nLab Selmer group

Idea

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.

Definitions

The reference for this section is #LoefflerZerbes18.

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

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

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

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

Definition

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

An important example of a local condition is the unramified local condition. Letting $G_{K_{v}}$ be the absolute Galois group of $K_{v}$ and letting $I_{v}$ be its inertia subgroup, the unramified local condition is defined as follows:

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

Definition

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

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

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

References

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