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| In [[mathematics]], [[Helmut Hasse]]'s '''local-global principle''', also known as the '''Hasse principle''', is the idea that one can find an [[diophantine equation|integer solution to an equation]] by using the [[Chinese remainder theorem]] to piece together solutions [[modular arithmetic|modulo]] powers of each different [[prime number]]. This is handled by examining the equation in the [[Completion_(ring_theory)|completions]] of the [[rational number]]s: the [[real number]]s and the [[p-adic number|''p''-adic numbers]]. A more formal version of the Hasse principle states that certain types of equations have a rational solution [[if and only if]] they have a solution in the [[real number]]s ''and'' in the ''p''-adic numbers for each prime ''p''.
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| ==Intuition==
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| Given a polynomial equation with rational coefficients, if it has rational solution, then this also yields a real solution and a ''p''-adic solution, as the rationals embed in the reals and ''p''-adics: a global solution yields local solutions at each prime. The Hasse principle asks when the reverse can be done, or rather, asks what the obstruction is: when can you patch together solutions over the reals and ''p''-adics to yield a solution over the rationals: when can local solutions be joined to form a global solution?
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| One can ask this for other rings or fields: integers, for instance, or [[number field]]s. For number fields, rather than reals and ''p''-adics, one uses complex embeddings and <math>\mathfrak p</math>-adics, for [[prime ideal]]s <math>\mathfrak p</math>.
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| ==Forms representing 0==
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| ===Quadratic forms===
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| The [[Hasse–Minkowski theorem]] states that the local-global principle holds for the problem of representing 0 by [[quadratic form]]s over the [[rational number]]s (which is [[Hermann Minkowski|Minkowski]]'s result); and more generally over any [[number field]] (as proved by Hasse), when one uses all the appropriate [[local field]] necessary conditions. [[Hasse's theorem on cyclic extensions]] states that the local-global principle applies to the condition of being a relative norm for a cyclic extension of number fields.
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| ===Cubic forms===
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| A counterexample by [[Ernst S. Selmer]] shows that the Hasse–Minkowski theorem cannot be extended to forms of degree 3: The cubic equation 3''x''<sup>3</sup> + 4''y''<sup>3</sup> + 5''z''<sup>3</sup> = 0 has a solution in real numbers, and in all p-adic fields, but it has no solution in which ''x'', ''y'', and ''z'' are all rational numbers.<ref>{{cite journal | author=Ernst S. Selmer | title=The Diophantine equation ''ax''<sup>3</sup> + ''by''<sup>3</sup> + ''cz''<sup>3</sup> = 0 | journal=Acta Mathematica | volume=85 | pages=203–362 | year=1957 | doi=10.1007/BF02395746 }}</ref>
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| [[Roger Heath-Brown]] showed<ref>{{cite journal | author=D.R. Heath-Brown | authorlink=Roger Heath-Brown | title=Cubic forms in 14 variables | journal=Invent. Math. | volume=170 | pages=199–230 | year=2007 | doi=10.1007/s00222-007-0062-1}}</ref> that every cubic form over the integers in at least 14 variables represents 0, improving on earlier results of [[Harold Davenport|Davenport]].<ref>{{cite journal | author=H. Davenport | title=Cubic forms in sixteen variables | journal=[[Proceedings of the Royal Society A]] | volume=272 | pages=285–303 | year=1963 | doi=10.1098/rspa.1963.0054 | issue=1350 }}</ref> Hence the local-global principle holds trivially for cubic forms over the rationals in at least 14 variables.
