Are $p$-adic groups also Lie groups? What is a $p$-adic group specifically?

Asked Jan 25 '21 at 13:29

Active Jan 25 '21 at 13:29

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I am a bit confused by the definition of a $p$-adic group. I initially thought that a $p$-adic group was simply a group with $\mathbb{Q}_p$ as the underlying manifold, however this seems to be too vague, as I have seen definitions that mention Galois groups, local Archimedian fields, etc... I have also noticed that these groups seem closely related to typical Lie groups, like $GL_n ({\mathbb{R}})$. My question is, are $p$-adic groups also Lie groups? Also, does anyone know of a reputable textbook with a concise and self-contained definition of a $p$-adic group?

Thanks

asked Jan 25 '21 at 13:29

Natasha

1

Concerning $GL_n(\Bbb Q_p)$, see also this post. Then, a book is this one. Do a search yourself! – Dietrich Burde Jan 25 '21 at 13:35
I think there's no formal notion of "$p$-adic group". It might be a sloppy way to mean a "$p$-adic Lie group", or an algebraic $\mathbf{Q}_p$-group, which is not exactly the same. – YCor Jan 28 '21 at 01:03
Maybe pro-$p$-groups would count too? – Lubin Feb 15 '21 at 04:06

Are $p$-adic groups also Lie groups? What is a $p$-adic group specifically?

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