Galois module
Encyclopedia
In mathematics
, a Galois module is a G-module
where G is the Galois group
of some extension
of fields
. The term Galois representation is frequently used when the G-module is a vector space
over a field
or a free module
over a ring
, but can also be used as a synonym for G-module. The study of Galois modules for extensions of local
or global field
s is an important tool in number theory
.
with Galois group G. For an extension w of v to L, let Iw denote its inertia group. A Galois module ρ : G → Aut(V) is said to be unramified if ρ(Iw) = {1}.
, let L be a Galois extension of a field K, and let G be the corresponding Galois group. Then the ring OL of algebraic integer
s of L can be considered as an OK[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 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 OL such that its conjugate elements under G give a free basis for OL over OK. This is an interesting question even (perhaps especially) when K is the rational number
field Q.
For example, if L = Q(√-3), is there a normal integral basis? The answer is yes, as one sees by identifying it with Q(ζ) where
In fact all the subfields of the cyclotomic field
s for p-th roots of unity for p a prime number have normal integral bases (over Z), as can be deduced from the theory of Gaussian period
s (the Hilbert–Speiser theorem). On the other hand the Gaussian field
does not. This is an example of a necessary condition found by Emmy Noether
(perhaps known earlier?). What matters here is tame ramification
. In terms of the discriminant
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 OL to be a projective module
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.
of a number field K, the ring of integers
OL of L is a Galois module over OK 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. For global class field theory, the union of the idele class groups of all finite separable extension
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 ℓ-adic Tate modules
of abelian varieties
.
introduced a class of Galois representations of the absolute Galois group GK of K, now called Artin representations. These are the continuous
finite-dimensional linear representations of GK on complex vector space
s. Artin's study of these representations led him to formulate the Artin reciprocity law 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 GK and the usual (Euclidean) topology on complex vector spaces, the image
of an Artin representation is always finite.
. An ℓ-adic representation of GK is a continuous group homomorphism
where M is either a finite-dimensional vector space over ℓ (the algebraic closure of the ℓ-adic numbers
Qℓ) or a finitely generated ℓ-module (where ℓ is the integral closure of Zℓ in ℓ). The first examples to arise were the ℓ-adic cyclotomic character and the ℓ-adic Tate modules of abelian varieties over K.
Unlike Artin representations, ℓ-adic representations can have infinite image. For example, the image of GQ under the ℓ-adic cyclotomic character is
. ℓ-adic representations with finite image are often called Artin representations. Via an isomorphism of ℓ with C they can be identified with bona fide Artin representations.
For an extension L/K of global fields with Galois group G, a finite place v of K, and place w of L above v, a Galois module ρ : G → Aut(V) is said to be unramified at w if ρ(Iw) = {1} (where Iw is the inertia subgroup of w). Equivalently, the extension Lker(ρ)/K is unramified at w. If ρ is not unramified at w it is called ramified at w.
s attaches to K its Weil group WK, a continuous group homomorphism , and an isomorphism
of topological group
s
where CK is K× or the idele class group IK/K× (depending on whether K is local or global) and is the abelianization of the Weil group of K. Via φ, any representation of GK can be considered as a representation of WK. However, WK can have strictly more representations than GK. For example, via rK the continuous complex characters of WK are in bijection with those of CK. Thus, the absolute value character on CK yields a character of WK whose image is infinite and therefore is not a character of GK (as all such have finite image).
An ℓ-adic representation of WK is defined in the same way as for GK. These arise naturally from geometry: if X is a smooth projective variety over K, then the ℓ-adic cohomology of the geometric fibre of X is an ℓ-adic representation of GK which, via φ, induces an ℓ-adic representation of WK. If K is a local field of residue characteristic p ≠ ℓ, then it is simpler to study the so-called Weil–Deligne representations 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 ℓ, Grothendieck's ℓ-adic monodromy theorem sets up a bijection between ℓ-adic representations of WK (over ℓ) and Weil–Deligne representations of WK over ℓ (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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Galois module is a G-module
G-module
In mathematics, given a group G, a G-module is an abelian group M on which G acts compatibly with the abelian group structure on M. This widely applicable notion generalizes that of a representation of G...
where G is the Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
of some extension
Field extension
In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties...
of fields
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
. The term Galois representation is frequently used when the G-module is a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
or a free module
Free module
In mathematics, a free module is a free object in a category of modules. Given a set S, a free module on S is a free module with basis S.Every vector space is free, and the free vector space on a set is a special case of a free module on a set.-Definition:...
over a ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
, but can also be used as a synonym for G-module. The study of Galois modules for extensions of local
Local field
In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...
or global field
Global field
In mathematics, the term global field refers to either of the following:*an algebraic number field, i.e., a finite extension of Q, or*a global function field, i.e., the function field of an algebraic curve over a finite field, equivalently, a finite extension of Fq, the field of rational functions...
s is an important tool in number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
.
