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Galois module (changes)

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A Galois module is a GG-module for a Galois group? GG; i.e. it is an abelian group on which a Galois group acts in a way compatible with the abelian group structure.

The category of GG-modules is equivalent to the category of modules? over the group ring ]G]\mathbb{Z}]G].

As always is the case, a group representation G×AAG\times A\to A can equivalently be written as GAut(A)G\to Aut(A). This is why Galois modules are frequently called Galois representations.

Properties

Proposition

Let KLK\hookrightarrow L be a Galois extension? of a number field KK.

Then the ring of integers O LO_L of this extension is a Galois module of Gal(K/hookrightarrowL)Gal(K/hookrightarrow L).

(see also Hilbert-Speiser theorem?)

Examples

Example

(ll-adic representation)

Let ll be a prime number. Let Gal(kk¯)Gal(k\hookrightarrow \overline k) be the absolute Galois group? of a number field? kk. Then a morphism of groups

Gal(kk¯)Aut(M)Gal(k\hookrightarrow \overline k)\to Aut (M)

is called an ll-adic representation of Gal(kk¯)Gal(k\hookrightarrow \overline k). Here MM is either a unite dimensional vector space? over the algebraic closure ¯ l\overline \mathbb{Q}_l or a finitely generated module over the integral closure? ¯ l\overline \mathbb{Z}_l.

In particular the ll-adic Tate-module is of this kind.

Example

(ll-adic Tate module?) Let ll be a prime number. Let AA be an abelian group. The ll-adic Tate module is defined to be the limit

T l(A)=lim nker(l n)T_l(A)=lim_n \;ker (l^n)

i.e. it is the limit over the directed diagram? ker(p n+1)ker(p n)ker(p^{n+1})\to ker(p^n). Here the kernel? ker(p n)ker(p^n) of the multiplication-with-p np^n map p n:AAp^n:A\to A is called p np^n-torsion? of AA.

Example

(the Tate-module)

Let k Sk_S denote the separable closure of kk. Let AA be the group of roots of unity? of k sk_s in kk. Then the ll-adic Tate-module of the absolute Galois group Gal(kk s)Gal(k\hookrightarrow k_s) is called the ll-adic Tate module of kk or the ll-adic cyclotomic character of kk.

It is equivalently the Tate-module of the multiplicative group scheme μ k\mu_k.

The Tate-module is endowed with the structure of a \mathbb{Z}-module by z(a n) n=((zmodulop n)a n) nz(a_n)_n=((z\; modulo\; p^n)a_n)_n.

Example

(ll-adic Tate module? of an abelian variety)

Let ll be a prime number. Let GG be an abelian variety? over a field kk. Let k sk_s denote the separable closure of kk. The k sk_s-valued points of GG assemble to an abelian group.

Then there are classical results on the rank? of the Tate-module T l(G)T_l(G): For example if the characteristic of kk is a prime number plp\neq l we have that T l(G)T_l(G) is a free l\mathbb{Z}_l module of rank 2dim(G)2dim(G).

A special case of the Tate conjecture? can be formulated via Tate-modules:

Let kk be finitely generated over its prime field of characteristic plp\neq l. Let A,BA,B be two abelian varieties over kk. Then the conjecture states that

hom(A,B) phom(T l(A),T l(B))hom(A,B)\otimes \mathbb{Z}_p\simeq hom(T_l(A),T_l(B))

If kk is a finite field or a number field the conjecture is true.


Galois modules and quasi finite fields

A Galois module is a GG-module for a Galois group? GG; i.e. it is an abelian group on which a Galois group acts in a way compatible with the abelian group structure.

The category of GG-modules is equivalent to the category of modules? over the group ring ]G]\mathbb{Z}]G].

As always is the case, a group representation G×AAG\times A\to A can equivalently be written as GAut(A)G\to Aut(A). This is why Galois modules are frequently called Galois representations.

Properties

Proposition

Let KLK\hookrightarrow L be a Galois extension? of a number field KK.

Then the ring of integers O LO_L of this extension is a Galois module of Gal(K/hookrightarrowL)Gal(K/hookrightarrow L).

(see also Hilbert-Speiser theorem?)

Examples

Example

(ll-adic representation)

Let ll be a prime number. Let Gal(kk¯)Gal(k\hookrightarrow \overline k) be the absolute Galois group? of a number field? kk. Then a morphism of groups

Gal(kk¯)Aut(M)Gal(k\hookrightarrow \overline k)\to Aut (M)

is called an ll-adic representation of Gal(kk¯)Gal(k\hookrightarrow \overline k). Here MM is either a unite dimensional vector space? over the algebraic closure ¯ l\overline \mathbb{Q}_l or a finitely generated module over the integral closure? ¯ l\overline \mathbb{Z}_l.

In particular the ll-adic Tate-module is of this kind.

Example

(ll-adic Tate module?) Let ll be a prime number. Let AA be an abelian group. The ll-adic Tate module is defined to be the limit

T l(A)=lim nker(l n)T_l(A)=lim_n \;ker (l^n)

i.e. it is the limit over the directed diagram? ker(p n+1)ker(p n)ker(p^{n+1})\to ker(p^n). Here the kernel? ker(p n)ker(p^n) of the multiplication-with-p np^n map p n:AAp^n:A\to A is called p np^n-torsion? of AA.

Example

(the Tate-module)

Let k Sk_S denote the separable closure of kk. Let AA be the group of roots of unity? of k sk_s in kk. Then the ll-adic Tate-module of the absolute Galois group Gal(kk s)Gal(k\hookrightarrow k_s) is called the ll-adic Tate module of kk or the ll-adic cyclotomic character of kk.

It is equivalently the Tate-module of the multiplicative group scheme μ k\mu_k.

The Tate-module is endowed with the structure of a \mathbb{Z}-module by z(a n) n=((zmodulop n)a n) nz(a_n)_n=((z\; modulo\; p^n)a_n)_n.

Example

(ll-adic Tate module? of an abelian variety)

Let ll be a prime number. Let GG be an abelian variety? over a field kk. Let k sk_s denote the separable closure of kk. The k sk_s-valued points of GG assemble to an abelian group.

Then there are classical results on the rank? of the Tate-module T l(G)T_l(G): For example if the characteristic of kk is a prime number plp\neq l we have that T l(G)T_l(G) is a free l\mathbb{Z}_l module of rank 2dim(G)2dim(G).

A special case of the Tate conjecture? can be formulated via Tate-modules:

Let kk be finitely generated over its prime field of characteristic plp\neq l. Let A,BA,B be two abelian varieties over kk. Then the conjecture states that

hom(A,B) phom(T l(A),T l(B))hom(A,B)\otimes \mathbb{Z}_p\simeq hom(T_l(A),T_l(B))

If kk is a finite field or a number field the conjecture is true.


Bondarko has a few papers about Galois modules over local fields.

Burns is also interested in Galois modules I think

Many things by Snaith et al, for example the book Galois module structure, Fields Inst Monographs 2.

nLab page on Galois module

Last revised on June 12, 2012 at 10:56:18. See the history of this page for a list of all contributions to it.