Tate module

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

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

\mathrm {Hom} _{K}(A,B)\otimes \mathbf {Z} _{p}\cong \mathrm {Hom} _{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.^[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

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 ${\hat {k}}(C)$ .^[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, ${\hat {K}}$ the union of the K_m and A^(p) the maximal unramified abelian p-extension of ${\hat {K}}$ . Let

T_{p}(K)=\mathrm {Gal} (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.^[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

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

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

Notes

^ Murty 2000, Proposition 13.4
^ Murty 2000, §13.8
^ Tate 1966
^ Faltings 1983
^ ^a ^b Manin & Panchishkin 2007, p. 245
^ Manin & Panchishkin 2007, p. 246

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