Examples
- Given a field K, the multiplicative group (Ks)× of a separable closure of K is a Galois module for the absolute Galois groupAbsolute Galois groupIn mathematics, the absolute Galois group GK of a field K is the Galois group of Ksep over K, where Ksep is a separable closure of K. Alternatively it is the group of all automorphisms of the algebraic closure of K that fix K. The absolute Galois group is unique up to isomorphism...
. Its second cohomology group is isomorphic to the Brauer groupBrauer groupIn mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras of finite rank over K and addition is induced by the tensor product of algebras. It arose out of attempts to classify division algebras over a field and is...
of K (by Hilbert's theorem 90Hilbert's Theorem 90In abstract algebra, Hilbert's Theorem 90 refers to an important result on cyclic extensions of fields that leads to Kummer theory...
, its first cohomology groupGroup cohomologyIn abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper, group cohomology is a way to study groups using a sequence of functors H n. The study of fixed points of groups acting on modules and quotient modules...
is zero).
- If X is a smooth proper scheme over a field K then the ℓ-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 Galois extensionGalois extension
In mathematics, a Galois extension is an algebraic field extension E/F satisfying certain conditions ; one also says that the extension is Galois. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.The definition...
with Galois group G. For an extension w of v to L, let Iw denote its inertia group. A Galois module ρ : G → Aut(V) is said to be unramified if ρ(Iw) = {1}.
Galois module structure of algebraic integers
In classical algebraic number theoryAlgebraic number theory
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...
, let L be a Galois extension of a field K, and let G be the corresponding Galois group. Then the ring OL of algebraic integer
Algebraic integer
In number theory, an algebraic integer is a complex number that is a root of some monic polynomial with coefficients in . The set of all algebraic integers is closed under addition and multiplication and therefore is a subring of complex numbers denoted by A...
s of L can be considered as an OK[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 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 OL such that its conjugate elements under G give a free basis for OL over OK. This is an interesting question even (perhaps especially) when K is the rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
field Q.
For example, if L = Q(√-3), is there a normal integral basis? The answer is yes, as one sees by identifying it with Q(ζ) where
- ζ = exp(2πi/3).
In fact all the subfields of the cyclotomic field
Cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to Q, the field of rational numbers...
s for p-th roots of unity for p a prime number have normal integral bases (over Z), as can be deduced from the theory of Gaussian period
Gaussian period
In mathematics, in the area of number theory, a Gaussian period is a certain kind of sum of roots of unity. The periods permit explicit calculations in cyclotomic fields connected with Galois theory and with harmonic analysis . They are basic in the classical theory called cyclotomy...
s (the Hilbert–Speiser theorem). On the other hand the Gaussian field
Gaussian rational
In mathematics, a Gaussian rational number is a complex number of the form p + qi, where p and q are both rational numbers....
does not. This is an example of a necessary condition found by Emmy Noether
Emmy Noether
Amalie Emmy Noether was an influential German mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics. Described by David Hilbert, Albert Einstein and others as the most important woman in the history of mathematics, she revolutionized the theories of...
(perhaps known earlier?). What matters here is tame ramification
Ramification
In mathematics, ramification is a geometric term used for 'branching out', in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign...
. In terms of the discriminant
Discriminant of an algebraic number field
In mathematics, the discriminant of an algebraic number field is a numerical invariant that, loosely speaking, measures the size of the algebraic number field...
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 OL to be a projective module
Projective module
In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module...
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.
Galois representations in number theory
Many objects that arise in number theory are naturally Galois representations. For example, if L is a Galois extensionGalois extension
In mathematics, a Galois extension is an algebraic field extension E/F satisfying certain conditions ; one also says that the extension is Galois. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.The definition...
of a number field K, the ring of integers
Ring of integers
In mathematics, the ring of integers is the set of integers making an algebraic structure Z with the operations of integer addition, negation, and multiplication...
