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.

**Authors:**

**Knowl status:**

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

**Referred to by:**

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

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

**Differences**(show/hide)