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  "body": "\n## Summary\n\nA $p$-adic Lie group is a [[topological group|92700QAB5Y]] that is also an analytic manifold over the field $\\Q_p$ of [[p-adic numbers|BT6T702C28]], with analytic group operations.\nIt is the $p$-adic analogue of a real [[Lie group|7JBRGCVJRB]].\nSuch groups are totally disconnected, so their structure is studied through compact open subgroups, which are [[pro-p groups|C13D6W3MM9]] up to finite index.\nThey appear as Galois groups in number theory and as the groups $G(\\Q_p)$ of points of algebraic groups.\n\n## Explanations\n\n### L1 human\n\nA real Lie group looks near each point like a piece of $\\R^n$, and its multiplication is given by convergent power series.\nA $p$-adic Lie group is the same idea with $\\R$ replaced by the $p$-adic numbers $\\Q_p$.\nThe additive group $\\Z_p$ of $p$-adic integers is the simplest example: it is an open piece of $\\Q_p$, and addition is a polynomial.\nThe matrix group $GL_n(\\Q_p)$ is another example, since products and inverses of matrices are given by rational functions of the entries.\nUnlike real Lie groups, these groups fall apart into tiny open and closed pieces, because $\\Q_p$ itself is totally disconnected.\n",
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