Abstract
A profinite group is index-stable if any two isomorphic open subgroups have the same index. Let p be a prime, and let G be a compact p-adic analytic group with associated \(\mathbb {Q}_p\)-Lie algebra \(\mathcal {L}(G)\). We prove that G is index-stable whenever \(\mathcal {L}(G)\) is semisimple. In particular, a just-infinite compact p-adic analytic group is index-stable if and only if it is not virtually abelian. Within the category of compact p-adic analytic groups, this gives a positive answer to a question of C. Reid. In the appendix, J-P. Serre proves that G is index-stable if and only if the determinant of any automorphism of \(\mathcal {L}(G)\) has p-adic norm 1.
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With an appendix by Jean-Pierre Serre.
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Ilir Snopce supported by CNPq and FAPERJ.
Appendix: A letter from Jean-Pierre Serre
Appendix: A letter from Jean-Pierre Serre
Dear MM. Noseda and Snopce,
I have just seen your arXiv paper on “Index-stable compact p-adic analytic groups”. Your proof uses Lazard’s very nice results, but in fact these results are not necessary: p-adic integration is enough. Let me explain:
Let L be a finite-dimensional Lie algebra over \(\mathbb {Q}_p\). Denote by |x| the p-adic norm of \(\mathbb {Q}_p\). Consider the following property of L:
For every automorphism s of L, we have \(|\mathrm {det}(s)| = 1\) (*).
This is true, for instance, if L is semisimple since \(\mathrm {det}(s)=\pm 1\), which is the case you consider.
Assume property (*). Let \(n = \dim L.\) Let u be a non-zero element of \(\wedge ^nL^*\), where \(L^*\) is the \({\mathbb {Q}}_p\)-dual of L. Let G be a compact p-adic analytic group with Lie algebra L. Then u defines a right-invariant differential form \(\omega _u\) on G of degree n. The corresponding measure \(\mu _u = |\omega _u|\) is a non-zero right-invariant positive measure on G, hence is a Haar measure since G is compact. Property (*) implies that \(\mu _u\) is invariant by every automorphism of L. Hence every isomorphism of G onto another group \(G'\) carries \(\mu _u\) (for G) into \(\mu _u\) (for \(G'\)). This implies that, if \(G, G'\) are open subgroups of some compact p-adic group \(G''\), then they have the same index - as wanted.
Conversely, if (*) is not true for some s, and if \(G''\) is compact with Lie algebra L, s defines a local automorphism of \(G''\), and if G is a small enough open subgroup of \(G''\), it is transformed by s into another open subgroup \(G'\), and the ratio \((G'':G)/(G'':G')\) is equal to \(|\mathrm {det} (s)|\), which is \(\ne 1\).
Best wishes,
J-P. Serre
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Noseda, F., Snopce, I. Index-stable compact \({\varvec{p}}\)-adic analytic groups. Arch. Math. 116, 153–160 (2021). https://doi.org/10.1007/s00013-020-01543-3
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DOI: https://doi.org/10.1007/s00013-020-01543-3