nLab Selmer group



The Selmer group is a subgroup of the Galois cohomology satisfying certain local conditions.

It appears in the statement of the Bloch-Kato conjecture on special values of L-functions, generalizing the Birch and Swinnerton-Dyer conjecture.


The reference for this section is #LoefflerZerbes18.

Let KK be a number field. Let VV be a Galois representation, i.e. a representation of Gal(K¯/K)\Gal(\overline{K}/K). We have the first Galois cohomology group, which we will denote with the following shortcut notation

H 1(K,V):=H 1(Gal(K¯/K),V)H^{1}(K,V):=H^{1}(\Gal(\overline{K}/K),V)

For any prime vv of KK, let K vK_{v} be the completion. We have “localization” maps (we are also using similar notation as above for the local Galois cohomology groups)

loc v:H 1(K,V)H 1(K v,V)\loc_{v}:H^{1}(K,V)\to H^{1}(K_{v},V)


A local condition on VV at a prime vv is a subspace vH 1(K v,V)\mathcal{F}_{v}\subseteq H^{1}(K_{v},V).

An important example of a local condition is the unramified local condition. Letting G K vG_{K_{v}} be the absolute Galois group of K vK_{v} and letting I vI_{v} be its inertia subgroup, the unramified local condition is defined as follows:

v,unr:=im(H 1(G K v/I v,V I v)H 1(K v,V))\mathcal{F}_{v,\unr}:=\im(H^{1}(G_{K_{v}}/I_{v},V^{I_{v}})\to H^{1}(K_{v},V))


A Selmer structure is a collection ={ v} v\mathcal{F}=\lbrace\mathcal{F}_{v}\rbrace_{v} of local conditions v\mathcal{F}_{v}, such that v= v,unr\mathcal{F}_{v}=\mathcal{F}_{v,\unr} for all but finitely many primes vv.

If \mathcal{F} is a Selmer structure, we define the corresponding Selmer group as

Sel (K,V)={xH 1(K,V):loc v vv}.\mathrm{Sel}_{\mathcal{F}}(K,V)=\lbrace x\in H^{1}(K,V):\loc_{v}\in\mathcal{F}_{v}\forall v\rbrace.


David Loeffler and Sarah Zerbes, Euler Systems, Arizona Winter School 2018 Notes (pdf)

Last revised on August 4, 2023 at 01:55:36. See the history of this page for a list of all contributions to it.