Tate module
In
Definition
Given an abelian group A and a prime number p, the p-adic Tate module of A is
where A[pn] is the
Examples
The Tate module
When the abelian group A is the group of
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
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
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.[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 Km denote the extension by pm-power roots of unity, the union of the Km and A(p) the maximal unramified abelian p-extension of . Let
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.[5]
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
The Ferrero–Washington theorem states that μ is zero.[6]
See also
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
- S2CID 121049418
- "Tate module", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- 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, ISBN 978-0-8218-6994-9
- Tate, John (1966), "Endomorphisms of abelian varieties over finite fields", Inventiones Mathematicae, 2 (2): 134–144, S2CID 245902