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UNRAMIFIEDNESS OF GALOIS REPRESENTATIONS ARISING FROM HILBERT MODULAR SURFACES

Part of: Arithmetic problems. Diophantine geometry Arithmetic algebraic geometry Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  23 November 2017

MATTHEW EMERTON ,
DAVIDE REDUZZI  and
LIANG XIAO
MATTHEW EMERTON
Affiliation:
Department of Mathematics, The University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637, USA; [email protected], [email protected]
DAVIDE REDUZZI
Affiliation:
Department of Mathematics, The University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637, USA; [email protected], [email protected]
LIANG XIAO
Affiliation:
Department of Mathematics, University of Connecticut, Storrs, 341 Mansfield Road, Storrs, Connecticut 06269, USA; [email protected]

Abstract

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Let $p$ be a prime number and $F$ a totally real number field. For each prime $\mathfrak{p}$ of $F$ above $p$ we construct a Hecke operator $T_{\mathfrak{p}}$ acting on $(\text{mod}\,p^{m})$ Katz Hilbert modular classes which agrees with the classical Hecke operator at $\mathfrak{p}$ for global sections that lift to characteristic zero. Using these operators and the techniques of patching complexes of Calegari and Geraghty we prove that the Galois representations arising from torsion Hilbert modular classes of parallel weight $\mathbf{1}$ are unramified at $p$ when $[F:\mathbb{Q}]=2$. Some partial and some conjectural results are obtained when $[F:\mathbb{Q}]>2$.

MSC classification

Primary: 11F80: Galois representations
Secondary: 11F33: Congruences for modular and $p$-adic modular forms 11F41: Automorphic forms on $(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 14G35: Modular and Shimura varieties 11G18: Arithmetic aspects of modular and Shimura varieties
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 5 , 2017 , e29
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2017

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