In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a Galois module is a
''G''-module, with ''G'' being the
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...
of some
extension of
fields
Fields may refer to:
Music
*Fields (band), an indie rock band formed in 2006
* Fields (progressive rock band), a progressive rock band formed in 1971
* ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010)
* "Fields", a song by ...
. The term Galois representation is frequently used when the ''G''-module is a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
over a field or a
free module
In mathematics, a free module is a module that has a ''basis'', that is, a generating set that is linearly independent. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commu ...
over a
ring
(The) Ring(s) may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
Arts, entertainment, and media Film and TV
* ''The Ring'' (franchise), a ...
in
representation theory
Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
, but can also be used as a synonym for ''G''-module. The study of Galois modules for extensions of
local
Local may refer to:
Geography and transportation
* Local (train), a train serving local traffic demand
* Local, Missouri, a community in the United States
Arts, entertainment, and media
* ''Local'' (comics), a limited series comic book by Bria ...
or
global field
In mathematics, a global field is one of two types of fields (the other one is local fields) that are characterized using valuations. There are two kinds of global fields:
*Algebraic number field: A finite extension of \mathbb
*Global functio ...
s and their
group cohomology
In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology ...
is an important tool in
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
.
Examples
*Given a field ''K'', the
multiplicative group
In mathematics and group theory, the term multiplicative group refers to one of the following concepts:
*the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referre ...
(''K
s'')
× of a
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'' is a Galois module for 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'' ...
. Its second
cohomology group
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to the
Brauer group
In mathematics, the Brauer group of a field ''K'' is an abelian group whose elements are Morita equivalence classes of central simple algebras over ''K'', with addition given by the tensor product of algebras. It was defined by the algebraist ...
of ''K'' (by
Hilbert's theorem 90
In abstract algebra, Hilbert's Theorem 90 (or Satz 90) is an important result on cyclic extensions of field (mathematics), fields (or to one of its generalizations) that leads to Kummer theory. In its most basic form, it states that if ''L''/''K'' ...
, its first cohomology group is zero).
*If ''X'' is a
smooth proper scheme over a field ''K'' then the
â„“-adic cohomology groups of its
geometric fibre are Galois modules for the absolute Galois group of ''K''.
Ramification theory
Let ''K'' be a
valued field (with valuation denoted ''v'') and let ''L''/''K'' be a
finite
Finite may refer to:
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked for person and/or tense or aspect
* "Finite", a song by Sara Gr ...
Galois extension
In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base field ...
with Galois group ''G''. For an
extension ''w'' of ''v'' to ''L'', let ''I
w'' denote its
inertia group. A Galois module ρ : ''G'' → Aut(''V'') is said to be unramified if ρ(''I
w'') = .
Galois module structure of algebraic integers
In classical
algebraic number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...
, let ''L'' be a Galois extension of a field ''K'', and let ''G'' be the corresponding Galois group. Then the ring ''O''
''L'' of
algebraic integer
In algebraic number theory, an algebraic integer is a complex number that is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients ...
s of ''L'' can be considered as an ''O''
''K'' 'G''module, and one can ask what its structure is. This is an arithmetic question, in that by the
normal basis theorem one knows that ''L'' is a free ''K''
'G''module of
rank
A rank is a position in a hierarchy. It can be formally recognized—for example, cardinal, chief executive officer, general, professor—or unofficial.
People Formal ranks
* Academic rank
* Corporate title
* Diplomatic rank
* Hierarchy ...
1. If the same is true for the integers, that is equivalent to the existence of a normal integral basis, i.e. of α in ''O''
''L'' such that its
conjugate elements under ''G'' give a free basis for ''O''
''L'' over ''O''
''K''. This is an interesting question even (perhaps especially) when ''K'' is the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example,
The set of all ...
field Q.
For example, if ''L'' = Q(), is there a normal integral basis? The answer is yes, as one sees by identifying it with Q(''ζ'') where
: ''ζ'' = exp(2''i''/3).
