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The Breuil-Schneider Conjecture: A Survey

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Advances in the Theory of Numbers

Part of the book series: Fields Institute Communications ((FIC,volume 77))

Abstract

This note is a survey of the Breuil-Schneider conjecture, based on the authors 30 min talk at the 13th conference of the Canadian Number Theory Association (CNTA XIII) held at Carleton University, June 16–20, 2014. We give an overview of the problem, and describe certain recent developments by the author and others.

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Notes

  1. 1.

    This is the notation we will eventually use in Sect. 5 below, to distinguish it from \(F/\mathbb{Q}_{p}\).

  2. 2.

    Here we tacitly fix algebraic closures \(\bar{F}\hookrightarrow \bar{F}_{w}\) extending F ↪ F w , in order to identify the decomposition group \(\varGamma _{F_{w}} =\mathop{ \mathrm{Gal}}\nolimits (\bar{F}_{w}/F_{w})\) with a subgroup of Γ F . We say r w is unramified if its restriction to the inertia group \(I_{F_{w}} =\mathop{ \mathrm{Gal}}\nolimits (\bar{F}_{w}/F_{w}^{nr})\) is trivial.

  3. 3.

    Meaning the representation π sm (ρ) embeds into \(\text{Ind}_{U_{n}}^{\mathop{\mathrm{GL}}\nolimits _{n}}(\psi )\) for some additive character ψ ≠ 1.

  4. 4.

    Not to be confused with automorphic forms. For instance, the inner forms of \(\mathop{\mathrm{GL}}\nolimits (2)\) are the algebraic groups D ×, for quaternion algebras D, whereas the outer forms of \(\mathop{\mathrm{GL}}\nolimits (2)\) are the unitary groups in two variables.

  5. 5.

    We use the notation \(\tilde{F}\) to distinguish it from our p-adic base field \(F/\mathbb{Q}_{p}\).

  6. 6.

    An anti-equivalence between compact \(\mathcal{O}_{E}\)-modules and \(\varpi _{E}\)-adically complete \(\mathcal{O}_{E}\)-modules.

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Acknowledgements

I would like to thank the organizers of CNTA XIII for a wonderful event in beautiful Ottawa, and for the opportunity to speak there. S.W. Shin read an early draft of this paper, and made comments; his feedback is greatly appreciated—as is the thorough reading of the anonymous referee.

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Authors and Affiliations

  1. Department of Mathematics, UC San Diego, La Jolla, CA, USA

    Claus M. Sorensen

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  1. Claus M. Sorensen
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Correspondence to Claus M. Sorensen .

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Editors and Affiliations

  1. School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada

    Ayşe Alaca

  2. School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada

    Şaban Alaca

  3. School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada

    Kenneth S. Williams

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Sorensen, C.M. (2015). The Breuil-Schneider Conjecture: A Survey. In: Alaca, A., Alaca, Ş., Williams, K. (eds) Advances in the Theory of Numbers. Fields Institute Communications, vol 77. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-3201-6_10

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