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5.9
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
=
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)}
.
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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