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| If we confine ourselves to non-singular forms, one can do better than this: Heath-Brown proved that every non-singular cubic form over the rational numbers in at least 10 variables represents 0,<ref>{{cite journal | author=D. R. Heath-Brown | authorlink=Roger Heath-Brown | title=Cubic forms in ten variables | journal=Proceedings of the London Mathematical Society| volume=47 | pages=225–257 | year=1983 | doi=10.1112/plms/s3-47.2.225 | issue=2 }}</ref> thus trivially establishing the Hasse principle for this class of forms. It is known that Heath-Brown's result is best possible in the sense that there exist non-singular cubic forms over the rationals in 9 variables that don't represent zero.<ref>{{cite journal | author=L. J. Mordell | authorlink=Louis Mordell | title=A remark on indeterminate equations in several variables | journal=Journal of the London Mathematical Society | volume=12 |pages=127–129 | year=1937 | doi=10.1112/jlms/s1-12.1.127 | issue=2 }}</ref> However, [[Christopher Hooley|Hooley]] showed that the Hasse principle holds for the representation of 0 by non-singular cubic forms over the rational numbers in at least nine variables.<ref>{{cite journal | author=C. Hooley | authorlink=Christopher Hooley | title=On nonary cubic forms | journal=J. Für die reine und angewandte Mathematik | volume=386 |pages=32–98 | year=1988 }}</ref> Davenport, Heath-Brown and Hooley all used the [[Hardy–Littlewood circle method]] in their proofs. According to an idea of [[Yuri Ivanovitch Manin|Manin]], the obstructions to the Hasse principle holding for cubic forms can be tied into the theory of the [[Brauer group]]; this is the [[Brauer–Manin obstruction]], which accounts completely for the failure of the Hasse principle for some classes of variety. However, [[Alexei Skorobogatov|Skorobogatov]] has shown that this is not the complete story.<ref>{{cite journal | author=Alexei N. Skorobogatov | title=Beyond the Manin obstruction | journal=Invent. Math. | volume=135 | issue=2 | pages=399–424 | year=1999 | doi=10.1007/s002220050291 }}</ref>
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| ===Forms of higher degree===
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| Counterexamples by [[Masahiko Fujiwara|Fujiwara]] and [[Masaki Sudo|Sudo]] show that the Hasse–Minkowski theorem is not extensible to forms of degree 10''n'' + 5, where ''n'' is a non-negative integer.<ref>{{cite journal | author=M. Fujiwara | authorlink=Masahiko Fujiwara | coauthors=[[Masaki Sudo|M. Sudo]] | title=Some forms of odd degree for which the Hasse principle fails | journal=Pacific Journal of Mathematics | volume=67 | year=1976 | issue=1 | pages=161–169}}</ref>
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| On the other hand, [[Birch's theorem]] shows that if ''d'' is any odd natural number, then there is a number ''N''(''d'') such that any form of degree ''d'' in more than ''N''(''d'') variables represents 0: the Hasse principle holds trivially.
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| ==Albert–Brauer–Hasse–Noether theorem==
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| The [[Albert–Brauer–Hasse–Noether theorem]] establishes a local-global principle for the splitting of a [[central simple algebra]] ''A'' over an algebraic number field ''K''. It states that if ''A'' splits over every [[local field|completion]] ''K''<sub>''v''</sub> then it is isomorphic to a [[matrix ring|matrix algebra]] over ''K''.
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| ==Hasse principle for algebraic groups==
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| The Hasse principle for algebraic groups states that if ''G'' is a simply-connected algebraic group defined over the global field ''k'' then the map from
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| :<math> H^1(k,G)\rightarrow\prod_s H^1(k_s,G)</math>
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| is injective, where the product is over all places ''s'' of ''k''.
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| The Hasse principle for orthogonal groups is closely related to the Hasse principle for the corresponding quadratic forms.
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| {{harvtxt|Kneser|1966}} and several others verified the Hasse principle by case-by-case proofs for each group. The last case was the group ''E''<sub>8</sub> which was only completed by {{harvtxt|Chernousov|1989}} many years after the other cases.
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| The Hasse principle for algebraic groups was used in the proofs of the [[Weil conjecture for Tamagawa numbers]] and the [[strong approximation theorem]].
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| ==See also==
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| * [[Local analysis]]
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| * [[Grunwald–Wang theorem]]
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| * [[Grothendieck–Katz p-curvature conjecture]]
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| ==Notes==
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| <references/>
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| ==References==
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| *{{citation|last= Chernousov|first= V. I. |title=The Hasse principle for groups of type E8 |journal= Soviet Math. Dokl. |volume= 39 |year=1989|pages= 592–596|mr= 1014762}}
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| *{{Citation | last1=Kneser | first1=Martin | title=Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) | publisher=[[American Mathematical Society]] | location=Providence, R.I. | id={{MR|0220736}} | year=1966 | chapter=Hasse principle for H¹ of simply connected groups | pages=159–163}}
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| * {{cite book | author=Serge Lang | authorlink=Serge Lang | title=Survey of Diophantine geometry | publisher=[[Springer-Verlag]] | year=1997 | isbn=3-540-61223-8 | pages=250–258 }}
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| * {{cite book | title=Torsors and rational points | author=Alexei Skorobogatov | series=Cambridge Tracts in Mathematics | volume=144 | year=2001 | isbn=0-521-80237-7 | pages=1–7,112 | publisher=Cambridge Univ. Press | location=Cambridge }}
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| == External links ==
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| * {{springer|title=Hasse principle|id=p/h046670}}
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| * [http://planetmath.org/encyclopedia/HassePrinciple.html PlanetMath article]
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| * Swinnerton-Dyer, ''Diophantine Equations: Progress and Problems'', [http://swc.math.arizona.edu/notes/files/DLSSw-Dyer1.pdf online notes]
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| [[Category:Algebraic number theory]]
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| [[Category:Diophantine equations]]
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| [[Category:Localization (mathematics)]]
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| [[Category:Mathematical principles]]
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