Unramifiedness of Galois representations arising from Hilbert modular surfaces

2014/10/22 by Matthew Emerton, Emerton, Matthew, Davide A. Reduzzi +3
Mathematics · #11F33 11F41 14G35 11G18 (secondary) #11F80 (primary) #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1410.6203

openalex publication_date 2014/10/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

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