Let $A$ be an abelian variety over a number field $K$.
Let $n \ge 1$.
Let $G_K$ be the absolute Galois group of $K$.
There is a connecting homomorphism $A(K)/nA(K) \to H^1(G_K,A[n])$, and likewise a homomorphism $A(K_v)/nA(K_v) \to H^1(G_{K_v},A[n])$ for each place $v$ of $K$.
The **$n$-Selmer group** $\operatorname{Sel}_n A$ is the set of classes in $H^1(G_K, A[n])$ whose image in $H^1(G_{K_v},A[n])$ lies in the image of $A(K_v)/nA(K_v) \to H^1(G_{K_v},A[n])$ for every place $v$, including the archimedean places.

One may also define the Selmer group for any isogeny from $A$ to another abelian variety over a global field.

Any Selmer group is finite and effectively computable. In fact, all known techniques for computing the Mordell-Weil group of $A$ involve computing a Selmer group.

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- Review status: reviewed
- Last edited by Bjorn Poonen on 2022-03-24 17:34:40

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**History:**(expand/hide all)

- 2022-03-24 17:34:40 by Bjorn Poonen (Reviewed)
- 2019-04-27 16:19:32 by Jennifer Paulhus (Reviewed)
- 2015-12-03 18:53:43 by Kiran S. Kedlaya

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