In fact all the subfields of the
cyclotomic field
In algebraic number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to \Q, the field of rational numbers.
Cyclotomic fields played a crucial role in the development of modern algebra and number theory ...
s for ''p''-th
roots of unity
In mathematics, a root of unity 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 in number theory, the theory of group char ...
for ''p'' a ''prime number'' have normal integral bases (over Z), as can be deduced from the theory of
Gaussian period
In mathematics, in the area of number theory, a Gaussian period is a certain kind of sum of root of unity, roots of unity. The periods permit explicit calculations in cyclotomic fields connected with Galois theory and with harmonic analysis (discre ...
s (the
Hilbert–Speiser theorem
In mathematics, the Hilbert–Speiser theorem is a result on cyclotomic fields, characterising those with a normal integral basis. More generally, it applies to any finite abelian extension of , which by the Kronecker–Weber theorem are isomorphi ...
). On the other hand, the
Gaussian field does not. This is an example of a ''necessary'' condition found by
Emmy Noether
Amalie Emmy Noether (23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She also proved Noether's theorem, Noether's first and Noether's second theorem, second theorems, which ...
(''perhaps known earlier?''). What matters here is ''tame''
ramification. In terms of the
discriminant
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the zero of a function, roots without computing them. More precisely, it is a polynomial function of the coef ...
''D'' of ''L'', and taking still ''K'' = Q, no prime ''p'' must divide ''D'' to the power ''p''. Then Noether's theorem states that tame ramification is necessary and sufficient for ''O
L'' to be a
projective module
In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, keeping some of the main properties of free modules. Various equivalent characterizati ...
over Z
'G'' It is certainly therefore necessary for it to be a ''free'' module. It leaves the question of the gap between free and projective, for which a large theory has now been built up.
A classical result, based on a result of
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time.
Hilbert discovered and developed a broad range of fundamental idea ...
, is that a tamely ramified
abelian number field has a normal integral basis. This may be seen by using the
Kronecker–Weber theorem
In algebraic number theory, it can be shown that every cyclotomic field is an abelian extension of the rational number field Q, having Galois group of the form modular arithmetic, (\mathbb Z/n\mathbb Z)^\times. The Kronecker–Weber theorem provide ...
to embed the abelian field into a cyclotomic field.
Galois representations in number theory
Many objects that arise in number theory are naturally Galois representations. For example, if ''L'' is a Galois extension of a
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 ...
''K'', the
ring of integers
In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often de ...
''O
L'' of ''L'' is a Galois module over ''O
K'' for the Galois group of ''L''/''K'' (see Hilbert–Speiser theorem). If ''K'' is a local field, the multiplicative group of its separable closure is a module for the absolute Galois group of ''K'' and its study leads to
local class field theory In mathematics, local class field theory, introduced by Helmut Hasse, is the study of abelian extensions of local fields; here, "local field" means a field which is complete with respect to an absolute value or a discrete valuation with a finite re ...
. For
global class field theory, the union of the
idele class group
In abstract algebra, an adelic algebraic group is a semitopological group defined by an algebraic group ''G'' over a number field ''K'', and the adele ring ''A'' = ''A''(''K'') of ''K''. It consists of the points of ''G'' having values in ''A''; th ...
s of all finite
separable extension
In field theory (mathematics), field theory, a branch of algebra, an algebraic field extension E/F is called a separable extension if for every \alpha\in E, the minimal polynomial (field theory), minimal polynomial of \alpha over is a separable po ...
s of ''K'' is used instead.
There are also Galois representations that arise from auxiliary objects and can be used to study Galois groups. An important family of examples are the
â„“-adic Tate modules of
abelian varieties
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular f ...
.
Artin representations
Let ''K'' be a number field.
Emil Artin
Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrians, Austrian mathematician of Armenians, Armenian descent.
Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number t ...
introduced a class of Galois representations of the absolute Galois group ''G
K'' of ''K'', now called Artin representations. These are the
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
finite-dimensional linear representations of ''G
K'' on
complex vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', can be added together and multiplied ("scaled") by numbers called ''scalars''. The operations of vector addition and sc ...
s. Artin's study of these representations led him to formulate the
Artin reciprocity law The Artin reciprocity law, which was established by Emil Artin in a series of papers (1924; 1927; 1930), is a general theorem in number theory that forms a central part of global class field theory
In mathematics, class field theory (CFT) is the f ...
and conjecture what is now called the
Artin conjecture concerning the
holomorphy of
Artin ''L''-functions.
Because of the incompatibility of the
profinite topology on ''G
K'' and the usual (Euclidean) topology on complex vector spaces, the
image
An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...
of an Artin representation is always finite.
â„“-adic representations
Let â„“ be a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), 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 ...
. An â„“-adic representation of ''G
K'' is a continuous
group homomorphism
In mathematics, given two groups, (''G'',∗) and (''H'', ·), a group homomorphism from (''G'',∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that
: h(u*v) = h(u) \cdot h(v)
whe ...
where ''M'' is either a finite-dimensional vector space over
â„“ (the algebraic closure of the
â„“-adic numbers Q
â„“) or a
finitely generated â„“-module (where
â„“ is the
integral closure
In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over a subring ''A'' of ''B'' if ''b'' is a root of some monic polynomial over ''A''.
If ''A'', ''B'' are fields, then the notions of "integral over" and ...
of Z
â„“ in
â„“). The first examples to arise were the
â„“-adic cyclotomic character and the â„“-adic Tate modules of abelian varieties over ''K''. Other examples come from the Galois representations of modular forms and automorphic forms, and the Galois representations on â„“-adic cohomology groups of algebraic varieties.
Unlike Artin representations, â„“-adic representations can have infinite image. For example, the image of ''G''
Q under the â„“-adic cyclotomic character is
. â„“-adic representations with finite image are often called Artin representations. Via an isomorphism of
â„“ with C they can be identified with ''bona fide'' Artin representations.
Mod â„“ representations
These are representations over a finite field of characteristic â„“. They often arise as the reduction mod â„“ of an â„“-adic representation.
Local conditions on representations
There are numerous conditions on representations given by some property of the representation restricted to a decomposition group of some prime. The terminology for these conditions is somewhat chaotic, with different authors inventing different names for the same condition and using the same name with different meanings. Some of these conditions include:
*Abelian representations. This means that the image of the Galois group in the representations is
abelian.
*Absolutely irreducible representations. These remain irreducible over an
algebraic 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 the field.
*Barsotti–Tate representations. These are similar to finite flat representations.
*Crystabelline representations
*Crystalline representations.
*de Rham representations.
*Finite flat representations. (This name is a little misleading, as they are really profinite rather than finite.) These can be constructed as a projective limit of representations of the Galois group on a finite flat
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 hav ...
.
*Good representations. These are related to the representations of
elliptic curves
In mathematics, an elliptic curve is a Smoothness, smooth, Projective variety, projective, algebraic curve of Genus of an algebraic curve, genus one, on which there is a specified point . An elliptic curve is defined over a field (mathematics), ...
with good reduction.
*Hodge–Tate representations.
*
Irreducible representation
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W, ...
s. These are irreducible in the sense that the only subrepresentation is the whole space or zero.
*Minimally ramified representations.
*Modular representations. These are representations coming from a
modular form
In mathematics, a modular form is a holomorphic function on the complex upper half-plane, \mathcal, that roughly satisfies a functional equation with respect to the group action of the modular group and a growth condition. The theory of modul ...
, but can also refer to
representations over fields of positive characteristic.
*Ordinary representations. These are related to the representations of elliptic curves with ordinary (non-supersingular) reduction. More precisely, they are 2-dimensional representations that are reducible with a 1-dimensional subrepresentation, such that the inertia group acts in a certain way on the submodule and the quotient. The exact condition depends on the author; for example it might act trivially on the quotient and by the character ε on the submodule.
*Potentially ''something'' representations. This means that the representations restricted to an open subgroup of finite index has some specified property.
