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New Horizons in pro-p Groups pp 329–348Cite as

Birkhäuser

p-adic Galois Representations and pro-p Galois Groups

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Part of the book series: Progress in Mathematics ((PM,volume 184))

Abstract

We study the intimate interactions between the theory of p-adic Galois representations and the structure of pro-p Galois groups. In particular, information passes in both directions. Algebraic geometry, for instance in the guise of elliptic curves and modular forms, yields naturally occurring Galois representations, whereas on the other side, co-homological techniques and variants on class field theory tell us about the generators and relations of the pro-p Galois groups. In the case of pro-p extensions ramified at (primes above) p, this combination works together rather well to elucidate the structure of the set of Galois representations. In the case of pro-p extensions unramified at p,both sides are poorly understood, but there is the fundamental conjecture of Fontaine—Mazur claiming that such representations should have finite image (since algebraic geometry can produce no others). This has very interesting consequences for the corresponding pro-p Galois groups, possibly producing a new family of just-infinite pro-p groups.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Illinois, Urbana, Illinois, 61801, USA

    Nigel Boston

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  1. Nigel Boston
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Editor information

Editors and Affiliations

  1. Centre for Mathematical Sciences, DPMMS, Cambridge, CB3 0WB, UK

    Marcus du Sautoy

  2. All Souls College, Oxford, OX1 4AL, UK

    Dan Segal

  3. Institute of Mathematics, Hebrew University, Jerusalem, 91904, Israel

    Aner Shalev

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Boston, N. (2000). p-adic Galois Representations and pro-p Galois Groups. In: du Sautoy, M., Segal, D., Shalev, A. (eds) New Horizons in pro-p Groups. Progress in Mathematics, vol 184. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1380-2_11

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  • DOI: https://doi.org/10.1007/978-1-4612-1380-2_11

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-7122-2

  • Online ISBN: 978-1-4612-1380-2

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