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Template:Distinguish 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, K^s is the separable closure of K, and A = G(K^s) (the K^s-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

$T_p(A)=\underset{⟵}{\lim }A[p^n]$

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 Z_p-module via

$z(a_n{)}_n=((z\text{ mod }{p}^n)a_n{)}_n.$

Examples

The Tate module

When the abelian group A is the group of roots of unity in a separable closure K^s 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 Z_p with a linear action of the absolute Galois group G_K 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 G_m,K over K.

The Tate module of an abelian variety

Given an abelian variety G over a field K, the K^s-valued points of G are an abelian group. The p-adic Tate module T_p(G) of G is a Galois representation (of the absolute Galois group, G_K, of K).

Classical results on abelian varieties show that if K has characteristic zero, or characteristic ℓ where the prime number p ≠ ℓ, then T_p(G) is a free module over Z_p 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 $H_{\text{et}}^1(G{\times }_K{K}^s,{\mathbf{Z}}_p)$.

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

${\mathrm{H}\mathrm{o}\mathrm{m}}_K(A,B)\otimes {\mathbf{Z}}_p\cong {\mathrm{H}\mathrm{o}\mathrm{m}}_{G_K}(T_p(A),T_p(B))$

where Hom_K(A, B) is the group of morphisms of abelian varieties from A to B, and the right-hand side is the group of G_K-linear maps from T_p(A) to T_p(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

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

where $\hat{k}$ is an extension of k containing all p-power roots of unity and A^(p) is the maximal unramified abelian p-extension of $\hat{k}(C)$.

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 Iwasawa. For a number field K we let K_m denote the extension by p^m-power roots of unity, $\hat{K}$ the union of the K_m and A^(p) the maximal unramified abelian p-extension of $\hat{K}$. Let

$T_p(K)=\mathrm{G}\mathrm{a}\mathrm{l}(A^{(p)}/\hat{K}).$

Then T_p(K) is a pro-p-group and so a Z_p-module. Using class field theory one can describe T_p(K) as isomorphic to the inverse limit of the class groups C_m of the K_m under norm.

Iwasawa exhibited T_p(K) as a module over the completion Z_p[[T]] and this implies a formula for the exponent of p in the order of the class groups C_m of the form

$\lambda m+\mu p^m+\kappa .$

The Ferrero–Washington theorem states that μ is zero.

See also

• Tate conjecture
• Tate twist

Notes

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References

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  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
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  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010