Tate module

{{Short description|Algebraic structure}} {{distinguish|Hodge–Tate module}} In mathematics, a '''Tate module''' of an abelian group, named for John Tate, is a module constructed from an abelian group ''A''. Often, this construction is made in the following situation: ''G'' is a commutative group scheme over a field ''K'', ''K^s'' is the separable closure of ''K'', and ''A'' = ''G''(''K^s'') (the ''K^s''-valued points of ''G''). In this case, the Tate module of ''A'' is equipped with an action of the absolute Galois group of ''K'', and it is referred to as the Tate module of ''G''.

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

:<math>T_p(A)=\underset{\longleftarrow}{\lim} A[p^n]</math>

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''[''p''^''n''+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

:<math>z(a_n)_n=((z\text{ mod }p^n)a_n)_n.</math>

==Examples== ===''The'' Tate module=== 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''.

===The Tate module of an abelian variety=== 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 2''d'', where ''d'' is the dimension of ''G''.<ref>{{harvnb|Murty|2000|loc=Proposition 13.4}}</ref> 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 <math>H^1_{\text{et}}(G\times_KK^s,\mathbf{Z}_p)</math>.

A special case of the Tate conjecture can be phrased in terms of Tate modules.<ref>{{harvnb|Murty|2000|loc=§13.8}}</ref> 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 :<math>\mathrm{Hom}_K(A,B)\otimes\mathbf{Z}_p\cong\mathrm{Hom}_{G_K}(T_p(A),T_p(B))</math> 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.<ref>{{harvnb|Tate|1966}}</ref> Gerd Faltings proved the case where ''K'' is a number field in his celebrated "Mordell paper".<ref>{{harvnb|Faltings|1983}}</ref>

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 :<math>k(C) \subset \hat k (C) \subset A^{(p)} \ </math> where <math> \hat k </math> is an extension of ''k'' containing all ''p''-power roots of unity and ''A''^(''p'') is the maximal unramified abelian ''p''-extension of <math>\hat k (C)</math>.<ref name=MP245>{{harvnb|Manin|Panchishkin|2007|p=245}}</ref>

==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 ''K''_''m'' denote the extension by ''p''^''m''-power roots of unity, <math>\hat K</math> the union of the ''K''_''m'' and ''A''^(''p'') the maximal unramified abelian ''p''-extension of <math>\hat K</math>. Let :<math>T_p(K) = \mathrm{Gal}(A^{(p)}/\hat K) \ . </math> 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.<ref name=MP245/>

Iwasawa exhibited ''T''_''p''(''K'') as a module over the completion '''Z'''_''p''<nowiki></nowiki>''T''<nowiki></nowiki> and this implies a formula for the exponent of ''p'' in the order of the class groups ''C''_''m'' of the form :<math> \lambda m + \mu p^m + \kappa \ . </math> The Ferrero–Washington theorem states that μ is zero.<ref name=MP246>{{harvnb|Manin|Panchishkin|2007|p=246}}</ref>

==See also== *Tate conjecture *Tate twist *Iwasawa theory

==Notes== {{reflist}}

==References== *{{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