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Tate module

释义

Definition
Examples
The Tate module The Tate module of an abelian variety
Tate module of a number field
See also
Notes
References

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

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

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.^[1] 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 .

A special case of the Tate conjecture can be phrased in terms of Tate modules.^[2] 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

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.^[3] Gerd Faltings proved the case where K is a number field in his celebrated "Mordell paper".^[4]

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

where is an extension of k containing all p-power roots of unity and A^(p) is the maximal unramified abelian p-extension of .^[5]

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 K_m denote the extension by p^m-power roots of unity, the union of the K_m and A^(p) the maximal unramified abelian p-extension of . Let

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.^[5]

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

The Ferrero–Washington theorem states that μ is zero.^[6]

Notes

1. ^{{harvnb|Murty|2000|loc=Proposition 13.4}}
2. ^{{harvnb|Murty|2000|loc=§13.8}}
3. ^{{harvnb|Tate|1966}}
4. ^{{harvnb|Faltings|1983}}
5. ^¹{{harvnb|Manin|Panchishkin|2007|p=245}}
6. ^{{harvnb|Manin|Panchishkin|2007|p=246}}

References

{{Citation

| last=Faltings
| first=Gerd
| author-link=Gerd Faltings
| year=1983
| title=Endlichkeitssätze für abelsche Varietäten über Zahlkörpern
| journal=Inventiones Mathematicae
| volume=73
| issue=3
| pages=349–366
| doi=10.1007/BF01388432
}}

{{Springer|id=t/t092270|title=Tate module}}
{{Citation

| last=Murty
| first=V. Kumar
| title=Introduction to abelian varieties
| series=CRM Monograph Series
| volume=3
| publisher=American Mathematical Society
| isbn=978-0-8218-1179-5
| year=2000
}}

Section 13 of {{Citation

| last=Rohrlich
| first=David
| contribution=Elliptic curves and the Weil–Deligne group
| title=Elliptic curves and related topics
| editor-last=Kisilevsky
| editor-first=Hershey
| editor2-last=Murty
| editor2-first=M. Ram
| year=1994
| isbn=978-0-8218-6994-9
| volume=4
| series=CRM Proceedings and Lecture Notes
| publisher=American Mathematical Society
}}

{{Citation

| last=Tate
| first=John
| title=Endomorphisms of abelian varieties over finite fields
| journal=Inventiones Mathematicae
| volume=2
| mr=0206004
| year=1966
| pages=134–144
| doi=10.1007/bf01404549
}}

1 : Abelian varieties

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