Difference between revisions of "Tate module"
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− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> J.T. Tate, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227061.png" />-divisible groups" T.A. Springer (ed.) et al. (ed.) , ''Proc. Conf. local fields (Driebergen, 1966)'' , Springer (1967) pp. 158–183 {{MR|0231827}} {{ZBL|0157.27601}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> I.R. Shafarevich, "The zeta-function" , Moscow (1969) (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> K. Iwasawa, "On the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227062.png" />-invariants of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227063.png" />-extensions" , ''Number theory, algebraic geometry and commutative algebra'' , Kinokuniya (1973) pp. 1–11 {{MR|357371}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> B. Ferrero, L.C. Washington, "The Iwasawa invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227064.png" /> vanishes for abelian number fields" ''Ann. of Math.'' , '''109''' (1979) pp. 377–395 {{MR|528968}} {{ZBL|0443.12001}} </TD></TR></table> |
Revision as of 21:57, 30 March 2012
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, "![]() |
[2] | I.R. Shafarevich, "The zeta-function" , Moscow (1969) (In Russian) |
[3] | K. Iwasawa, "On the ![]() ![]() |
[4] | B. Ferrero, L.C. Washington, "The Iwasawa invariant ![]() |
Tate module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tate_module&oldid=23993