Unramifiedness of weight one Hilbert Hecke algebras

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 non-cuspidal Hecke algebra at places above $p$ whose ramification index is not divisible by $p-1$. A novel geometric ingredient, which is also of an independent interest, is the construction and study, in the case when $p$ ramifies in $F$, of generalised $\Theta$-operators using Reduzzi--Xiao's generalised Hasse invariants, including especially an injectivity criterion in terms of minimal weights.

arXiv comment33 pages; accepted for publication in Algebra & Number Theory

References in corpus (7)

• Hilbert modular forms and p-adic Hodge theory
• On Ihara's lemma for Hilbert Modular Varieties
• A Serre weight conjecture for geometric Hilbert modular forms in characteristic p
• Minimal weights of Hilbert modular forms in characteristic p
• Unramifiedness of Galois representations attached to weight one Hilbert modular eigenforms mod p
• The cone of minimal weights for mod $p$ Hilbert modular forms
• Geometric weight-shifting operators on Hilbert modular forms in characteristic p

Cited by in corpus (1)

• Kodaira-Spencer isomorphisms and degeneracy maps on Iwahori-level Hilbert modular varieties: the saving trace