Editing Hasse principle

{{Short description|Solving integer equations  from all modular solutions}}
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''.

==Intuition==
Given a polynomial equation with rational coefficients, if it has a 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?

One can ask this for other [[ring (algebra)|rings]] or [[field (algebra)|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>.

==Forms representing 0==
===Quadratic forms===
The [[Hasse–Minkowski theorem]] states that the local–global principle holds for the problem of [[Isotropic quadratic form|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.

===Cubic forms===
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>&nbsp;+&nbsp;4''y''<sup>3</sup>&nbsp;+&nbsp;5''z''<sup>3</sup>&nbsp;=&nbsp;0 has a solution in real numbers, and in all p-adic fields, but it has no nontrivial 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>&nbsp;+&nbsp;''by''<sup>3</sup>&nbsp;+&nbsp;''cz''<sup>3</sup>&nbsp;=&nbsp;0 | journal=Acta Mathematica | volume=85 | pages=203–362 | year=1951 | doi=10.1007/BF02395746 | doi-access=free }}</ref>

[[Roger Heath-Brown]] showed<ref name=HB>{{cite journal | author=D.R. Heath-Brown | authorlink=Roger Heath-Brown | title=Cubic forms in 14 variables | journal=Invent. Math. | volume=170 | issue=1 | pages=199–230 | year=2007 | doi=10.1007/s00222-007-0062-1| bibcode=2007InMat.170..199H | s2cid=16600794 }}</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 | bibcode=1963RSPSA.272..285D | s2cid=122443854 }}</ref> Since every cubic form over the p-adic numbers with at least ten variables represents 0,<ref name=HB/> the local–global principle holds trivially for cubic forms over the rationals in at least 14 variables.

Restricting 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 do not 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=Journal 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 the Brauer–Manin obstruction cannot explain all the failures of the Hasse principle.<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 | arxiv=alg-geom/9711006 | bibcode=1999InMat.135..399S | s2cid=14285244 }}</ref>

===Forms of higher degree===
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 |author2=M. Sudo |authorlink2=Masaki 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 | doi=10.2140/pjm.1976.67.161| doi-access=free }}</ref>

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.

==Albert–Brauer–Hasse–Noether theorem==
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''.

==Hasse principle for algebraic groups==

The Hasse principle for [[algebraic group]]s states that if ''G'' is a simply-connected algebraic group defined over the [[global field]] ''k'' then the map
:<math> H^1(k,G)\rightarrow\prod_s H^1(k_s,G)</math>
is injective, where the product is over all places ''s'' of ''k''.

The Hasse principle for orthogonal groups is closely related to the Hasse principle for the corresponding quadratic forms.

{{harvtxt|Kneser|1966}} and several others verified the Hasse principle by case-by-case proofs for each group. The last case was the group [[E8 (mathematics)|''E''<sub>8</sub>]] which was only completed by {{harvtxt|Chernousov|1989}} many years after the other cases.

The Hasse principle for algebraic groups was used in the proofs of the [[Weil conjecture for Tamagawa numbers]] and the [[strong approximation theorem]].

==See also==

* [[Local analysis]]
* [[Grunwald–Wang theorem]]
* [[Grothendieck–Katz p-curvature conjecture]]

==Notes==
<references/>

==References==

*{{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}}
*{{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. |mr=0220736 | year=1966 | chapter=Hasse principle for H¹ of simply connected groups | pages=159–163}}
* {{cite book | author=Serge Lang | authorlink=Serge Lang | title=Survey of Diophantine geometry | url=https://archive.org/details/surveyondiophant00lang | url-access=limited | publisher=[[Springer-Verlag]] | year=1997 | isbn=3-540-61223-8 | pages=[https://archive.org/details/surveyondiophant00lang/page/n263 250]–258 }}
* {{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=[https://archive.org/details/torsorsrationalp0000skor/page/1 1–7,112] | publisher=Cambridge Univ. Press | location=Cambridge | url=https://archive.org/details/torsorsrationalp0000skor/page/1 }}

== External links ==
* {{springer|title=Hasse principle|id=p/h046670}}
* {{PlanetMath|urlname=hasseprinciple|title=Hasse Principle}}
* Swinnerton-Dyer, ''Diophantine Equations: Progress and Problems'', [https://web.archive.org/web/20081012163238/http://swc.math.arizona.edu/notes/files/DLSSw-Dyer1.pdf online notes]
* {{cite journal|first1=J.|last1=Franklin|url=http://web.maths.unsw.edu.au/~jim/globallocalfinal.pdf|title=Globcal and local |journal=Mathematical Intelligencer|volume=36|number=4|year=2014|pages=4–9|doi=10.1007/s00283-014-9482-0}}

[[Category:Algebraic number theory]]
[[Category:Diophantine equations]]
[[Category:Localization (mathematics)]]
[[Category:Mathematical principles]]