Galois representation: Difference between revisions
→Mod ℓ representations: link for modular reps Tags: Mobile edit Mobile web edit Advanced mobile edit |
SafariScribe (talk | contribs) Adding local short description: "Mathematical terminology", overriding Wikidata description "module over the group algebra of some Galois group" |
||
(13 intermediate revisions by 11 users not shown) | |||
Line 1: | Line 1: | ||
{{Short description|Mathematical terminology}} |
|||
{{technical|date=March 2016}} |
|||
In [[mathematics]], a '''Galois module''' is a [[G-module|''G''-module]], with ''G'' being the [[Galois group]] of some [[field extension|extension]] of [[Field (mathematics)|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 (mathematics)|ring]] in [[representation theory]], but can also be used as a synonym for ''G''-module. The study of Galois modules for extensions of [[local field|local]] or [[global field]]s and their [[Galois cohomology|group cohomology]] is an important tool in [[number theory]]. |
In [[mathematics]], a '''Galois module''' is a [[G-module|''G''-module]], with ''G'' being the [[Galois group]] of some [[field extension|extension]] of [[Field (mathematics)|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 (mathematics)|ring]] in [[representation theory]], but can also be used as a synonym for ''G''-module. The study of Galois modules for extensions of [[local field|local]] or [[global field]]s and their [[Galois cohomology|group cohomology]] is an important tool in [[number theory]]. |
||
Line 10: | Line 10: | ||
==Galois module structure of algebraic integers== |
==Galois module structure of algebraic integers== |
||
In classical [[algebraic number theory]], let ''L'' be a Galois extension of a field ''K'', and let ''G'' be the corresponding Galois group. Then the ring ''O''<sub>''L''</sub> of [[algebraic integer]]s of ''L'' can be considered as an ''O''<sub>''K''</sub>[''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 |
In classical [[algebraic number theory]], let ''L'' be a Galois extension of a field ''K'', and let ''G'' be the corresponding Galois group. Then the ring ''O''<sub>''L''</sub> of [[algebraic integer]]s of ''L'' can be considered as an ''O''<sub>''K''</sub>[''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 [[Free_module#Definition|rank]] 1. If the same is true for the integers, that is equivalent to the existence of a '''normal integral basis''', i.e. of |
||
For example, if ''L'' = '''Q'''({{radic|−3}}), is there a normal integral basis? The answer is yes, as one sees by identifying it with '''Q'''('' |
For example, if ''L'' = '''Q'''({{radic|−3}}), is there a normal integral basis? The answer is yes, as one sees by identifying it with '''Q'''('' |
||
Line 18: | Line 18: | ||
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 rational|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 (mathematics)|ramification]]. In terms of the [[discriminant of an algebraic number field|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 ''O<sub>L</sub>'' 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. |
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 rational|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 (mathematics)|ramification]]. In terms of the [[discriminant of an algebraic number field|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 ''O<sub>L</sub>'' 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. |
||
A classical result, based on a result of [[David Hilbert]], is that a tamely ramified [[abelian number field]] has a normal integral basis. This may be seen by using the [[Kronecker–Weber theorem]] to embed the abelian field into a cyclotomic field.<ref name=F8>Fröhlich |
A classical result, based on a result of [[David Hilbert]], is that a tamely ramified [[abelian number field]] has a normal integral basis. This may be seen by using the [[Kronecker–Weber theorem]] to embed the abelian field into a cyclotomic field.<ref name=F8>{{harvnb|Fröhlich|1983|p=8}}</ref> |
||
==Galois representations in number theory== |
==Galois representations in number theory== |
||
Line 26: | Line 26: | ||
===<span id="ArtinReps"></span>Artin representations=== |
===<span id="ArtinReps"></span>Artin representations=== |
||
{{ |
{{Further|Artin conductor#Artin representation and Artin character}} |
||
Let ''K'' be a number field. [[Emil Artin]] introduced a class of Galois representations of the absolute Galois group ''G<sub>K</sub>'' of ''K'', now called '''Artin representations'''. These are the [[continuous function|continuous]] finite-dimensional linear representations of ''G<sub>K</sub>'' 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 (L-functions)|Artin conjecture]] concerning the [[holomorph (mathematics)|holomorph]]y of [[Artin L-function|Artin ''L''-functions]]. |
Let ''K'' be a number field. [[Emil Artin]] introduced a class of Galois representations of the absolute Galois group ''G<sub>K</sub>'' of ''K'', now called '''Artin representations'''. These are the [[continuous function|continuous]] finite-dimensional linear representations of ''G<sub>K</sub>'' 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 (L-functions)|Artin conjecture]] concerning the [[holomorph (mathematics)|holomorph]]y of [[Artin L-function|Artin ''L''-functions]]. |
||
Line 37: | Line 37: | ||
===Mod ℓ representations=== |
