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| {{distinguish|Hodge–Tate module}}
| | Hello, my title is Andrew and my spouse doesn't like it at all. For years he's been living in Mississippi and he doesn't strategy on changing it. The favorite hobby for him and his kids is to play lacross and he would never give it up. My working day job is a travel agent.<br><br>Here is my site ... love psychics ([http://medialab.Zendesk.com/entries/54181460-Will-You-Often-End-Up-Bored-Try-One-Of-These-Hobby-Ideas- related website]) |
| In [[mathematics]], a '''Tate module''' of an abelian group, named for [[John Tate]], is a [[module (mathematics)|module]] constructed from an [[abelian group]] ''A''. Often, this construction is made in the following situation: ''G'' is a [[group scheme|commutative group scheme]] over a [[field (mathematics)|field]] ''K'', ''K<sup>s</sup>'' is the [[separable closure]] of ''K'', and ''A'' = ''G''(''K<sup>s</sup>'') (the [[Glossary of scheme theory#point|''K<sup>s</sup>''-valued points of ''G'']]). In this case, the Tate module of ''A'' is equipped with an [[group action|action]] of the [[absolute Galois group]] of ''K'', and it is referred to as the Tate module of ''G''.
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| ==Definition==
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| Given an abelian group ''A'' and a [[prime number]] ''p'', the '''''p''-adic Tate module of ''A''''' is
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| :<math>T_p(A)=\underset{\longleftarrow}{\lim} A[p^n]</math>
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| where ''A''[''p<sup>n</sup>''] is the [[torsion subgroup|''p<sup>n</sup>'' torsion]] of ''A'' (i.e. the [[kernel (algebra)|kernel]] of the multiplication-by-''p<sup>n</sup>'' map), and the [[inverse limit]] is over [[positive integer]]s ''n'' with [[transition morphism]]s given by the multiplication-by-''p'' map ''A''[''p<sup>n''+1</sup>] → ''A''[''p<sup>n</sup>'']. Thus, the Tate module encodes all the ''p''-power torsion of ''A''. It is equipped with the structure of a [[p-adic integer|'''Z'''<sub>''p''</sub>]]-module via
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| :<math>z(a_n)_n=((z\text{ mod }p^n)a_n)_n.</math>
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| ==Examples==
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| ===''The'' Tate module===
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| When the abelian group ''A'' is the group of [[roots of unity]] in a separable closure ''K<sup>s</sup>'' of ''K'', the ''p''-adic Tate module of ''A'' is sometimes referred to as ''the'' Tate module (where the choice of ''p'' and ''K'' are tacitly understood). It is a [[free module|free rank one module]] over ''Z''<sub>''p''</sub> with a linear action of the absolute Galois group ''G<sub>K</sub>'' of ''K''. Thus, it is a [[Galois representation]] also referred to as the [[cyclotomic character|''p''-adic cyclotomic character]] of ''K''. It can also be considered as the Tate module of the [[algebraic torus|multiplicative group scheme]] '''G'''<sub>''m'',''K''</sub> over ''K''.
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| ===The Tate module of an abelian variety===
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| Given an [[abelian variety]] ''G'' over a field ''K'', the ''K<sup>s</sup>''-valued points of ''G'' are an abelian group. The ''p''-adic Tate module ''T''<sub>''p''</sub>(''G'') of ''G'' is a Galois representation (of the absolute Galois group, ''G<sub>K</sub>'', of ''K'').
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| Classical results on abelian varieties show that if ''K'' has [[characteristic zero]], or characteristic ℓ where the prime number ''p'' ≠ ℓ, then ''T''<sub>''p''</sub>(''G'') is a free module over ''Z''<sub>''p''</sub> of rank 2''d'', where ''d'' is the dimension of ''G''.<ref>{{harvnb|Murty|2000|loc=Proposition 13.4}}</ref> In the other case, it is still free, but the rank may take any value from 0 to ''d'' (see for example [[Hasse–Witt matrix]]).
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| In the case where ''p'' is not equal to the characteristic of ''K'', the ''p''-adic Tate module of ''G'' is the [[Duality (mathematics)|dual]] of the [[étale cohomology]] <math>H^1_{\text{et}}(G\times_KK^s,\mathbf{Z}_p)</math>.
