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A predicted distribution for Galois groups of maximal unramified extensions

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We consider the distribution of the Galois groups \(\operatorname {Gal}(K^{\operatorname{un}}/K)\) of maximal unramified extensions as \(K\) ranges over \(\Gamma \)-extensions of ℚ or \({{\mathbb{F}}}_{q}(t)\). We prove two properties of \(\operatorname {Gal}(K^{\operatorname{un}}/K)\) coming from number theory, which we use as motivation to build a probability distribution on profinite groups with these properties. In Part I, we build such a distribution as a limit of distributions on \(n\)-generated profinite groups. In Part II, we prove as \(q\rightarrow \infty \), agreement of \(\operatorname {Gal}(K^{\operatorname{un}}/K)\) as \(K\) varies over totally real \(\Gamma \)-extensions of \({{\mathbb{F}}}_{q}(t)\) with our distribution from Part I, in the moments that are relatively prime to \(q(q-1)|\Gamma |\). In particular, we prove for every finite group \(\Gamma \), in the \(q\rightarrow \infty \) limit, the prime-to-\(q(q-1)|\Gamma |\)-moments of the distribution of class groups of totally real \(\Gamma \)-extensions of \({{\mathbb{F}}}_{q}(t)\) agree with the prediction of the Cohen–Lenstra–Martinet heuristics.

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Acknowledgements

The authors would like to thank Brandon Alberts, Nigel Boston, Michael Bush, Brian Conrad, Jordan Ellenberg, Joseph Gunther, Jack Hall, Aaron Landesman, Jonah Leshin, Akshay Venkatesh, and Weitong Wang for important and fruitful conversations regarding the work in this paper, and Alex Bartel and Michael Bush for comments on an earlier draft. We are deeply indebted to the anonymous referees for extensive comments that improved greatly the exposition of the paper. The first author was partially supported by NSF grants DMS-1301690, DMS-1652116 and DMS-2200541. The second author was partially supported by an American Institute of Mathematics Five-Year Fellowship, a Packard Fellowship for Science and Engineering, a Sloan Research Fellowship, NSF grants DMS-1301690 and DMS-2052036, NSF Waterman Award DMS-2140043, a Radcliffe Fellowship at the Radcliffe Institute for Advanced Study at Harvard University, and a MacArthur Fellowship. The third author was partially supported by NSF grants DMS-1555048 and DMS-2302356.

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  1. Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W Green St, Urbana, IL, 61801, USA

    Yuan Liu

  2. Department of Mathematics, Harvard University, Science Center Room 325, 1 Oxford Street, Cambridge, MA, 02138, USA

    Melanie Matchett Wood

  3. Department of Mathematics and Statistics, Amherst College, 31 Quadrangle, Amherst, MA, 01002, USA

    David Zureick-Brown

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  1. Yuan Liu
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Liu, Y., Wood, M.M. & Zureick-Brown, D. A predicted distribution for Galois groups of maximal unramified extensions. Invent. math. (2024). https://doi.org/10.1007/s00222-024-01257-1

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