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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.

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