Unramifiedness of weight 1 Hilbert Hecke algebras

Deo, Shaunak V.; Dimitrov, Mladen; Wiese, Gabor

doi:10.2140/ant.2024.18.1465

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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

DOI: 10.2140/ant.2024.18.1465

Abstract

We prove that the Galois pseudo-representation valued in the mod $p^{n}$ 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 $p - 1$ . 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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Vol. 18, No. 8, 2024