Tate module

A free -module associated to a -divisible group defined over a complete discrete valuation ring of characteristic 0 with residue field of characteristic . Let , , and , where is the algebraic closure of the quotient field of the ring ; the limit is taken with respect to the mappings for which . Then , where is the height of the group and has the natural structure of a -module. The functor allows one to reduce a number of questions about the group to simpler questions about -modules.

The Tate module is defined similarly for an Abelian variety. Let be an Abelian variety defined over , and let be the group of points of order in . Then is defined as . The Tate module of a curve is the Tate module of its Jacobi variety.

The construction of the module can be extended to number fields. Let be an algebraic number field and let be a -extension of the field (an extension with Galois group isomorphic to ). For the intermediate field of degree over , let be the -component of the ideal class group of the field . Then , where the limit is taken with respect to norm-mappings for . The module is characterized by its Iwasawa invariants , and , defined by

where for all sufficiently large . For cyclotomic -extensions the invariant is equal to 0. This was also proved for Abelian fields [4]. Examples are known of non-cyclotomic -extensions with (see [3]). Even in the case when , is not necessarily a free -module.

References

[1]	J.T. Tate, "-divisible groups" T.A. Springer (ed.) et al. (ed.) , Proc. Conf. local fields (Driebergen, 1966) , Springer (1967) pp. 158–183 MR0231827 Zbl 0157.27601
[2]	I.R. Shafarevich, "The zeta-function" , Moscow (1969) (In Russian)
[3]	K. Iwasawa, "On the -invariants of -extensions" , Number theory, algebraic geometry and commutative algebra , Kinokuniya (1973) pp. 1–11 MR357371
[4]	B. Ferrero, L.C. Washington, "The Iwasawa invariant vanishes for abelian number fields" Ann. of Math. , 109 (1979) pp. 377–395 MR528968 Zbl 0443.12001