*Reducible representations. These have a proper non-zero sub-representation.
*Semistable representations. These are two dimensional representations related to the representations coming from
semistable elliptic curves.
*Tamely ramified representations. These are trivial on the (first)
ramification group.
*Trianguline representations.
*Unramified representations. These are trivial on the inertia group.
*Wildly ramified representations. These are non-trivial on the (first) ramification group.
Representations of the Weil group
If ''K'' is a local or global field, the theory of
class formation In mathematics, a class formation is a topological group acting on a module satisfying certain conditions. Class formations were introduced by Emil Artin and John Tate to organize the various Galois groups and modules that appear in class fiel ...
s attaches to ''K'' its
Weil group
In mathematics, a Weil group, introduced by , is a modification of the absolute Galois group of a local or global field, used in class field theory. For such a field ''F'', its Weil group is generally denoted ''WF''. There also exists "finite leve ...
''W
K'', a continuous group homomorphism , and an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
of
topological group
In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures ...
s
:
where ''C
K'' is ''K''
× or the idele class group ''I
K''/''K''
× (depending on whether ''K'' is local or global) and is the
abelianization
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.
The commutator subgroup is important because it is the smallest normal s ...
of the Weil group of ''K''. Via φ, any representation of ''G
K'' can be considered as a representation of ''W
K''. However, ''W
K'' can have strictly more representations than ''G
K''. For example, via ''r
K'' the continuous complex characters of ''W
K'' are in bijection with those of ''C
K''. Thus, the absolute value character on ''C
K'' yields a character of ''W
K'' whose image is infinite and therefore is not a character of ''G
K'' (as all such have finite image).
An â„“-adic representation of ''W
K'' is defined in the same way as for ''G
K''. These arise naturally from geometry: if ''X'' is a smooth
projective variety
In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in \mathbb^n of some finite family of homogeneous polynomials that generate a prime ideal, th ...
over ''K'', then the â„“-adic cohomology of the geometric fibre of ''X'' is an â„“-adic representation of ''G
K'' which, via φ, induces an ℓ-adic representation of ''W
K''. If ''K'' is a local field of residue characteristic ''p'' ≠ℓ, then it is simpler to study the so-called Weil–Deligne representations of ''W
K''.
Weil–Deligne representations
Let ''K'' be a local field. Let ''E'' be a field of characteristic zero. A Weil–Deligne representation over ''E'' of ''W
K'' (or simply of ''K'') is a pair (''r'', ''N'') consisting of
* a continuous group homomorphism , where ''V'' is a finite-dimensional vector space over ''E'' equipped with the
discrete topology
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
,
* a
nilpotent
In mathematics, an element x of a ring (mathematics), ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0.
The term, along with its sister Idempotent (ring theory), idem ...
endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a g ...
such that ''r''(''w'')N''r''(''w'')
−1= , , ''w'', , ''N'' for all ''w'' ∈ ''W
K''.
[Here , , ''w'', , is given by where ''qK'' is the size of the residue field of ''K'' and ''v''(''w'') is such that ''w'' is equivalent to the −''v''(''w'')th power of the (arithmetic) Frobenius of ''WK''.]
These representations are the same as the representations over ''E'' of the
Weil–Deligne group of ''K''.
If the residue characteristic of ''K'' is different from â„“,
Grothendieck
Alexander Grothendieck, later Alexandre Grothendieck in French (; ; ; 28 March 1928 – 13 November 2014), was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. His research ext ...
's
â„“-adic monodromy theorem sets up a bijection between â„“-adic representations of ''W
K'' (over
ℓ) and Weil–Deligne representations of ''W
K'' over
â„“ (or equivalently over C). These latter have the nice feature that the continuity of ''r'' is only with respect to the discrete topology on ''V'', thus making the situation more algebraic in flavor.
See also
*
Compatible system of â„“-adic representations
*
Arboreal Galois representation
Notes
References
*
*
*
Further reading
*
*
{{Authority control
Algebraic number theory
Galois theory