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Unramifiedness of weight 1 Hilbert Hecke algebras

Shaunak V. Deo, Mladen Dimitrov and Gabor Wiese

Vol. 18 (2024), No. 8, 1465–1496
Abstract

We prove that the Galois pseudo-representation valued in the mod pn cuspidal Hecke algebra for GL (2) over a totally real number field F, of parallel weight 1 and level prime to p, is unramified at any place above p. The same is true for the noncuspidal Hecke algebra at places above p whose ramification index is not divisible by p1. A novel geometric ingredient, which is also of independent interest, is the construction and study, in the case when p ramifies in F, of generalised Θ-operators using Reduzzi and Xiao’s generalised Hasse invariants, including especially an injectivity criterion in terms of minimal weights.

Keywords
Hilbert modular forms, Galois representations, Hecke algebras, Theta-operators, weight one
Mathematical Subject Classification
Primary: 11F80
Secondary: 11F25, 11F33, 11F41, 11G18, 14G35
Milestones
Received: 11 November 2021
Revised: 11 July 2023
Accepted: 18 September 2023
Published: 18 September 2024
Authors
Shaunak V. Deo
Department of Mathematics
Indian Institute of Science
Bangalore
India
Mladen Dimitrov
University of Lille
CNRS, UMR 8524 – Laboratoire Paul Painleve
Lille
France
Gabor Wiese
Department of Mathematics
University of Luxembourg
Grand Duchy of Luxembourg

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