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en>Bbb23: Reverted edits by 82.4.96.75 (talk) to last version by Michael Bednarek - not notable - no article

2014-12-17T00:28:41Z

Reverted edits by 82.4.96.75 (talk) to last version by Michael Bednarek - not notable - no article

← Older revision		Revision as of 02:28, 17 December 2014
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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		I am Declan and was born on 1 November 1979. My hobbies are Rugby league football and Mineral collecting.<br><br>Here is my web site - [http://www.Lukezg.com/plus/guestbook.php how to get free fifa 15 coins]

	~~:<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>~~

	~~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]]~~

	~~: 0 → ''B''(''K'')/''f''(''A''(''K'')) → Sel~~<~~sup~~>~~(f)~~<~~/sup~~>~~(''A''/''K'') → Ш(''A''/''K'')[''f''] → 0.~~

	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.

	~~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]].~~

	~~== References ==~~

	*{{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}}
	*{{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}}~~
	*{{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}}
	* {{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}}
	* {{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}}
	*{{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}}~~
	*{{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 }}
	*{{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}}
	*{{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}}
	*{{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}}

	~~[[Category:Number theory]~~]

en>AnomieBOT: Dating maintenance tags: {{Citation needed}}

2014-01-30T00:24:07Z

Dating maintenance tags: {{Citation needed}}

← Older revision		Revision as of 02:24, 30 January 2014
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	~~I woke up another day~~ ~~and luke bryan tour date~~ ([~~http:~~//~~lukebryantickets.citizenswebcasting.com http:~~//~~lukebryantickets~~.~~citizenswebcasting.com~~]) ~~realised - I've been solitary for a while at~~ the ~~moment and following much bullying~~ from ~~friends I now locate myself registered~~ for ~~on line dating~~. ~~They promised me~~ that ~~there are plenty~~ of ~~regular,~~ [~~http~~://~~www~~.~~Reddit~~.~~com~~/~~r/howto/search?q=pleasant pleasant~~] ~~and fun visitors to meet up~~, ~~so here goes the toss!<br>I try to maintain~~ as ~~toned as possible being at~~ the ~~gym several~~-~~times weekly~~. ~~I love my sports~~ and ~~make an effort~~ to ~~play or see since many a potential. Being winter I~~'~~ll often at Hawthorn suits. Note: In case that you contemplated shopping a sport I really don~~'~~t mind, I have observed~~ the ~~carnage~~ of ~~wrestling fits at stocktake revenue~~.~~<br>My pals~~ and ~~family are magnificent and hanging out together at tavern gigabytes or dinners is obviously essential~~. ~~As I discover that you [~~http://~~www~~.~~cinemaudiosociety~~.~~org luke bryan cheap concert tickets~~] ~~could never get a nice conversation together with the sound~~ I ~~have never been into night clubs~~. ~~I likewise have two definitely cheeky~~ and [http://~~lukebryantickets~~.~~sgs-suparco~~.org ~~lukebrya~~] ~~very cunning puppies who are invariably enthusiastic to meet new individuals~~.<br><br>~~my web page~~; [http://~~okkyunglee~~.~~com luke bryan today show concert~~]		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

			:<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>

			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]]

			: 0 → ''B''(''K'')/''f''(''A''(''K'')) → Sel<sup>(f)</sup>(''A''/''K'') → Ш(''A''/''K'')[''f''] → 0.

			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.

			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]].

			== References ==

			*{{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}}
			*{{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}}
			*{{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}}
			* {{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}}
			* {{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}}
			*{{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}}
			*{{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 }}
			*{{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}}
			*{{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}}
			*{{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}}

			[[Category:Number theory]]

en>GoingBatty: Disambiguated: Someone Like You → Someone like You (Adele song)

2012-08-01T00:10:49Z

Disambiguated: Someone Like You → Someone like You (Adele song)

