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A free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t0922701.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t0922702.png" /> associated to a [[P-divisible group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t0922703.png" />-divisible group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t0922704.png" /> defined over a complete discrete valuation ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t0922705.png" /> of characteristic 0 with residue field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t0922706.png" /> of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t0922707.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t0922708.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t0922709.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227011.png" /> is the algebraic closure of the quotient field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227012.png" /> of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227013.png" />; the limit is taken with respect to the mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227014.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227015.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227016.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227017.png" /> is the height of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227019.png" /> has the natural structure of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227020.png" />-module. The functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227021.png" /> allows one to reduce a number of questions about the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227022.png" /> to simpler questions about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227023.png" />-modules.
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The Tate module is defined similarly for an [[Abelian variety|Abelian variety]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227024.png" /> be an Abelian variety defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227025.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227026.png" /> be the group of points of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227027.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227028.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227029.png" /> is defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227030.png" />. The Tate module of a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227031.png" /> is the Tate module of its [[Jacobi variety|Jacobi variety]].
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The construction of the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227032.png" /> can be extended to number fields. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227033.png" /> be an algebraic number field and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227034.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227035.png" />-extension of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227036.png" /> (an extension with Galois group isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227037.png" />). For the intermediate field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227038.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227039.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227040.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227041.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227042.png" />-component of the ideal class group of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227043.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227044.png" />, where the limit is taken with respect to norm-mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227045.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227046.png" />. The module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227047.png" /> is characterized by its Iwasawa invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227050.png" />, defined by
+
A free  $  \mathbf Z _ {p} $-
 +
module $  T ( G) $
 +
associated to a [[P-divisible group| $  p $-
 +
divisible group]]  $  G $
 +
defined over a complete discrete valuation ring  $  R $
 +
of characteristic 0 with residue field $  k $
 +
of characteristic  $  p $.  
 +
Let  $  G = \{ G _  \nu  , i _  \nu  \} $,
 +
$  \nu \geq  0 $,
 +
and  $  T ( G) = \lim\limits _  \leftarrow  G _  \nu  ( \overline{K}\; ) $,  
 +
where  $  \overline{K}\; $
 +
is the algebraic closure of the quotient field  $  K $
 +
of the ring  $  R $;
 +
the limit is taken with respect to the mappings $  j _  \nu  : G _ {\nu + 1 }  \rightarrow G _  \nu  $
 +
for which  $  i _  \nu  \circ j _  \nu  = p $.  
 +
Then  $  T ( G) = \mathbf Z _ {p}  ^ {h} $,
 +
where  $  h $
 +
is the height of the group  $  G $
 +
and $  T ( G) $
 +
has the natural structure of a  $  G ( \overline{K}\; /K) $-
 +
module. The functor  $  G \rightarrow T ( G) $
 +
allows one to reduce a number of questions about the group  $  G $
 +
to simpler questions about  $  G ( \overline{K}\; /K) $-
 +
modules.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227051.png" /></td> </tr></table>
+
The Tate module is defined similarly for an [[Abelian variety|Abelian variety]]. Let  $  A $
 +
be an Abelian variety defined over  $  k $,
 +
and let  $  A _ {p  ^ {n}  } $
 +
be the group of points of order  $  p  ^ {n} $
 +
in  $  A ( \overline{k}\; ) $.  
 +
Then  $  T ( A) $
 +
is defined as  $  \lim\limits _  \leftarrow  A _ {p  ^ {n}  } $.  
 +
The Tate module of a curve  $  X $
 +
is the Tate module of its [[Jacobi variety|Jacobi variety]].
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227052.png" /> for all sufficiently large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227053.png" />. For cyclotomic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227054.png" />-extensions the invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227055.png" /> is equal to 0. This was also proved for Abelian fields [[#References|[4]]]. Examples are known of non-cyclotomic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227056.png" />-extensions with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227057.png" /> (see [[#References|[3]]]). Even in the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227059.png" /> is not necessarily a free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227060.png" />-module.
+
The construction of the module  $  T _ {p} ( X) $
 +
can be extended to number fields. Let  $  K $
 +
be an algebraic number field and let  $  k _  \infty  $
 +
be a  $  \mathbf Z _ {p} $-
 +
extension of the field  $  k $(
 +
an extension with Galois group isomorphic to  $  \mathbf Z _ {p} $).  
 +
For the intermediate field  $  k _ {n} $
 +
of degree  $  p  ^ {n} $
 +
over  $  k $,
 +
let  $  \mathop{\rm Cl} ( k _ {n} ) _ {p} $
 +
be the  $  p $-
 +
component of the ideal class group of the field  $  k _ {n} $.  
 +
Then  $  T _ {p} ( k _  \infty  ) = \lim\limits _  \leftarrow    \mathop{\rm Cl} ( k _ {n} ) _ {p} $,
 +
where the limit is taken with respect to norm-mappings  $  \mathop{\rm Cl} ( k _ {m} ) _ {p} \rightarrow  \mathop{\rm Cl} ( k _ {n} ) _ {p} $
 +
for  $  m > n $.
 +
The module  $  T _ {p} ( k _  \infty  ) $
 +
is characterized by its Iwasawa invariants  $  \lambda $,
 +
$  \mu $
 +
and  $  \nu $,
 +
defined by
 +
 
