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  "body": "\n## Definition\n\nLet $p$ be a prime and let $\\Q_p$ be the field of [[p-adic numbers|BT6T702C28]].\nLet $U \\subseteq \\Q_p^n$ be open.\nA map $f \\colon U \\to \\Q_p^m$ is *analytic* if each point of $U$ has an open neighbourhood on which each coordinate of $f$ is a convergent [[power series|0MZB06ST2B]].\nA *$p$-adic analytic manifold* of dimension $n$ is a Hausdorff space $M$ with an atlas of charts.\nThe charts are homeomorphisms from open sets of $M$ onto open sets of $\\Q_p^n$, and their transition maps are analytic.\nA *$p$-adic Lie group* is a group $G$ with the structure of a $p$-adic analytic manifold such that:\n\n1. the multiplication $G \\times G \\to G$, $(x, y) \\mapsto xy$, is analytic;\n2. the inversion $G \\to G$, $x \\mapsto x^{-1}$, is analytic.\n\n## Remarks\n\nExample: the group $\\Z_p$ of $p$-adic integers under addition is an open subset of $\\Q_p$, so the identity map is a chart.\nAddition $(x, y) \\mapsto x + y$ and negation $x \\mapsto -x$ are polynomials, hence analytic.\nSo $\\Z_p$ is a $p$-adic Lie group of dimension $1$.\nThe group $GL_n(\\Q_p)$ is open in the space $\\Q_p^{n^2}$ of matrices.\nIts product and inverse are rational in the entries with nonvanishing denominators, so it is a $p$-adic Lie group of dimension $n^2$.\n\nNon-example: the additive group $\\R$ is not a $p$-adic Lie group with its usual topology.\nEvery open subset of $\\Q_p^n$ is totally disconnected, so every $p$-adic analytic manifold is totally disconnected.\nBut $\\R$ is connected and has more than one point.\n\nSuch groups are also called $p$-adic analytic groups.\nEvery $p$-adic Lie group is a totally disconnected locally compact Hausdorff group.\n",
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