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In mathematics, a Tate module of an abelian group, named for John Tate, is a
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Mo ...
constructed from an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is com ...
''A''. Often, this construction is made in the following situation: ''G'' is a commutative group scheme over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
''K'', ''Ks'' is the
separable closure In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky ...
of ''K'', and ''A'' = ''G''(''Ks'') (the ''Ks''-valued points of ''G''). In this case, the Tate module of ''A'' is equipped with an
action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...
of the
absolute Galois group In mathematics, the absolute Galois group ''GK'' of a field ''K'' is the Galois group of ''K''sep over ''K'', where ''K''sep is a separable closure of ''K''. Alternatively it is the group of all automorphisms of the algebraic closure of ''K'' ...
of ''K'', and it is referred to as the Tate module of ''G''.


Definition

Given an abelian group ''A'' and a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...
''p'', the ''p''-adic Tate module of ''A'' is :T_p(A)=\underset A ^n/math> where ''A'' 'pn''is the ''pn'' torsion of ''A'' (i.e. the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine lea ...
of the multiplication-by-''pn'' map), and the
inverse limit In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits ...
is over
positive integer In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
s ''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 :z(a_n)_n=((z\textp^n)a_n)_n.


Examples


''The'' Tate module

When the abelian group ''A'' is the group of
roots of unity In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important i ...
in a separable closure ''Ks'' 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 ''GK'' 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 In mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups ha ...
G''m'',''K'' over ''K''.


The Tate module of an abelian variety

Given an
abelian variety In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functi ...
''G'' over a field ''K'', the ''Ks''-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, ''GK'', of ''K''). Classical results on abelian varieties show that if ''K'' has
characteristic zero In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive ide ...
, 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''. 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 In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conject ...
H^1_(G\times_KK^s,\mathbf_p). A special case of the
Tate conjecture In number theory and algebraic geometry, the Tate conjecture is a 1963 conjecture of John Tate that would describe the algebraic cycles on a variety in terms of a more computable invariant, the Galois representation on étale cohomology. The co ...
can be phrased in terms of Tate modules. Suppose ''K'' is finitely generated over its
prime field In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive id ...
(e.g. a
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subt ...
, an
algebraic number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...
, 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 :\mathrm_K(A,B)\otimes\mathbf_p\cong\mathrm_(T_p(A),T_p(B)) 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 ''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.
Gerd Faltings Gerd Faltings (; born 28 July 1954) is a German mathematician known for his work in arithmetic geometry. Education From 1972 to 1978, Faltings studied mathematics and physics at the University of Münster. In 1978 he received his PhD in mathema ...
proved the case where ''K'' is a number field in his celebrated "Mordell paper". 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 :k(C) \subset \hat k (C) \subset A^ \ where \hat k is an extension of ''k'' containing all ''p''-power roots of unity and ''A''(''p'') is the maximal unramified abelian ''p''-extension of \hat k (C).


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 In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...
, the other class of
global field In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields: *Algebraic number field: A finite extension of \mathbb *Global function fi ...
, introduced by
Kenkichi Iwasawa Kenkichi Iwasawa ( ''Iwasawa Kenkichi'', September 11, 1917 – October 26, 1998) was a Japanese mathematician who is known for his influence on algebraic number theory. Biography Iwasawa was born in Shinshuku-mura, a town near Kiryū, in Gun ...
. For a number field ''K'' we let ''K''''m'' denote the extension by ''p''''m''-power roots of unity, \hat K the union of the ''K''''m'' and ''A''(''p'') the maximal unramified abelian ''p''-extension of \hat K. Let :T_p(K) = \mathrm(A^/\hat K) \ . 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. Iwasawa exhibited ''T''''p''(''K'') as a module over the completion Z''p'' ''T'' and this implies a formula for the exponent of ''p'' in the order of the class groups ''C''''m'' of the form : \lambda m + \mu p^m + \kappa \ . The
Ferrero–Washington theorem In algebraic number theory, the Ferrero–Washington theorem, proved first by and later by , states that Iwasawa's μ-invariant vanishes for cyclotomic Z''p''-extensions of abelian algebraic number field In mathematics, an algebraic number fie ...
states that μ is zero.


See also

*
Tate conjecture In number theory and algebraic geometry, the Tate conjecture is a 1963 conjecture of John Tate that would describe the algebraic cycles on a variety in terms of a more computable invariant, the Galois representation on étale cohomology. The co ...
*
Tate twist In number theory and algebraic geometry, the Tate twist, 'The Tate Twist', https://ncatlab.org/nlab/show/Tate+twist named after John Tate, is an operation on Galois modules. For example, if ''K'' is a field, ''GK'' is its absolute Galois grou ...
*
Iwasawa theory In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by (), as part of the theory of cyclotomic fields. In the ...


Notes


References

* * * *Section 13 of *{{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 Abelian varieties