Let $G$ be a $p$-adic Lie group, $\text{Lie}(G)$ its Lie algebra.
Is there any reasonable notion of exponential map $\text{exp} : \text{Lie}(G)\to G$?
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11$\begingroup$ Yes, at least locally (the exponential is defined on a compact open subgroup of the Lie algebra). This is developed in detail in Chapter 2 of Bourbaki, Lie algebras and Lie groups. $\endgroup$– YCorCommented Dec 2, 2017 at 12:54
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9$\begingroup$ To be a little more precise: there is a compact open subring of the Lie algebra on which the exponential is defined, defines a homeomorphism onto a compact open subgroup of $G$, and on which the Baker-Campbell-Hausdorff formula converges and computes $\log(\exp(x)\exp(y))$. $\endgroup$– YCorCommented Dec 2, 2017 at 16:39
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2 Answers
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Besides the books already mentioned, I highly recommend Michel Lazard's Groupes analytiques p-adiques, which is the original source for a lot of the material in both Dixon-DuSautoy-Mann-Segal's Analytic pro-p groups and Schneider's p-adic Lie groups. Lazard's text was most probably written in close collaboration with Serre and is freely available online: http://www.numdam.org/item?id=PMIHES_1965__26__5_0
The exponential map and the Hausdorff formula are treated in particular (with a look towards $\mathbb{Z}_p$-integrality) in section 3.2.
answered Dec 18, 2017 at 18:39
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$\begingroup$ What makes you think of collaboration with Bourbaki? There are warm thanks to Jean-Pierre Serre (p10), but I don't see the point with Bourbaki. $\endgroup$– YCorCommented Dec 18, 2017 at 20:08
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$\begingroup$ @YCor: I was under the impression that the Bourbaki treatment of the relation between Lie Groups and Lie Algebras (chapter 3 in particular), including the ultrametric case, bears much more than a coincidental resemblance with Serre's 1965 notes on "Lie algebras and Lie groups". In his latest comment on his answer, Jim Humphreys also suggests that Serre had "a major role" in the Bourbaki approach. That is what I intended to say, Lazard worked with Serre directly, and thus at least indirectly with Bourbaki. But if that's too speculative, I can remove it. $\endgroup$ Commented Dec 18, 2017 at 20:26
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$\begingroup$ Besides that, Lazard took part in Séminaires Bourbaki at least in the 1950s. Here, in 1953, he already investigates what he calls the "formule de Campbell-Pascal-Baker-Poincaré-Hausdorff (...?)", although, if I'm not mistaken, only w.r.t. archimedean values: numdam.org/article/SB_1951-1954__2__255_0.pdf $\endgroup$ Commented Dec 18, 2017 at 20:42
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$\begingroup$ Precisely, he thanks Serre as Serre, not Serre as member of Bourbaki. Since Serre is the person in Bourbaki who is most associated to this field, why speculate on other links with Bourbaki? You're talking of probable "close collaboration with Bourbaki"; it seems purely speculative (and taking part in Séminaire Bourbaki implies by no means "collaborating" with Bourbaki). $\endgroup$– YCorCommented Dec 18, 2017 at 20:45
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$\begingroup$ Would you say that EGA was written in close collaboration with Bourbaki? Dieudonné was coauthor, Serre was deeply involved around Grothendieck... but these people have their own existence beyond Bourbaki. In this better-documented case, the historical context certainly cannot ignore Bourbaki, but "collaboration with Bourbaki" would be inappropriate. $\endgroup$– YCorCommented Dec 18, 2017 at 20:53
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This is just a comment (in community-wiki format). I don't know how to cite an article efficiently otherwise.
As YCor points out, this notion has become fairly standard in the development of Lie groups over $p$-adic fields. On the other hand, there is not much explicit literature along these lines beyond Bourbaki. One paper that might be of interest is here. This suggests however that it's not easy to say much about the use of exponentials in $p$-adic groups.
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2$\begingroup$ Serre's book called something like "Lie algebras and Lie groups" also has a self-contained discussion of this stuff (including applications to versions of Lie's three theorems), so that could be another reference (maybe not "explicit literature", whatever that may mean?). $\endgroup$– nfdc23Commented Dec 2, 2017 at 20:47
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3$\begingroup$ Schneider has a book on p-adic Lie groups that grew out of his lectures. P-adic analytic groups are also treated in Dixon-Sautoy-Segal's "Analytic Pro-p-Groups" book. Both treat the exponential map and much more. $\endgroup$– M.G.Commented Dec 2, 2017 at 22:40
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3$\begingroup$ I'm not sure what is meant by "it's not easy to say much about the use of exponentials in $p$-adic Lie groups". Typically there are many problems about general totally disconnected locally compact groups for which the case of $p$-adic Lie groups is a natural easy test-case, precisely because of the use of the Lie algebra. And it's not only useful for compact $p$-adic Lie groups. For instance, the classification of topologically simple non-discrete $p$-adic Lie groups is easy to carry out using the Lie algebras. $\endgroup$– YCorCommented Dec 2, 2017 at 23:11
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$\begingroup$ @YCor: I was only referring to the narrower question about exponentiation raised by the paper I was linking to. Naturally there are many roles for $p$-adic Lie groups and their Lie algebras. $\endgroup$ Commented Dec 3, 2017 at 18:48
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1$\begingroup$ @nfdc23: The 1964 Harvard lectures Lie Algebras and Lie Groups by Serre (published in typewritten large format by W.A. Benjamin in 1965, with some symbols added by hand) are as you say another useful reference. This is close to the formal treatment in Bourbaki's treatise, Chap. 1-3. (Apparently Serre himself had a major role in developing the Bourbaki approach.) $\endgroup$ Commented Dec 3, 2017 at 18:59