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Euler system
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<h2 id="definition">Definition</h2> <p>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: </p> <ul><li>An Euler system is given by collection of elements <i>c</i><i>F</i>. These elements are often indexed by certain <a href="/facts/Number_field/e0K0TXSh">number fields</a> <i>F</i> containing some fixed number field <i>K</i>, or by something closely related such as square-free integers. The elements <i>c</i><i>F</i> are typically elements of some Galois cohomology group such as H1(<i>F</i>, <i>T</i>) where <i>T</i> is a <i>p</i>-adic representation of the <a href="/facts/Absolute_Galois_group/fvcDwzWq">absolute Galois group</a> of <i>K</i>.</li> <li>The most important condition is that the elements <i>c</i><i>F</i> and <i>c</i><i>G</i> for two different fields <i>F</i> ⊆ <i>G</i> are related by a simple formula, such as</li></ul> c o r G / F ( c G ) = ∏ q ∈ Σ ( G / F ) P ( F r q − 1 | H o m O ( T , O ( 1 ) ) ; F r q − 1 ) c F {\displaystyle {\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}} Here the "Euler factor" <i>P</i>(τ|<i>B</i>;<i>x</i>) is defined to be the element det(1-τ<i>x</i>|<i>B</i>) considered as an element of O[<i>x</i>], which when <i>x</i> happens to act on <i>B</i> is not the same as det(1-τ<i>x</i>|<i>B</i>) considered as an element of O. <ul><li>There may be other conditions that the <i>c</i><i>F</i> have to satisfy, such as congruence conditions.</li></ul> <p><a href="/facts/Kazuya_Kato/F05g52S9">Kazuya Kato</a> 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".<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p> <h2 id="examples">Examples</h2> <h3>Cyclotomic units</h3> <p>For every square-free positive integer <i>n</i> pick an <i>n</i>-th root ζ<i>n</i> of 1, with ζ<i>mn</i> = ζ<i>m</i>ζ<i>n</i> for <i>m</i>,<i>n</i> coprime. Then the cyclotomic Euler system is the set of numbers α<i>n</i> = 1 − ζ<i>n</i>. These satisfy the relations </p> N Q ( ζ n l ) / Q ( ζ l ) ( α n l ) = α n F l − 1 {\displaystyle N_{Q(\zeta _{nl})/Q(\zeta _{l})}(\alpha _{nl})=\alpha _{n}^{F_{l}-1}} α n l ≡ α n {\displaystyle \alpha _{nl}\equiv \alpha _{n}} modulo all primes above <i>l</i> <p>where <i>l</i> is a prime not dividing <i>n</i> and <i>F</i><i>l</i> is a Frobenius automorphism with <i>F</i><i>l</i>(ζ<i>n</i>) = ζ<i>l</i><i>n</i>. Kolyvagin used this Euler system to give an elementary proof of the <a href="/facts/Gras_conjecture/H1BXMKEk">Gras conjecture</a>. </p> <h3>Gauss sums</h3> <h3>Elliptic units</h3> <h3>Heegner points</h3> <p>Kolyvagin constructed an Euler system from the <a href="/facts/Heegner_point/WYRrdqLj">Heegner points</a> of an elliptic curve, and used this to show that in some cases the <a href="/facts/Tate-Shafarevich_group/yCDadEcM">Tate-Shafarevich group</a> is finite. </p> <h3>Kato's Euler system</h3> <p>Kato's Euler system consists of certain elements occurring in the <a href="/facts/Algebraic_K-theory/0RY5M49W">algebraic K-theory</a> of <a href="/facts/Modular_curve/UE3kT5SW">modular curves</a>. These elements—named Beilinson elements after <a href="/facts/Alexander_Beilinson/dDfhNJzs">Alexander Beilinson</a> who introduced them in Beilinson (1984)—were used by Kazuya Kato in Kato (2004) to prove one divisibility in Barry Mazur's <a href="/facts/Main_conjecture_of_Iwasawa_theory/WfPLvMTT">main conjecture of Iwasawa theory</a> for <a href="/facts/Elliptic_curve/60jvaHSA">elliptic curves</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p> <h2 id="notes">Notes</h2> <ul><li>Banaszak, Grzegorz (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Euler_systems_for_number_fields">"Euler systems for number fields"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li> <li><a href="/facts/Alexander_Beilinson/dDfhNJzs">Beilinson, Alexander</a> (1984), "Higher regulators and values of L-functions", in R. V. Gamkrelidze (ed.), <i>Current problems in mathematics</i> (in Russian), vol. 24, pp. 181–238, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0760999">0760999</a></li> <li><a href="/facts/John_Coates_(mathematician)/P7XvMrHY">Coates, J.H.</a>; Greenberg, R.; <a href="/facts/Kenneth_Alan_Ribet/6hU9bzw8">Ribet, K.A.</a>; <a href="/facts/Karl_Rubin/3xhVe6yE">Rubin, K.</a> (1999), <i>Arithmetic Theory of Elliptic Curves</i>, Lecture Notes in Mathematics, vol. 1716, <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-66546-3</li> <li><a href="/facts/John_Coates_(mathematician)/P7XvMrHY">Coates, J.</a>; <a href="/facts/Sujatha_Ramdorai/70rf2Tqu">Sujatha, R.</a> (2006), "Euler systems", <i>Cyclotomic Fields and Zeta Values</i>, Springer Monographs in Mathematics, Springer-Verlag, pp. 71–87, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-33068-2</li> <li><a href="/facts/Kazuya_Kato/F05g52S9">Kato, Kazuya</a> (2004), "<i>p</i>-adic Hodge theory and values of zeta functions of modular forms", in Pierre Berthelot; Jean-Marc Fontaine; Luc Illusie; Kazuya Kato; Michael Rapoport (eds.), <i>Cohomologies p-adiques et applications arithmétiques. III.