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  "body": "\n## Definition\n\nLet $K$ be an [[algebraic number field|MWBDFS2KJ5]] with absolute [[Galois group|0XTY86BZKY]] $G_K$.\nLet $\\phi \\colon E \\to E'$ be an isogeny of [[elliptic curves|427N6S6QQW]] over $K$.\nThat is, $\\phi$ is a nonconstant morphism over $K$ that is also a [[group homomorphism|0JYTG5K9Y4]].\nLet $E[\\phi] = \\ker(\\phi \\colon E(\\overline{K}) \\to E'(\\overline{K}))$, a finite $G_K$-module.\nFor each [[place|BT5V3TSNY6]] $v$ of $K$, let $\\delta_v \\colon E'(K_v) \\to H^1(K_v, E[\\phi])$ be the connecting map of the [[Galois cohomology|9MPCZ3EAJM]] sequence of $0 \\to E[\\phi] \\to E \\xrightarrow{\\phi} E' \\to 0$ over the completion $K_v$.\nThe *$\\phi$-Selmer group* of $E$ over $K$ is\n$$\\mathrm{Sel}^{(\\phi)}(E/K) = \\set{\\xi \\in H^1(K, E[\\phi]) : \\mathrm{res}_v(\\xi) \\in \\delta_v(E'(K_v)) \\text{ for every place } v \\text{ of } K},$$\nwhere $\\mathrm{res}_v \\colon H^1(K, E[\\phi]) \\to H^1(K_v, E[\\phi])$ is restriction.\nFor an integer $n \\ge 2$, the *$n$-Selmer group* $\\mathrm{Sel}^{(n)}(E/K)$ is the case $\\phi = [n] \\colon E \\to E$.\n\n## Remarks\n\nEquivalently, $\\mathrm{Sel}^{(\\phi)}(E/K)$ is the kernel of $H^1(K, E[\\phi]) \\to \\prod_v H^1(K_v, E)$.\nThere is an exact sequence\n$$0 \\to E'(K)/\\phi(E(K)) \\to \\mathrm{Sel}^{(\\phi)}(E/K) \\to \\text{Ш}(E/K)[\\phi] \\to 0,$$\nwhere $\\text{Ш}(E/K)$ is the [[Tate–Shafarevich group|BK9WBMF56G]].\nThe Selmer group is finite and effectively computable, which bounds the rank of $E(K)$.\n\nExample: for $E \\colon y^2 = x^3 - x$ over $\\Q$ and $n = 2$, one has $E(\\Q) = \\set{O, (0, 0), (1, 0), (-1, 0)}$ and $2E(\\Q) = \\set{O}$.\nSo $E(\\Q)/2E(\\Q)$ has order $4$, and by the exact sequence $\\mathrm{Sel}^{(2)}(E/\\Q)$ has order at least $4$.\n\nNon-example: a class of $H^1(\\Q, E[2])$ whose restriction to $H^1(\\R, E[2])$ does not come from $E(\\R)$ fails the local condition at the real place, so it is not in $\\mathrm{Sel}^{(2)}(E/\\Q)$.\n",
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