===Mod ℓ representations=== |
||
{{Further|Modular representation theory}} |
|||
These are representations over a finite field of characteristic ℓ. They often arise as the reduction mod ℓ of an ℓ-adic representation. |
|||
===Local conditions on representations=== |
===Local conditions on representations=== |
||
Line 46: | Line 47: | ||
*Absolutely irreducible representations. These remain irreducible over an [[algebraic closure]] of the field. |
*Absolutely irreducible representations. These remain irreducible over an [[algebraic closure]] of the field. |
||
*Barsotti–Tate representations. These are similar to finite flat representations. |
*Barsotti–Tate representations. These are similar to finite flat representations. |
||
*Crystabelline representations |
|||
*Crystalline representations. |
*Crystalline representations. |
||
*de Rham representations. |
*de Rham representations. |
||
Line 51: | Line 53: | ||
*Good representations. These are related to the representations of [[elliptic curves]] with good reduction. |
*Good representations. These are related to the representations of [[elliptic curves]] with good reduction. |
||
*Hodge–Tate representations. |
*Hodge–Tate representations. |
||
*Irreducible |
*[[Irreducible representation]]s. These are irreducible in the sense that the only subrepresentation is the whole space or zero. |
||
*Minimally ramified representations. |
*Minimally ramified representations. |
||
*Modular representations. These are representations coming from a [[modular form]] |
*Modular representations. These are representations coming from a [[modular form]], but can also refer to [[Modular representation theory|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 |
*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 |
||
*Potentially something representations. This means that the representations restricted to an open subgroup of finite index has some property. |
*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. |
*Reducible representations. These have a proper non-zero sub-representation. |
||
*Semistable representations. These are two dimensional representations related to the representations coming from [[ |
*Semistable representations. These are two dimensional representations related to the representations coming from [[Semistable abelian variety#Semistable elliptic curve|semistable elliptic curves]]. |
||
*Tamely ramified representations. These are trivial on the (first) [[ramification group]]. |
*Tamely ramified representations. These are trivial on the (first) [[ramification group]]. |
||
*Trianguline representations. |
|||
*Unramified representations. These are trivial on the inertia group. |
*Unramified representations. These are trivial on the inertia group. |
||
*Wildly ramified representations. These are non-trivial on the (first) ramification group. |
*Wildly ramified representations. These are non-trivial on the (first) ramification group. |
||
Line 79: | Line 82: | ||
==See also== |
==See also== |
||
*[[Compatible system of ℓ-adic representations]] |
*[[Compatible system of ℓ-adic representations]] |
||
*[[Arboreal Galois representation]] |
|||
==Notes== |
==Notes== |
||
{{ |
{{Reflist}} |
||
==References== |
==References== |
||
Line 101: | Line 105: | ||
|last=Tate |
|last=Tate |
||
|first=John |
|first=John |
||
|author-link=John Tate |
|author-link=John Tate (mathematician) |
||
|contribution=Number theoretic background |
|contribution=Number theoretic background |
||
|url= |
|url=https://www.ams.org/online_bks/pspum332/ |
||
|title=Automorphic forms, representations, and L-functions, Part 2 |
|title=Automorphic forms, representations, and L-functions, Part 2 |
||
|pages=3–26 |
|pages=3–26 |
||
Line 117: | Line 121: | ||
* {{citation | last=Snaith | first=Victor P. | title=Galois module structure | series=Fields Institute monographs | location=Providence, RI | publisher=[[American Mathematical Society]] | year=1994 | isbn=0-8218-0264-X | zbl=0830.11042 }} |
* {{citation | last=Snaith | first=Victor P. | title=Galois module structure | series=Fields Institute monographs | location=Providence, RI | publisher=[[American Mathematical Society]] | year=1994 | isbn=0-8218-0264-X | zbl=0830.11042 }} |
||
* {{citation | last=Fröhlich | first=Albrecht | authorlink=Albrecht Fröhlich | title=Galois module structure of algebraic integers | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge | volume=1 | location=Berlin-Heidelberg-New York-Tokyo | publisher=[[Springer-Verlag]] | year=1983 | isbn=3-540-11920-5 | zbl=0501.12012 }} |
* {{citation | last=Fröhlich | first=Albrecht | authorlink=Albrecht Fröhlich | title=Galois module structure of algebraic integers | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge | volume=1 | location=Berlin-Heidelberg-New York-Tokyo | publisher=[[Springer-Verlag]] | year=1983 | isbn=3-540-11920-5 | zbl=0501.12012 }} |
||
{{Authority control}} |
|||
[[Category:Algebraic number theory]] |
[[Category:Algebraic number theory]] |
Latest revision as of 19:44, 5 August 2024
In mathematics, a Galois module is a G-module, with G being 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 in representation theory, but can also be used as a synonym for G-module. The study of Galois modules for extensions of local or global fields and their group cohomology is an important tool in number theory.