OL of L is a Galois module over OK 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. For global class field theory, the union of the idele class groups of all finite separable extension
Separable extension
In modern algebra, an algebraic field extension E\supseteq F is a separable extension if and only if for every \alpha\in E, the minimal polynomial of \alpha over F is a separable polynomial . Otherwise, the extension is called inseparable...
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 ℓ-adic Tate modules
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, Ks is the separable closure of K, and A = G...
of abelian varieties
Abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and 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 functions...
.
Artin representations
Let K be a number field. Emil ArtinEmil Artin
Emil Artin was an Austrian-American mathematician of Armenian descent.-Parents:Emil Artin was born in Vienna to parents Emma Maria, née Laura , a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of Armenian descent...
introduced a class of Galois representations of the absolute Galois group GK of K, now called Artin representations. These are the continuous
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
finite-dimensional linear representations of GK on complex vector space
Complex vector space
A complex vector space is a vector space over the complex numbers. It can also refer to:* a vector space over the real numbers with a linear complex structure...
s. Artin's study of these representations led him to formulate the Artin reciprocity law and conjecture what is now called the Artin conjecture concerning the holomorphy of Artin L-functions
Artin L-function
In mathematics, an Artin L-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group G. These functions were introduced in the 1923 by Emil Artin, in connection with his research into class field theory. Their fundamental properties, in particular the Artin...
.
Because of the incompatibility of the profinite topology on GK and the usual (Euclidean) topology on complex vector spaces, the image
Image (mathematics)
In mathematics, an image is the subset of a function's codomain which is the output of the function on a subset of its domain. Precisely, evaluating the function at each element of a subset X of the domain produces a set called the image of X under or through the function...
of an Artin representation is always finite.
ℓ-adic representations
Let ℓ be a prime numberPrime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
. An ℓ-adic representation of GK is a continuous group homomorphism
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
where M is either a finite-dimensional vector space over ℓ (the algebraic closure of the ℓ-adic numbers
P-adic number
In mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...
Qℓ) or a finitely generated ℓ-module (where ℓ is the integral closure of Zℓ in ℓ). The first examples to arise were the ℓ-adic cyclotomic character and the ℓ-adic Tate modules of abelian varieties over K.
Unlike Artin representations, ℓ-adic representations can have infinite image. For example, the image of GQ under the ℓ-adic cyclotomic character is
. ℓ-adic representations with finite image are often called Artin representations. Via an isomorphism of ℓ with C they can be identified with bona fide Artin representations.For an extension L/K of global fields with Galois group G, a finite place v of K, and place w of L above v, a Galois module ρ : G → Aut(V) is said to be unramified at w if ρ(Iw) = {1} (where Iw is the inertia subgroup of w). Equivalently, the extension Lker(ρ)/K is unramified at w. If ρ is not unramified at w it is called ramified at w.
Representations of the Weil group
If K is a local or global field, the theory of class formationClass formation
In mathematics, a class formation is a structure used to organize the various Galois groups and modules that appear in class field theory. They were invented by Emil Artin and John Tate.-Definitions:...
s attaches to K its Weil group WK, a continuous group homomorphism , and an isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
of topological group
Topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
s
where CK is K× or the idele class group IK/K× (depending on whether K is local or global) and is the abelianization of the Weil group of K. Via φ, any representation of GK can be considered as a representation of WK. However, WK can have strictly more representations than GK. For example, via rK the continuous complex characters of WK are in bijection with those of CK. Thus, the absolute value character on CK yields a character of WK whose image is infinite and therefore is not a character of GK (as all such have finite image).
An ℓ-adic representation of WK is defined in the same way as for GK. These arise naturally from geometry: if X is a smooth projective variety over K, then the ℓ-adic cohomology of the geometric fibre of X is an ℓ-adic representation of GK which, via φ, induces an ℓ-adic representation of WK. If K is a local field of residue characteristic p ≠ ℓ, then it is simpler to study the so-called Weil–Deligne representations of WK.
Weil–Deligne representations
Let K be a local field. Let E be a field of characteristic zero. A Weil–Deligne representation over E of WK (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,
- a nilpotentNilpotentIn mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0....
endomorphismEndomorphismIn mathematics, an endomorphism is a morphism from a mathematical object to itself. For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about...
such that r(w)Nr(w)−1= ||w||N for all w ∈ 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 ℓ, Grothendieck's ℓ-adic monodromy theorem sets up a bijection between ℓ-adic representations of WK (over ℓ) and Weil–Deligne representations of WK over ℓ (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.