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For $A$ an abelian variety over a number field $K$ and $p$ a prime number, the $p$-Selmer group of $A$ is the kernel of the map \[ H^1(G_K, A[p](\overline{K})) \to \prod_v H^1(G_{K_v}, A[p](\overline{K}_v)), \] where $v$ runs over the completions of $K$, and $G_K$ and $G_{K_v}$ denote the respective absolute Galois groups of $K$ and $K_v$. One may similarly define the Selmer group for any isogeny from $A$ to another abelian variety; any such group is finite and effectively computable. In fact, all known techniques for computing the Mordell-Weil group of $A$ involve computing Selmer groups as a key step.

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  • Review status: reviewed
  • Last edited by Jennifer Paulhus on 2019-04-27 16:19:32
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