Examples
[edit]- Given a field K, the multiplicative group (Ks)× of a separable closure of K is a Galois module for the absolute Galois group. Its second cohomology group is isomorphic to the Brauer group of K (by Hilbert's theorem 90, its first cohomology group 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
[edit]Let K be a valued field (with valuation denoted v) and let L/K be a finite Galois extension with Galois group G. For an extension w of v to L, let Iw denote its inertia group. A Galois module
Galois module structure of algebraic integers
[edit]In classical algebraic number theory, let L be a Galois extension of a field K, and let G be the corresponding Galois group. Then the ring OL of algebraic integers 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
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 = exp(2π i/3).
In fact all the subfields of the cyclotomic fields 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 periods (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.
A classical result, based on a result of David Hilbert, is that a tamely ramified abelian number field has a normal integral basis. This may be seen by using the Kronecker–Weber theorem to embed the abelian field into a cyclotomic field.[1]
Galois representations in number theory
[edit]Many objects that arise in number theory are naturally Galois representations. For example, if L is a Galois extension 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 extensions 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.
Artin representations
[edit]Let K be a number field. Emil Artin 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 spaces. 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.
ℓ-adic representations
[edit]Let ℓ be a prime number. An ℓ-adic representation of GK is a continuous group homomorphism
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 Qℓ with C they can be identified with bona fide Artin representations.
Mod ℓ representations
[edit]These are representations over a finite field of characteristic ℓ. They often arise as the reduction mod ℓ of an ℓ-adic representation.
Local conditions on representations
[edit]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 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.
- Good representations. These are related to the representations of elliptic curves with good reduction.
- Hodge–Tate representations.
- Irreducible representations. 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, 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
[edit]If K is a local or global field, the theory of class formations attaches to K its Weil group WK, a continuous group homomorphism
where CK is K× or the idele class group IK/K× (depending on whether K is local or global) and W ab
K is the abelianization of the Weil group of K. Via
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
Weil–Deligne representations
[edit]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 r : WK → AutE(V), where V is a finite-dimensional vector space over E equipped with the discrete topology,
- a nilpotent endomorphism N : V → V such that r(w)Nr(w)−1= ||w||N for all w ∈ WK.[2]
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 Qℓ) and Weil–Deligne representations of WK over Qℓ (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
[edit]Notes
[edit]- ^ Fröhlich 1983, p. 8
- ^ Here ||w|| is given by q v(w)
K 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.
References
[edit]- Kudla, Stephen S. (1994), "The local Langlands correspondence: the non-archimedean case", Motives, Part 2, Proc. Sympos. Pure Math., vol. 55, Providence, R.I.: Amer. Math. Soc., pp. 365–392, ISBN 978-0-8218-1635-6
- Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften, vol. 323, Berlin: Springer-Verlag, ISBN 978-3-540-66671-4, MR 1737196, Zbl 0948.11001
- Tate, John (1979), "Number theoretic background", Automorphic forms, representations, and L-functions, Part 2, Proc. Sympos. Pure Math., vol. 33, Providence, R.I.: Amer. Math. Soc., pp. 3–26, ISBN 978-0-8218-1437-6
Further reading
[edit]- Snaith, Victor P. (1994), Galois module structure, Fields Institute monographs, Providence, RI: American Mathematical Society, ISBN 0-8218-0264-X, Zbl 0830.11042
- Fröhlich, Albrecht (1983), Galois module structure of algebraic integers, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 1, Berlin-Heidelberg-New York-Tokyo: Springer-Verlag, ISBN 3-540-11920-5, Zbl 0501.12012