Tate module GudangMovies21 Rebahinxxi LK21

      In mathematics, a Tate module of an abelian group, named for John Tate, is a module constructed from an abelian group A. Often, this construction is made in the following situation: G is a commutative group scheme over a field K, Ks is the separable closure of K, and A = G(Ks) (the Ks-valued points of G). In this case, the Tate module of A is equipped with an action of the absolute Galois group of K, and it is referred to as the Tate module of G.


      Definition


      Given an abelian group A and a prime number p, the p-adic Tate module of A is





      T

      p


      (
      A
      )
      =


      lim
      ⟵


      A
      [

      p

      n


      ]


      {\displaystyle T_{p}(A)={\underset {\longleftarrow }{\lim }}A[p^{n}]}


      where A[pn] is the pn torsion of A (i.e. the kernel of the multiplication-by-pn map), and the inverse limit is over positive integers n with transition morphisms given by the multiplication-by-p map A[pn+1] → A[pn]. Thus, the Tate module encodes all the p-power torsion of A. It is equipped with the structure of a Zp-module via




      z
      (

      a

      n



      )

      n


      =
      (
      (
      z

      mod


      p

      n


      )

      a

      n



      )

      n


      .


      {\displaystyle z(a_{n})_{n}=((z{\text{ mod }}p^{n})a_{n})_{n}.}



      Examples




      = The Tate module

      =
      When the abelian group A is the group of roots of unity in a separable closure Ks 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 rank one module over Zp with a linear action of the absolute Galois group GK of K. Thus, it is a Galois representation also referred to as the p-adic cyclotomic character of K. It can also be considered as the Tate module of the multiplicative group scheme Gm,K over K.


      = The Tate module of an abelian variety

      =
      Given an abelian variety G over a field K, the Ks-valued points of G are an abelian group. The p-adic Tate module Tp(G) of G is a Galois representation (of the absolute Galois group, GK, of K).
      Classical results on abelian varieties show that if K has characteristic zero, or characteristic ℓ where the prime number p ≠ ℓ, then Tp(G) is a free module over Zp of rank 2d, where d is the dimension of G. 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).
      In the case where p is not equal to the characteristic of K, the p-adic Tate module of G is the dual of the étale cohomology




      H

      et


      1


      (
      G

      ×

      K



      K

      s


      ,


      Z


      p


      )


      {\displaystyle H_{\text{et}}^{1}(G\times _{K}K^{s},\mathbf {Z} _{p})}

      .
      A special case of the Tate conjecture can be phrased in terms of Tate modules. Suppose K is 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






      H
      o
      m


      K


      (
      A
      ,
      B
      )
      ⊗


      Z


      p


      ≅


      H
      o
      m



      G

      K




      (

      T

      p


      (
      A
      )
      ,

      T

      p


      (
      B
      )
      )


      {\displaystyle \mathrm {Hom} _{K}(A,B)\otimes \mathbf {Z} _{p}\cong \mathrm {Hom} _{G_{K}}(T_{p}(A),T_{p}(B))}


      where HomK(A, B) is the group of morphisms of abelian varieties from A to B, and the right-hand side is the group of GK-linear maps from Tp(A) to Tp(B). The case where K is a finite field was proved by Tate himself in the 1960s. Gerd Faltings proved the case where K is a number field in his celebrated "Mordell paper".
      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




      k
      (
      C
      )
      ⊂



      k
      ^



      (
      C
      )
      ⊂

      A

      (
      p
      )





      {\displaystyle k(C)\subset {\hat {k}}(C)\subset A^{(p)}\ }


      where






      k
      ^





      {\displaystyle {\hat {k}}}

      is an extension of k containing all p-power roots of unity and A(p) is the maximal unramified abelian p-extension of






      k
      ^



      (
      C
      )


      {\displaystyle {\hat {k}}(C)}

      .


      Tate module of a number field


      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 Kenkichi Iwasawa. For a number field K we let Km denote the extension by pm-power roots of unity,






      K
      ^





      {\displaystyle {\hat {K}}}

      the union of the Km and A(p) the maximal unramified abelian p-extension of






      K
      ^





      {\displaystyle {\hat {K}}}

      . Let





      T

      p


      (
      K
      )
      =

      G
      a
      l

      (

      A

      (
      p
      )



      /




      K
      ^



      )

      .


      {\displaystyle T_{p}(K)=\mathrm {Gal} (A^{(p)}/{\hat {K}})\ .}


      Then Tp(K) is a pro-p-group and so a Zp-module. Using class field theory one can describe Tp(K) as isomorphic to the inverse limit of the class groups Cm of the Km under norm.
      Iwasawa exhibited Tp(K) as a module over the completion Zp[[T]] and this implies a formula for the exponent of p in the order of the class groups Cm of the form




      λ
      m
      +
      μ

      p

      m


      +
      κ

      .


      {\displaystyle \lambda m+\mu p^{m}+\kappa \ .}


      The Ferrero–Washington theorem states that μ is zero.


      See also


      Tate conjecture
      Tate twist
      Iwasawa theory


      Notes




      References


      Faltings, Gerd (1983), "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern", Inventiones Mathematicae, 73 (3): 349–366, Bibcode:1983InMat..73..349F, doi:10.1007/BF01388432, S2CID 121049418
      "Tate module", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
      Manin, Yu. I.; Panchishkin, A. A. (2007), Introduction to Modern Number Theory, Encyclopaedia of Mathematical Sciences, vol. 49 (Second ed.), ISBN 978-3-540-20364-3, ISSN 0938-0396, Zbl 1079.11002
      Murty, V. Kumar (2000), Introduction to abelian varieties, CRM Monograph Series, vol. 3, American Mathematical Society, ISBN 978-0-8218-1179-5
      Section 13 of Rohrlich, David (1994), "Elliptic curves and the Weil–Deligne group", in Kisilevsky, Hershey; Murty, M. Ram (eds.), Elliptic curves and related topics, CRM Proceedings and Lecture Notes, vol. 4, American Mathematical Society, ISBN 978-0-8218-6994-9
      Tate, John (1966), "Endomorphisms of abelian varieties over finite fields", Inventiones Mathematicae, 2 (2): 134–144, Bibcode:1966InMat...2..134T, doi:10.1007/bf01404549, MR 0206004, S2CID 245902

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