Unramifiedness of Galois representations arising from Hilbert modular surfaces

Abstract

Let $p$ be a prime number and $F$ a totally real number field. For each prime $\mathfrak{p}$ of $F$ above $p$ we construct a Hecke operator $T_{\mathfrak{p}}$ acting on $(\mathrm{mod}{p}^m)$ Katz Hilbert modular classes which agrees with the classical Hecke operator at $\mathfrak{p}$ 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 $1$ are unramified at $p$ when $[F:\mathbb{Q}]=2$. Some partial and some conjectural results are obtained when $[F:\mathbb{Q}]>2$.

Final version, to appear in Forum Sigma

References in corpus (2)

• Torsion in the Coherent Cohomology of Shimura Varieties and Galois Representations
• Unramifiedness of Galois representations attached to weight one Hilbert modular eigenforms mod p