New page

I woke up another day and luke bryan tour date ([http://lukebryantickets.citizenswebcasting.com http://lukebryantickets.citizenswebcasting.com]) realised - I've been solitary for a while at the moment and following much bullying from friends I now locate myself registered for on line dating. They promised me that there are plenty of regular, [http://www.Reddit.com/r/howto/search?q=pleasant pleasant] and fun visitors to meet up, so here goes the toss!<br>I try to maintain as toned as possible being at the gym several-times weekly. I love my sports and make an effort to play or see since many a potential. Being winter I'll often at Hawthorn suits. Note: In case that you contemplated shopping a sport I really don't mind, I have observed the carnage of wrestling fits at stocktake revenue.<br>My pals and family are magnificent and hanging out together at tavern gigabytes or dinners is obviously essential. As I discover that you [http://www.cinemaudiosociety.org luke bryan cheap concert tickets] could never get a nice conversation together with the sound I have never been into night clubs. I likewise have two definitely cheeky and [http://lukebryantickets.sgs-suparco.org lukebrya] very cunning puppies who are invariably enthusiastic to meet new individuals.<br><br>my web page; [http://okkyunglee.com luke bryan today show concert]

← Older revision		Revision as of 02:28, 17 December 2014
Line 1:		Line 1:
	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		I am Declan and was born on 1 November 1979. My hobbies are Rugby league football and Mineral collecting.<br><br>Here is my web site - [http://www.Lukezg.com/plus/guestbook.php how to get free fifa 15 coins]

	~~:<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>~~

	~~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]]~~

	~~: 0 → ''B''(''K'')/''f''(''A''(''K'')) → Sel~~<~~sup~~>~~(f)~~<~~/sup~~>~~(''A''/''K'') → Ш(''A''/''K'')[''f''] → 0.~~

	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.

	~~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]].~~

	~~== References ==~~

	*{{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}}
	*{{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}}~~
	*{{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}}
	* {{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}}
	* {{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}}
	*{{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}}~~
	*{{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 }}
	*{{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}}
	*{{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}}
	*{{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}}

	~~[[Category:Number theory]~~]

← Older revision		Revision as of 02:24, 30 January 2014
Line 1:		Line 1:
	~~I woke up another day~~ ~~and luke bryan tour date~~ ([~~http:~~//~~lukebryantickets.citizenswebcasting.com http:~~//~~lukebryantickets~~.~~citizenswebcasting.com~~]) ~~realised - I've been solitary for a while at~~ the ~~moment and following much bullying~~ from ~~friends I now locate myself registered~~ for ~~on line dating~~. ~~They promised me~~ that ~~there are plenty~~ of ~~regular,~~ [~~http~~://~~www~~.~~Reddit~~.~~com~~/~~r/howto/search?q=pleasant pleasant~~] ~~and fun visitors to meet up~~, ~~so here goes the toss!<br>I try to maintain~~ as ~~toned as possible being at~~ the ~~gym several~~-~~times weekly~~. ~~I love my sports~~ and ~~make an effort~~ to ~~play or see since many a potential. Being winter I~~'~~ll often at Hawthorn suits. Note: In case that you contemplated shopping a sport I really don~~'~~t mind, I have observed~~ the ~~carnage~~ of ~~wrestling fits at stocktake revenue~~.~~<br>My pals~~ and ~~family are magnificent and hanging out together at tavern gigabytes or dinners is obviously essential~~. ~~As I discover that you [~~http://~~www~~.~~cinemaudiosociety~~.~~org luke bryan cheap concert tickets~~] ~~could never get a nice conversation together with the sound~~ I ~~have never been into night clubs~~. ~~I likewise have two definitely cheeky~~ and [http://~~lukebryantickets~~.~~sgs-suparco~~.org ~~lukebrya~~] ~~very cunning puppies who are invariably enthusiastic to meet new individuals~~.<br><br>~~my web page~~; [http://~~okkyunglee~~.~~com luke bryan today show concert~~]		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

			:<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>

			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]]

			: 0 → ''B''(''K'')/''f''(''A''(''K'')) → Sel<sup>(f)</sup>(''A''/''K'') → Ш(''A''/''K'')[''f''] → 0.

			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.

			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]].

			== References ==

			*{{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}}
			*{{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}}
			*{{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}}
			* {{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}}
			* {{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}}
			*{{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}}
			*{{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 }}
			*{{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}}
			*{{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}}
			*{{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}}

			[[Category:Number theory]]