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{{Short description|Mathematical concept}}
In [[mathematics]], particularly [[number theory]], an '''Euler system''' is a collection of compatible elements of [[Galois cohomology]] groups indexed by [[Field extension|fields]].  They were introduced by {{harvs|txt|authorlink=Victor Kolyvagin|last=Kolyvagin|year=1990}} in his work on [[Heegner point]]s on [[modular elliptic curve]]s, which was motivated by his earlier paper {{harvtxt|Kolyvagin|1988}} and  the work of {{harvtxt|Thaine|1988}}.  Euler systems are named after [[Leonhard Euler]] because  the factors relating different elements of an Euler system resemble the [[Euler factor]]s of an [[Euler product]].

Euler systems can be used to construct annihilators of [[ideal class group]]s or [[Selmer group]]s, thus giving bounds on their orders, which in turn has led to deep theorems such as the finiteness of some [[Tate-Shafarevich group]]s. This led to [[Karl Rubin]]'s new proof of the [[main conjecture of Iwasawa theory]], considered simpler than the original proof due to [[Barry Mazur]] and [[Andrew Wiles]].

==Definition==
Although there are several definitions of special sorts of Euler system, there seems to be no published  definition of an Euler system that covers all known cases. But it is possible to say roughly what an Euler system is, as follows: 
*An Euler system is given by collection of elements ''c''<sub>''F''</sub>. These elements are often indexed by certain [[number field]]s ''F'' containing some fixed number field ''K'', or by something closely related such as square-free integers. The elements ''c''<sub>''F''</sub> are typically elements of  some Galois cohomology group such as H<sup>1</sup>(''F'', ''T'') where ''T'' is a ''p''-adic representation of the [[absolute Galois group]] of ''K''.
*The most important condition is that the elements ''c''<sub>''F''</sub> and ''c''<sub>''G''</sub> for two different fields ''F''&nbsp;⊆&nbsp;''G'' are related by a simple formula, such as
:<math> {\rm cor}_{G/F}(c_G) = \prod_{q\in \Sigma(G/F)}P(\mathrm{Fr}_q^{-1}|{\rm Hom}_O(T,O(1));\mathrm{Fr}_q^{-1})c_F</math>
:Here the "Euler factor" ''P''(&tau;|''B'';''x'') is defined to be the element det(1-&tau;''x''|''B'') considered as an element of O[''x''], which when ''x'' happens to act on ''B'' is not the same as det(1-&tau;''x''|''B'') considered as an element of O.
*There may be other conditions that the ''c''<sub>''F''</sub> have to satisfy, such as congruence conditions.

[[Kazuya Kato]] refers to the elements in an Euler system as "arithmetic incarnations of zeta" and describes the property of being an Euler system as "an arithmetic reflection of the fact that these incarnations are related to special values of Euler products".<ref>{{harvnb|Kato|2007|loc=§2.5.1}}</ref>

==Examples==

===Cyclotomic units===
For every square-free positive integer ''n'' pick an ''n''-th root ζ<sub>''n''</sub> of 1, with ζ<sub>''mn''</sub> = ζ<sub>''m''</sub>ζ<sub>''n''</sub> for ''m'',''n'' coprime. Then the cyclotomic Euler system is the set of numbers 
α<sub>''n''</sub> = 1 &minus; ζ<sub>''n''</sub>. These satisfy the relations
:<math>N_{Q(\zeta_{nl})/Q(\zeta_l)}(\alpha_{nl}) = \alpha_n^{F_l-1}</math>
:<math>\alpha_{nl}\equiv\alpha_n </math> modulo all primes above ''l''
where ''l'' is a prime not dividing ''n'' and ''F''<sub>''l''</sub> is a Frobenius automorphism with ''F''<sub>''l''</sub>(ζ<sub>''n''</sub>) = ζ{{su|b=''n''|p=''l''}}.
Kolyvagin used this Euler system to give an elementary proof of the [[Gras conjecture]].

===Gauss sums===
{{empty section|date=April 2025}}

===Elliptic units===
{{empty section|date=April 2025}}

===Heegner points===
Kolyvagin constructed an Euler system from the [[Heegner point]]s of an elliptic curve, and used this to show that in some cases the [[Tate-Shafarevich group]] is finite.

===Kato's Euler system===
'''Kato's Euler system''' consists of certain elements occurring in the [[algebraic K-theory]] of [[modular curve]]s. These elements&mdash;named '''Beilinson elements''' after [[Alexander Beilinson]] who introduced them in {{harvtxt|Beilinson|1984}}&mdash;were used by Kazuya Kato in {{harvtxt|Kato|2004}} to prove one divisibility in Barry Mazur's [[main conjecture of Iwasawa theory]] for [[elliptic curve]]s.<ref>{{harvnb|Kato|2007}}</ref>