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| A special case of the [[Tate conjecture]] can be phrased in terms of Tate modules.<ref>{{harvnb|Murty|2000|loc=§13.8}}</ref> Suppose ''K'' is [[finitely generated algebra|finitely generated]] over its [[prime field]] (e.g. a [[finite field]], an [[algebraic number field]], a [[global function field]]), of characteristic different from ''p'', and ''A'' and ''B'' are two abelian varieties over ''K''. The Tate conjecture then predicts that
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| :<math>\mathrm{Hom}_K(A,B)\otimes\mathbf{Z}_p\cong\mathrm{Hom}_{G_K}(T_p(A),T_p(B))</math>
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| where Hom<sub>''K''</sub>(''A'', ''B'') is the group of [[abelian variety|morphisms of abelian varieties]] from ''A'' to ''B'', and the right-hand side is the group of ''G<sub>K</sub>''-linear maps from ''T<sub>p</sub>''(''A'') to ''T<sub>p</sub>''(''B''). The case where ''K'' is a finite field was proved by Tate himself in the 1960s.<ref>{{harvnb|Tate|1966}}</ref> [[Gerd Faltings]] proved the case where ''K'' is a number field in his celebrated "Mordell paper".<ref>{{harvnb|Faltings|1983}}</ref>
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| In the case of a Jacobian over a curve ''C'' over a finite field ''k'' of characteristic prime to ''p'', the Tate module can be identified with the Galois group of the composite extension
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| :<math>k(C) \subset \hat k (C) \subset A^{(p)} \ </math>
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| where <math> \hat k </math> is an extension of ''k'' containing all ''p''-power roots of unity and ''A''<sup>(''p'')</sup> is the maximal unramified abelian ''p''-extension of <math>\hat k (C)</math>.<ref name=MP245>{{harvnb|Manin|Panchishkin|2007|p=245}}</ref>
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| ==Tate module of a number field==
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| The description of the Tate module for the function field of a curve over a finite field suggests a definition for a Tate module of an [[algebraic number field]], the other class of [[global field]], introduced by Iwasawa. For a number field ''K'' we let ''K''<sub>''m''</sub> denote the extension by ''p''<sup>''m''</sup>-power roots of unity, <math>\hat K</math> the union of the ''K''<sub>''m''</sub> and ''A''<sup>(''p'')</sup> the maximal unramified abelian ''p''-extension of <math>\hat K</math>. Let
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| :<math>T_p(K) = \mathrm{Gal}(A^{(p)}/\hat K) \ . </math>
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| Then ''T''<sub>''p''</sub>(''K'') is a pro-''p''-group and so a '''Z'''<sub>''p''</sub>-module. Using [[class field theory]] one can describe ''T''<sub>''p''</sub>(''K'') as isomorphic to the inverse limit of the class groups ''C''<sub>''m''</sub> of the ''K''<sub>''m''</sub> under norm.<ref name=MP245/>
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| Iwasawa exhibited ''T''<sub>''p''</sub>(''K'') as a module over the completion '''Z'''<sub>''p''</sub><nowiki>[[</nowiki>''T''<nowiki>]]</nowiki> and this implies a formula for the exponent of ''p'' in the order of the class groups ''C''<sub>''m''</sub> of the form
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| :<math> \lambda m + \mu p^m + \kappa \ . </math>
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| The [[Ferrero–Washington theorem]] states that μ is zero.<ref name=MP246>{{harvnb|Manin|Panchishkin|2007|p=246}}</ref>
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| ==See also==
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| *[[Tate conjecture]]
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| *[[Tate twist]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{Citation
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| | last=Faltings
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| | first=Gerd
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| | author-link=Gerd Faltings
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| | year=1983
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| | title=Endlichkeitssätze für abelsche Varietäten über Zahlkörpern
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| | journal=Inventiones Mathematicae
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| | volume=73
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| | issue=3
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| | pages=349–366
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| | doi=10.1007/BF01388432
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| }}
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| *{{Springer|id=t/t092270|title=Tate module}}
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| *{{Citation
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| | last=Murty
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| | first=V. Kumar
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| | title=Introduction to abelian varieties
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| | series=CRM Monograph Series
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| | volume=3
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| | publisher=American Mathematical Society
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| | isbn=978-0-8218-1179-5
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| | year=2000
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| }}
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| *Section 13 of {{Citation
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| | last=Rohrlich
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| | first=David
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| | contribution=Elliptic curves and the Weil–Deligne group
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| | title=Elliptic curves and related topics
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| | editor-last=Kisilevsky
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| | editor-first=Hershey
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| | editor2-last=Murty
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| | editor2-first=M. Ram
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| | year=1994
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| | isbn=978-0-8218-6994-9
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| | volume=4
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| | series=CRM Proceedings and Lecture Notes
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| | publisher=[[American Mathematical Society]]
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| }}
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| *{{Citation
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| | last=Tate
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| | first=John
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| | title=Endomorphisms of abelian varieties over finite fields
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| | journal=Inventiones Mathematicae
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| | volume=2
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| | mr=0206004
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| | year=1966
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| | pages=134–144
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| }}
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| [[Category:Abelian varieties]]
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Hello, my title is Andrew and my spouse doesn't like it at all. For years he's been living in Mississippi and he doesn't strategy on changing it. The favorite hobby for him and his kids is to play lacross and he would never give it up. My working day job is a travel agent.
Here is my site ... love psychics (related website)