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| In [[arithmetic geometry]], the '''Selmer group''', named in honor of the work of {{harvs|txt|authorlink=Ernst S. Selmer|last=Selmer|year=1951}} by {{harvtxt|Cassels|1962}}, is a group constructed from an [[isogeny]] of [[abelian varieties]]. The Selmer group of an abelian variety ''A'' with respect to an [[isogeny]] ''f'' : ''A'' → ''B'' of abelian varieties can be defined in terms of [[Galois cohomology]] as
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| :<math>\mathrm{Sel}^{(f)}(A/K)=\bigcap_v\mathrm{ker}(H^1(G_K,\mathrm{ker}(f))\rightarrow H^1(G_{K_v},A_v[f])/\mathrm{im}(\kappa_v))</math>
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| where ''A''<sub>v</sub>[''f''] denotes the ''f''-[[torsion (algebra)|torsion]] of ''A''<sub>v</sub> and <math>\kappa_v</math> is the local Kummer map <math>B_v(K_v)/f(A_v(K_v))\rightarrow H^1(G_{K_v},A_v[f])</math>. Note that <math>H^1(G_{K_v},A_v[f])/\mathrm{im}(\kappa_v)</math> is isomorphic to <math>H^1(G_{K_v},A_v)[f]</math>. Geometrically, the principal homogeneous spaces coming from elements of the Selmer group have ''K''<sub>v</sub>-rational points for all places ''v'' of ''K''. The Selmer group is finite. This implies that the part of the [[Tate–Shafarevich group]] killed by ''f'' is finite due to the following [[exact sequence]]
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| : 0 → ''B''(''K'')/''f''(''A''(''K'')) → Sel<sup>(f)</sup>(''A''/''K'') → Ш(''A''/''K'')[''f''] → 0.
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| The Selmer group in the middle of this exact sequence is finite and effectively computable. This implies the weak [[Mordell–Weil theorem]] that its subgroup ''B''(''K'')/''f''(''A''(''K'')) is finite. There is a notorious problem about whether this subgroup can be effectively computed: there is a procedure for computing it that will terminate with the correct answer if there is some prime ''p'' such that the ''p''-component of the Tate–Shafarevich group is finite. It is conjectured that the [[Tate–Shafarevich group]] is in fact finite, in which case any prime ''p'' would work. However, if (as seems unlikely) the [[Tate–Shafarevich group]] has an infinite ''p''-component for every prime ''p'', then the procedure may never terminate.
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| Ralph Greenberg has generalized the notion of Selmer group to more general ''p''-adic [[Galois representation]]s and to ''p''-adic variations of [[Motive (algebraic geometry)|motive]]s in the context of [[Iwasawa theory]].
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| == References ==
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| *{{Citation | last1=Cassels | first1=John William Scott | title=Arithmetic on curves of genus 1. III. The Tate–Šafarevič and Selmer groups | doi=10.1112/plms/s3-12.1.259 | id={{MR|0163913}} | year=1962 | journal=Proceedings of the London Mathematical Society. Third Series | issn=0024-6115 | volume=12 | pages=259–296}}
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| *{{Citation | last1=Cassels | first1=John William Scott | title=Lectures on elliptic curves | url=http://books.google.com/books?id=zgqUAuEJNJ4C | publisher=[[Cambridge University Press]] | series=London Mathematical Society Student Texts | isbn=978-0-521-41517-0 | id={{MR|1144763}} | year=1991 | volume=24}}
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| *{{Citation | last1=Châtelet | first1=François | title=Méthode galoisienne et courbes de genre un | id={{MR|0020575}} | year=1946 | journal=Annales de L'Université de Lyon Sect. A. (3) | volume=9 | pages=40–49}}
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| * {{Citation | last2=Silverman | first2=Joseph H. | author2-link=Joseph H. Silverman | last1=Hindry | first1=Marc | author1-link=Marc Hindry | title=Diophantine geometry: an introduction | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-98981-5 | year=2000 | volume=201}}
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| * {{Citation | last1=Greenberg | first1=Ralph | author1-link=Ralph Greenberg | editor1-last=Serre | editor1-first=Jean-Pierre | editor1-link=Jean-Pierre Serre | editor2-last=Jannsen | editor2-first=Uwe | editor3-last=Kleiman | editor3-first=Steven L. | title=Motives | publisher=[[American Mathematical Society]] | location=Providence, R.I. | isbn=978-0-8218-1637-0 | year=1994 | chapter=Iwasawa Theory and p-adic Deformation of Motives}}
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| *{{Citation | last1=Lang | first1=Serge | author1-link=Serge Lang | last2=Tate | first2=John | author2-link=John Tate | title=Principal homogeneous spaces over abelian varieties | url=http://www.jstor.org/stable/2372778 | id={{MR|0106226}} | year=1958 | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=80 | pages=659–684}}
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| *{{Citation | last1=Selmer | first1=Ernst S. | title=The Diophantine equation ''ax''<sup>3</sup> + ''by''<sup>3</sup> + ''cz''<sup>3</sup> = 0 | doi=10.1007/BF02395746 | id={{MR|0041871}} | year=1951 | journal=[[Acta Mathematica]] | issn=0001-5962 | volume=85 | pages=203–362 }}
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| *{{Citation | last1=Shafarevich | first1=I. R. | title=The group of principal homogeneous algebraic manifolds | language=Russian | id={{MR|0106227}} English translation in his collected mathematical papers | year=1959 | journal=Doklady Akademii Nauk SSSR | issn=0002-3264 | volume=124 | pages=42–43}}
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| *{{Citation | last1=Tate | first1=John | author1-link=John Tate | title=WC-groups over p-adic fields | url=http://www.numdam.org/item?id=SB_1956-1958__4__265_0 | publisher=Secrétariat Mathématique | location=Paris | series=Séminaire Bourbaki; 10e année: 1957/1958 | id={{MR|0105420}} | year=1958 | volume=13}}
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| *{{Citation | last1=Weil | first1=André | author1-link=André Weil | title=On algebraic groups and homogeneous spaces | url=http://www.jstor.org/stable/2372637 | id={{MR|0074084}} | year=1955 | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=77 | pages=493–512}}
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| [[Category:Number theory]]
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I am Declan and was born on 1 November 1979. My hobbies are Rugby league football and Mineral collecting.
Here is my web site - how to get free fifa 15 coins