Galois module



A Galois module VV 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.

If VV is a vector space then this is a linear representation of GG and one speaks of Galois representation.

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 action 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.



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(KL)Gal(K\hookrightarrow L).

(see also Hilbert-Speiser theorem?)



(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 unit 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.


(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.


(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.


(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.


(l-adic cohomology of a smooth variety)

Let ll be a prime number. Let XX be a smooth variety? over a field kk of characteristic prime to ll. Let k sk_s denote the separable closure of kk.

The ll-adic cohomology in degree ii is defined to be the directed limit lim nH et i(X k s,/l n)lim_n\; H^i_{et}(X_{k_s}, \mathbb{Z}/l^n\mathbb{Z}). It is a Galois module where the action is given by pullback.

More specifically, given σGal(kk s)\sigma\in Gal(k\hookrightarrow k_s) it acts on the X k s=X kk sX_{k_s}=X\otimes_k k_s via the second factor. This is an isomorphism, since σ\sigma is an automorphism, and hence σ *\sigma^* on cohomology is an isomorphism.

Note that since we have a equivalence T lAH et 1(A k s, l) T_l A\simeq H_{et}^1(A_{k_s}, \mathbb{Z}_l)^\vee, we have that the l-adic Tate module is a special case of the ll-adic cohomology.


Review of the fact that Galois representations encode local systems are are hence analogs in arithmetic geometry of flat connections in differential geometry includes

  • Tom Lovering, Étale cohomology and Galois Representations, 2012 (pdf)

See also at function field analogy.

Last revised on September 24, 2014 at 15:38:36. See the history of this page for a list of all contributions to it.