 +
$$
 +
|  \mathop{\rm Cl} ( k _ {n} ) _ {p} |  = \
 +
p ^ {e _ {n} } ,
 +
$$
 +
 
 +
where  $  e _ {n} = \lambda n + \mu p ^ {n + \nu } $
 +
for all sufficiently large $  n $.  
 +
For cyclotomic $  \mathbf Z _ {p} $-
 +
extensions the invariant $  \mu $
 +
is equal to 0. This was also proved for Abelian fields [[#References|[4]]]. Examples are known of non-cyclotomic $  \mathbf Z _ {p} $-
 +
extensions with $  \mu > 0 $(
 +
see [[#References|[3]]]). Even in the case when $  \mu = 0 $,  
 +
$  T _ {p} ( k _  \infty  ) $
 +
is not necessarily a free $  \mathbf Z _ {p} $-
 +
module.
  
 
====References====
 
====References====
 
<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>
 
<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>

Latest revision as of 08:25, 6 June 2020


A free $ \mathbf Z _ {p} $- module $ T ( G) $ associated to a $ p $- divisible group $ G $ defined over a complete discrete valuation ring $ R $ of characteristic 0 with residue field $ k $ of characteristic $ p $. Let $ G = \{ G _ \nu , i _ \nu \} $, $ \nu \geq 0 $, and $ T ( G) = \lim\limits _ \leftarrow G _ \nu ( \overline{K}\; ) $, where $ \overline{K}\; $ is the algebraic closure of the quotient field $ K $ of the ring $ R $; the limit is taken with respect to the mappings $ j _ \nu : G _ {\nu + 1 } \rightarrow G _ \nu $ for which $ i _ \nu \circ j _ \nu = p $. Then $ T ( G) = \mathbf Z _ {p} ^ {h} $, where $ h $ is the height of the group $ G $ and $ T ( G) $ has the natural structure of a $ G ( \overline{K}\; /K) $- module. The functor $ G \rightarrow T ( G) $ allows one to reduce a number of questions about the group $ G $ to simpler questions about $ G ( \overline{K}\; /K) $- modules.

The Tate module is defined similarly for an Abelian variety. Let $ A $ be an Abelian variety defined over $ k $, and let $ A _ {p ^ {n} } $ be the group of points of order $ p ^ {n} $ in $ A ( \overline{k}\; ) $. Then $ T ( A) $ is defined as $ \lim\limits _ \leftarrow A _ {p ^ {n} } $. The Tate module of a curve $ X $ is the Tate module of its Jacobi variety.

The construction of the module $ T _ {p} ( X) $ can be extended to number fields. Let $ K $ be an algebraic number field and let $ k _ \infty $ be a $ \mathbf Z _ {p} $- extension of the field $ k $( an extension with Galois group isomorphic to $ \mathbf Z _ {p} $). For the intermediate field $ k _ {n} $ of degree $ p ^ {n} $ over $ k $, let $ \mathop{\rm Cl} ( k _ {n} ) _ {p} $ be the $ p $- component of the ideal class group of the field $ k _ {n} $. Then $ T _ {p} ( k _ \infty ) = \lim\limits _ \leftarrow \mathop{\rm Cl} ( k _ {n} ) _ {p} $, where the limit is taken with respect to norm-mappings $ \mathop{\rm Cl} ( k _ {m} ) _ {p} \rightarrow \mathop{\rm Cl} ( k _ {n} ) _ {p} $ for $ m > n $. The module $ T _ {p} ( k _ \infty ) $ is characterized by its Iwasawa invariants $ \lambda $, $ \mu $ and $ \nu $, defined by

$$ | \mathop{\rm Cl} ( k _ {n} ) _ {p} | = \ p ^ {e _ {n} } , $$

where $ e _ {n} = \lambda n + \mu p ^ {n + \nu } $ for all sufficiently large $ n $. For cyclotomic $ \mathbf Z _ {p} $- extensions the invariant $ \mu $ is equal to 0. This was also proved for Abelian fields [4]. Examples are known of non-cyclotomic $ \mathbf Z _ {p} $- extensions with $ \mu > 0 $( see [3]). Even in the case when $ \mu = 0 $, $ T _ {p} ( k _ \infty ) $ is not necessarily a free $ \mathbf Z _ {p} $- 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
How to Cite This Entry:
Tate module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tate_module&oldid=23993
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article