are not empty, it follows that the set of rational
 
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
 
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
+
{{Cite|CaFr}}, and so it is valid for algebraic curves of genus
 
0 (see
 
0 (see
[[#References|[3]]]). For quadrics over a number field the Hasse
+
{{Cite|Ca}}). For quadrics over a number field the Hasse
 
principle was stated and proved by H. Hasse in
 
principle was stated and proved by H. Hasse in
[[#References|[1]]]. For cubic hypersurfaces the Hasse principle is
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{{Cite|Ha}}. For cubic hypersurfaces the Hasse principle is
 
not true, in general (see
 
not true, in general (see
[[#References|[3]]],
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{{Cite|Ca}},
[[#References|[4]]]); a counterexample (over $\Q$) is the projective
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{{Cite|Ma}}); a counterexample (over $\Q$) is the projective
 
curve $3x^3+4y^3+5z^3 = 0$ or the projective surface $5x^3+12y^3+9z^3+10t^3=0$.
 
curve $3x^3+4y^3+5z^3 = 0$ or the projective surface $5x^3+12y^3+9z^3+10t^3=0$.
  
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[[Galois cohomology|Galois cohomology]];
 
[[Galois cohomology|Galois cohomology]];
 
[[Weil–Châtelet group|Weil–Châtelet group]], and also
 
[[Weil–Châtelet group|Weil–Châtelet group]], and also
[[#References|[2]]],
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{{Cite|CaFr}},
[[#References|[3]]],
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{{Cite|Ca}},
[[#References|[5]]]). One says that the Hasse principle holds for $G$
+
{{Cite|Se}}). One says that the Hasse principle holds for $G$
 
if it holds for $M(G)$. The Hasse principle holds for simply-connected
 
if it holds for $M(G)$. The Hasse principle holds for simply-connected
 
and adjoint semi-simple algebraic groups over number fields
 
and adjoint semi-simple algebraic groups over number fields
([[#References|[5]]],
+
({{Cite|Se}},
[[#References|[6]]]). If $G$ is an Abelian variety, then the Hasse
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{{Cite|Ch}}). If $G$ is an Abelian variety, then the Hasse
 
principle holds for $G$ if and only if the Shafarevich–Tate group (cf.
 
principle holds for $G$ if and only if the Shafarevich–Tate group (cf.
 
[[Galois cohomology|Galois cohomology]]) of $G$ vanishes (see the
 
[[Galois cohomology|Galois cohomology]]) of $G$ vanishes (see the
 
examples in
 
examples in
[[#References|[7]]],
+
{{Cite|Ru}},
[[#References|[8]]]).
+
{{Cite|Ko}}).
  
 
====References====
 
====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
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|valign="top"|{{Ref|Ca}}||valign="top"| J.W.S. Cassels, "Diophantine equations with special reference to elliptic curves" ''J. London Math. Soc.'', '''41''' (1966) pp. 193–291
Angew. Math.'' , '''153''' (1924) pp. 113–130</TD></TR><TR><TD
+
|-
valign="top">[2]</TD> <TD valign="top"> J.W.S. Cassels (ed.)
+
|valign="top"|{{Ref|CaFr}}||valign="top"| J.W.S. Cassels (ed.) A. Fröhlich (ed.), ''Algebraic number theory'', Acad. Press (1967)
A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press
+
|-
(1967)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">
+
|valign="top"|{{Ref|Ch}}||valign="top"| V. Chernusov, "The Hasse principle for groups of type $E_8$", Minsk (1988) (In Russian)
J.W.S. Cassels, "Diophantine equations with special reference to
+
|-
elliptic curves" ''J. London Math. Soc.'' , '''41''' (1966)
+
|valign="top"|{{Ref|Ha}}||valign="top"| H. Hasse, "Darstellbarkeit von Zahlen durch quadratische Formen in einem beliebigen algebraischen Zahlkörper" ''J. Reine Angew. Math.'', '''153''' (1924) pp. 113–130
pp. 193–291</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">
+
|-
Yu.I. Manin, "Cubic forms. Algebra, geometry, arithmetic" ,
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|valign="top"|{{Ref|Ko}}||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)
North-Holland (1974) (Translated from Russian)</TD></TR><TR><TD
+
|-
valign="top">[5]</TD> <TD valign="top"> J.-P. Serre, "Cohomologie
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|valign="top"|{{Ref|Ma}}||valign="top"| Yu.I. Manin, "Cubic forms. Algebra, geometry, arithmetic", North-Holland (1974) (Translated from Russian)
Galoisienne" , Springer (1964)</TD></TR><TR><TD valign="top">[6]</TD>
+
|-
<TD valign="top"> V. Chernusov, "The Hasse principle for groups of
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|valign="top"|{{Ref|Ru}}||valign="top"| K. Rubin, "Tate–Shafarevich groups and $L$-functions of elliptic curves with complex multiplication" ''Invent. Math.'', '''89''' (1987) pp. 527–560
type $E_8$" , Minsk (1988) (In Russian)</TD></TR><TR><TD
+
|-
valign="top">[7]</TD> <TD valign="top"> K. Rubin, "Tate–Shafarevich
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|valign="top"|{{Ref|Se}}||valign="top"| J.-P. Serre, "Cohomologie Galoisienne", Springer (1964)
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>
 

Latest revision as of 15:48, 17 February 2012

2020 Mathematics Subject Classification: Primary: 11Dxx Secondary: 11Gxx [MSN][ZBL]

The Hasse principle is 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 [CaFr], and so it is valid for algebraic curves of genus 0 (see [Ca]). For quadrics over a number field the Hasse principle was stated and proved by H. Hasse in [Ha]. For cubic hypersurfaces the Hasse principle is not true, in general (see [Ca], [Ma]); 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; Weil–Châtelet group, and also [CaFr], [Ca], [Se]). 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 ([Se], [Ch]). If $G$ is an Abelian variety, then the Hasse principle holds for $G$ if and only if the Shafarevich–Tate group (cf. Galois cohomology) of $G$ vanishes (see the examples in [Ru], [Ko]).

References

[Ca] J.W.S. Cassels, "Diophantine equations with special reference to elliptic curves" J. London Math. Soc., 41 (1966) pp. 193–291
[CaFr] J.W.S. Cassels (ed.) A. Fröhlich (ed.), Algebraic number theory, Acad. Press (1967)
[Ch] V. Chernusov, "The Hasse principle for groups of type $E_8$", Minsk (1988) (In Russian)
[Ha] H. Hasse, "Darstellbarkeit von Zahlen durch quadratische Formen in einem beliebigen algebraischen Zahlkörper" J. Reine Angew. Math., 153 (1924) pp. 113–130
[Ko] 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)
[Ma] Yu.I. Manin, "Cubic forms. Algebra, geometry, arithmetic", North-Holland (1974) (Translated from Russian)
[Ru] K. Rubin, "Tate–Shafarevich groups and $L$-functions of elliptic curves with complex multiplication" Invent. Math., 89 (1987) pp. 527–560
[Se] J.-P. Serre, "Cohomologie Galoisienne", Springer (1964)
How to Cite This Entry:
Hasse principle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hasse_principle&oldid=19596
This article was adapted from an original article by Yu.G. Zarkhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article