Tate module

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

:$T\_p(A)=\backslash underset\{\backslash longleftarrow\}\{\backslash lim\}\; A[p^n]$

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

:$z(a\_n)\_n=((z\backslash text\{\; mod\; \}p^n)a\_n)\_n.$

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.

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.{{harvnb|Murty|2000|loc=Proposition 13.4}} 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 $H^1\_\{\backslash text\{et\}\}(G\backslash times\_KK^s,\backslash mathbf\{Z\}\_p)$.

A special case of the Tate conjecture can be phrased in terms of Tate modules.{{harvnb|Murty|2000|loc=§13.8}} 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

:$\backslash mathrm\{Hom\}\_K(A,B)\backslash otimes\backslash mathbf\{Z\}\_p\backslash cong\backslash mathrm\{Hom\}\_\{G\_K\}(T\_p(A),T\_p(B))$

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.{{harvnb|Tate|1966}} Gerd Faltings proved the case where K is a number field in his celebrated "Mordell paper".{{harvnb|Faltings|1983}}

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

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

where $\backslash hat\; k$ is an extension of k containing all p-power roots of unity and A^(p) is the maximal unramified abelian p-extension of $\backslash hat\; k\; (C)$.{{harvnb|Manin|Panchishkin|2007|p=245}}

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, $\backslash hat\; K$ the union of the K_m and A^(p) the maximal unramified abelian p-extension of $\backslash hat\; K$. Let

:$T\_p(K)\; =\; \backslash mathrm\{Gal\}(A^\{(p)\}/\backslash hat\; K)\; \backslash \; .$

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_pT and this implies a formula for the exponent of p in the order of the class groups C_m of the form

:$\backslash lambda\; m\; +\; \backslash mu\; p^m\; +\; \backslash kappa\; \backslash \; .$

The Ferrero–Washington theorem states that μ is zero.{{harvnb|Manin|Panchishkin|2007|p=246}}

• Tate conjecture
• Tate twist
• Iwasawa theory

{{reflist}}

• {{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

| bibcode=1983InMat..73..349F

| s2cid=121049418

}}

• {{Springer|id=t/t092270|title=Tate module}}
• {{citation | first1=Yu. I. | last1=Manin | authorlink1=Yuri I. Manin | first2=A. A. | last2=Panchishkin | title=Introduction to Modern Number Theory | series=Encyclopaedia of Mathematical Sciences | volume=49 | edition=Second | year=2007 | isbn=978-3-540-20364-3 | issn=0938-0396 | zbl=1079.11002 }}
• {{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

| ref=none

}}

• {{Citation

| last=Tate

| first=John

| title=Endomorphisms of abelian varieties over finite fields

| journal=Inventiones Mathematicae

| volume=2

| mr=0206004

| year=1966

| issue=2

| pages=134–144

| doi=10.1007/bf01404549

| bibcode=1966InMat...2..134T

| s2cid=245902

}}

Category:Abelian varieties