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Unramifiedness of weight one Hilbert Hecke algebras

Shaunak V. Deo, Mladen Dimitrov, Gabor Wiese

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 $Θ$-operators using Reduzzi--Xiao's generalised Hasse invariants, including especially an injectivity criterion in terms of minimal weights.

Unramifiedness of weight one Hilbert Hecke algebras

Abstract

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

Paper Structure

This paper contains 15 sections, 31 theorems, 74 equations, 1 figure.

Table of Contents

  1. Hilbert modular forms in finite characteristic
  2. Pappas--Rapoport splitting models for Hilbert modular varieties
  3. Generalised Hasse invariants
  4. Minimal weights
  5. Partial Frobenius operator
  6. Refined injectivity criterion for $\Theta$-operators
  7. Doubling and Hecke algebras
  8. Hilbert modular forms of parallel weight $1$
  9. Doubling
  10. Hecke algebras
  11. Pseudo-representations for weight $1$ Hecke algebras
  12. Pseudo-representations of degree $2$
  13. Existence of an ordinary Hecke algebra-valued pseudo-representation
  14. Proof of the main theorem
  15. Non-Eisenstein ideals

Key Result

Theorem 1

There exists a $\mathbb{T}^{(1)}$-valued pseudo-representation $P^{(1)}$ of $\mathrm{G}_F$ of degree $2$ which is unramified at all primes $\mathfrak{q}$ not dividing $\mathfrak{n} p$ and satisfies $P^{(1)}(\mathop{\mathrm{Frob}}\nolimits_\mathfrak{q})=(T_\mathfrak{q}, {\langle \mathfrak{q} \rangle}

Figures (1)

  • Figure 1: Weights of Hasse invariants.

Theorems & Definitions (70)

  • Theorem 1
  • Corollary : Corollary \ref{['cor:RT']}
  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 1.4
  • proof
  • Lemma 1.5
  • proof
  • Remark 1.6
  • ...and 60 more