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Topological Realisations of Absolute Galois Groups

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Cohomology of Arithmetic Groups (JS66 2016)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 245))

Abstract

Let F be a field of characteristic 0 containing all roots of unity. We construct a functorial compact Hausdorff space \(X_F\) whose profinite fundamental group agrees with the absolute Galois group of F, i.e. the category of finite covering spaces of \(X_F\) is equivalent to the category of finite extensions of F. The construction is based on the ring of rational Witt vectors of F. In the case of the cyclotomic extension of \(\mathbb {Q}\), the classical fundamental group of \(X_F\) is a (proper) dense subgroup of the absolute Galois group of F. We also discuss a variant of this construction when the field is not required to contain all roots of unity, in which case there are natural Frobenius-type automorphisms which encode the descent along the cyclotomic extension.

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Notes

  1. 1.

    It is this connection, as well as the observation that the Dennis trace map from K-theory to topological Hochschild homology factors canonically over the K-theory of endomorphisms, that led the second author to consider the rational Witt vectors.

  2. 2.

    An instance is the definition of a \(\Lambda \)-ring, which can be regarded as a commutative ring A with a map \(A\rightarrow \mathrm {W}(A)\) satisfying certain properties. In most natural examples, including \(\mathrm {K}_0\) of a commutative ring, the map \(A\rightarrow \mathrm {W}(A)\) actually factors through a map \(A\rightarrow \mathrm {W}_{\mathrm {rat}}(A)\).

  3. 3.

    The induced action on \(2\pi \mathrm {i}\mathbb {Z}\subset \mathbb {C}\) is trivial, as we are working over the cyclotomic extension. In fact, there can not be an action (except for complex conjugation) on \(2\pi \mathrm {i}\mathbb {Z}\), which presents an obstruction to extending this action beyond the cyclotomic extension.

  4. 4.

    In fact, one can combine the first and second observation, which leads to the observation that \(\mathrm {W}_{\mathrm {rat}}(F)\) is a \(\Lambda \)-ring; in fact, (almost tautologically) one for which the map \(\mathrm {W}_{\mathrm {rat}}(F)\rightarrow \mathrm {W}(\mathrm {W}_{\mathrm {rat}}(F))\) factors over \(\mathrm {W}_{\mathrm {rat}}(F)\rightarrow \mathrm {W}_{\mathrm {rat}}(\mathrm {W}_{\mathrm {rat}}(F))\).

  5. 5.

    In fancy language, the ‘dynamical system’ of the connected components of \({\text {Spec }}(\mathrm {W}_{\mathrm {rat}}(\mathbb {Q}(\zeta _\infty ))\otimes \mathbb {C})\) with its Frobenius operators is one form of the Bost–Connes system, [6].

  6. 6.

    Note that this convention is reverse to that prevalent in algebraic topology, but it is common in algebraic geometry and is more convenient when working with categories of covering spaces. Of course, these two conventions yield groups which are opposite groups of one another, hence related by a canonical isomorphism \([\gamma ]\mapsto [\gamma ]^{-1}\).

  7. 7.

    Note, however, the misprint there: the two inclusions \(\ker u\subset {\text {im}}u'\) and \(\ker u\supset {\text {im}}u'\) must be exchanged.

  8. 8.

    Here is the construction. Let \(\varphi \in {\text {End}}(\mathbb {Q}/\mathbb {Z})\cong {\text {End}}(\hat{\mathbb {Z}})\cong \hat{\mathbb {Z}}\cong \prod _{p\text { prime}}\mathbb {Z}_p\) be such that the component \(\varphi _p\in \mathbb {Z}_p\) at each p is transcendental over \(\mathbb {Q}\). Then we set \(A=\{ (a,b)\in \mathbb {Q}^2\mid \varphi (a\bmod \mathbb {Z})=b\bmod z \}\).

  9. 9.

    Note that inverting \([\zeta _\ell ]-1\) in particular inverts \(\ell \), so the condition that \(\varphi _\ell ^\sharp \) lifts Frobenius is vacuous.

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Acknowledgements

Part of this work was done while the second author was a Clay Research Fellow. All of it was done while the first author was supported by the Swiss National Science Foundation. The authors wish to thank Lennart Meier for asking a very helpful question, Markus Land and Thomas Nikolaus for a discussion about Proposition 7.10, and Eric Leichtnam for pointing out some typographical errors in an earlier version.

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Authors and Affiliations

  1. Département Mathematik, ETH Zürich, Rämistrasse 101, 8092, Zürich, Switzerland

    Robert A. Kucharczyk

  2. Mathematisches Institut, Rheinische Friedrich-Wilhelms-Universität Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    Peter Scholze

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Correspondence to Peter Scholze .

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  1. Department of Mathematics, Ohio State University, Columbus, USA

    James W. Cogdell

  2. Mathematical Institute, University of Bonn, Bonn, Germany

    Günter Harder

  3. Department of Mathematics, University of Toronto, Toronto, Canada

    Stephen Kudla

  4. Department of Mathematics, Purdue University, West Lafayette, USA

    Freydoon Shahidi

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Kucharczyk, R.A., Scholze, P. (2018). Topological Realisations of Absolute Galois Groups. In: Cogdell, J., Harder, G., Kudla, S., Shahidi, F. (eds) Cohomology of Arithmetic Groups. JS66 2016. Springer Proceedings in Mathematics & Statistics, vol 245. Springer, Cham. https://doi.org/10.1007/978-3-319-95549-0_8

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