==Notes==
{{reflist}}

==References==
* {{eom|id=e/e120240|first=Grzegorz|last= Banaszak|title=Euler systems for number fields}}
*{{Citation
| last=Beilinson
| first=Alexander
| author-link=Alexander Beilinson
| contribution=Higher regulators and values of L-functions
| editor=R. V. Gamkrelidze
| title=Current problems in mathematics
| volume=24
| year=1984
| pages=181–238
| mr=0760999
| language=Russian
}}
* {{citation | first1=J.H. | last1=Coates | authorlink1=John Coates (mathematician) | first2=R. | last2=Greenberg | first3=K.A. | last3=Ribet | authorlink3=Kenneth Alan Ribet | first4=K. | last4=Rubin | authorlink4=Karl Rubin | title=Arithmetic Theory of Elliptic Curves | series=Lecture Notes in Mathematics | volume=1716 | publisher=[[Springer-Verlag]] | year=1999 | isbn=3-540-66546-3 }} 
* {{citation | first1=J. | last1=Coates | authorlink1=John Coates (mathematician) | first2=R. | last2=Sujatha | authorlink2=Sujatha Ramdorai | title=Cyclotomic Fields and Zeta Values | series=Springer Monographs in Mathematics | publisher=Springer-Verlag | year=2006 | isbn=3-540-33068-2 | chapter=Euler systems | pages=71–87 }}
*{{Citation
| last=Kato
| first=Kazuya
| author-link=Kazuya Kato
| contribution=''p''-adic Hodge theory and values of zeta functions of modular forms
|editor=Pierre Berthelot |editor2=Jean-Marc Fontaine |editor3=Luc Illusie |editor4=Kazuya Kato |editor5=Michael Rapoport
| title=Cohomologies p-adiques et applications arithmétiques. III.
| year=2004
| publisher=Société Mathématique de France
| location=Paris
| series=Astérisque
| volume=295
| pages=117–290
| mr=2104361
}}
*{{Citation
| last=Kato
| first=Kazuya
| author-link=Kazuya Kato
| contribution=Iwasawa theory and generalizations
| editor1=Marta Sanz-Solé | editor1-link=Marta Sanz-Solé | editor2=Javier Soria | editor3=Juan Luis Varona |display-editors = 3 | editor4=Joan Verdera
| title=International Congress of Mathematicians
| year=2007
| publisher=European Mathematical Society
| location=Zürich
| volume=I
| pages=335–357
| url=http://www.icm2006.org/proceedings/Vol_I/18.pdf
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| mr=2334196
}}. Proceedings of the congress held in Madrid, August 22–30, 2006
*{{Citation | last1=Kolyvagin | first1=V. A. | title=The Mordell-Weil and Shafarevich-Tate groups for Weil elliptic curves | mr=984214 | year=1988 | journal=Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya | issn=0373-2436 | volume=52 | issue=6 | pages=1154–1180}}
*{{Citation | last1=Kolyvagin | first1=V. A. | title=The Grothendieck Festschrift, Vol. II | publisher=Birkhäuser Boston | location=Boston, MA | series=Progr. Math. | isbn=978-0-8176-3428-5 | doi=10.1007/978-0-8176-4575-5_11 | mr=1106906 | year=1990 | volume=87 | chapter=Euler systems | pages=435–483}}
*{{Citation | last1=Mazur | first1=Barry | author1-link=Barry Mazur | last2=Rubin | first2=Karl | title=Kolyvagin systems | url=https://archive.org/details/kolyvaginsystems0168_799mazu/page/ | isbn=978-0-8218-3512-8 | mr=2031496 | year=2004 | journal=Memoirs of the American Mathematical Society | issn=0065-9266 | volume=168 | issue=799 | pages=[https://archive.org/details/kolyvaginsystems0168_799mazu/page/ viii+96] | doi=10.1090/memo/0799 | doi-access=free }}
* {{Citation | last1=Rubin | first1=Karl | title=Euler systems | url=https://books.google.com/books?isbn=0691050767 | publisher=[[Princeton University Press]] | series=Annals of Mathematics Studies | mr=1749177 | year=2000 | volume=147}}
*{{Citation | last1=Scholl | first1=A. J. | title=Galois representations in arithmetic algebraic geometry (Durham, 1996) | url=https://books.google.com/books?isbn=9780521644198  | publisher=[[Cambridge University Press]] | series=London Math. Soc. Lecture Note Ser. | isbn=978-0-521-64419-8  | mr=1696501 | year=1998 | volume=254 | chapter=An introduction to Kato's Euler systems | pages=379–460}}
*{{Citation | last1=Thaine | first1=Francisco | title=On the ideal class groups of real abelian number fields | doi=10.2307/1971460 | mr=951505 | year=1988 | journal=[[Annals of Mathematics]] |series=Second Series | issn=0003-486X | volume=128 | issue=1 | pages=1–18| jstor=1971460 | url=http://www.repositorio.unicamp.br/jspui/handle/REPOSIP/68685 | url-access=subscription }}

==External links==
* Several papers on Kolyvagin systems are available at [http://abel.math.harvard.edu/~mazur/projects.html Barry Mazur's web page] {{Webarchive|url=https://web.archive.org/web/20110517121127/http://abel.math.harvard.edu/~mazur/projects.html |date=2011-05-17 }} (as of July 2005).

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[[Category:Algebraic number theory]]