</i>, Astérisque, vol. 295, Paris: Société Mathématique de France, pp. 117–290, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2104361">2104361</a></li> <li><a href="/facts/Kazuya_Kato/F05g52S9">Kato, Kazuya</a> (2007), "Iwasawa theory and generalizations", in <a href="/facts/Marta_Sanz-Sol%25C3%25A9/PWbRqKp3">Marta Sanz-Solé</a>; Javier Soria; Juan Luis Varona; et al. (eds.), <a href="http://www.icm2006.org/proceedings/Vol_I/18.pdf"><i>International Congress of Mathematicians</i></a> (PDF), vol. I, Zürich: European Mathematical Society, pp. 335–357, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2334196">2334196</a>, retrieved 2010-08-12. Proceedings of the congress held in Madrid, August 22–30, 2006</li> <li>Kolyvagin, V. A. (1988), "The Mordell-Weil and Shafarevich-Tate groups for Weil elliptic curves", <i>Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya</i>, 52 (6): 1154–1180, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0373-2436">0373-2436</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0984214">0984214</a></li> <li>Kolyvagin, V. A. (1990), "Euler systems", <i>The Grothendieck Festschrift, Vol. II</i>, Progr. Math., vol. 87, Boston, MA: Birkhäuser Boston, pp. 435–483, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-0-8176-4575-5_11">10.1007/978-0-8176-4575-5_11</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8176-3428-5, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1106906">1106906</a></li> <li><a href="/facts/Barry_Mazur/4tGbAnCX">Mazur, Barry</a>; Rubin, Karl (2004), <a href="https://archive.org/details/kolyvaginsystems0168_799mazu/page/">"Kolyvagin systems"</a>, <i>Memoirs of the American Mathematical Society</i>, 168 (799): <a href="https://archive.org/details/kolyvaginsystems0168_799mazu/page/">viii+96</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2Fmemo%2F0799">10.1090/memo/0799</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-3512-8, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0065-9266">0065-9266</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2031496">2031496</a></li> <li>Rubin, Karl (2000), <a href="https://books.google.com/books?isbn=0691050767"><i>Euler systems</i></a>, Annals of Mathematics Studies, vol. 147, <a href="/facts/Princeton_University_Press/kwvFlKEj">Princeton University Press</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1749177">1749177</a></li> <li>Scholl, A. J. (1998), "An introduction to Kato's Euler systems", <a href="https://books.google.com/books?isbn=9780521644198"><i>Galois representations in arithmetic algebraic geometry (Durham, 1996)</i></a>, London Math. Soc. Lecture Note Ser., vol. 254, <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, pp. 379–460, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-64419-8, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1696501">1696501</a></li> <li>Thaine, Francisco (1988), <a href="http://www.repositorio.unicamp.br/jspui/handle/REPOSIP/68685">"On the ideal class groups of real abelian number fields"</a>, <i><a href="/facts/Annals_of_Mathematics/CUYktUqD">Annals of Mathematics</a></i>, Second Series, 128 (1): 1–18, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1971460">10.2307/1971460</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0003-486X">0003-486X</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1971460">1971460</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0951505">0951505</a></li></ul> <h2 id="external-links">External links</h2> <ul><li>Several papers on Kolyvagin systems are available at <a href="http://abel.math.harvard.edu/~mazur/projects.html">Barry Mazur's web page</a> <a href="https://web.archive.org/web/20110517121127/http://abel.math.harvard.edu/~mazur/projects.html">Archived</a> 2011-05-17 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a> (as of July 2005).</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Kato 2007, §2.5.1 - Kato, Kazuya (2007), "Iwasawa theory and generalizations", in Marta Sanz-Solé; Javier Soria; Juan Luis Varona; et al. (eds.), International Congress of Mathematicians (PDF), vol. I, Zürich: European Mathematical Society, pp. 335–357, MR 2334196, retrieved 2010-08-12 <a href="http://www.icm2006.org/proceedings/Vol_I/18.pdf" target="_blank">http://www.icm2006.org/proceedings/Vol_I/18.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Kato 2007 - Kato, Kazuya (2007), "Iwasawa theory and generalizations", in Marta Sanz-Solé; Javier Soria; Juan Luis Varona; et al. (eds.), International Congress of Mathematicians (PDF), vol. I, Zürich: European Mathematical Society, pp. 335–357, MR 2334196, retrieved 2010-08-12 <a href="http://www.icm2006.org/proceedings/Vol_I/18.pdf" target="_blank">http://www.icm2006.org/proceedings/Vol_I/18.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> </ol>