Unramifiedness of Galois representations arising from Hilbert modular surfaces

TL;DR

This paper proves that Galois representations from torsion Hilbert modular classes of parallel weight one are unramified at primes dividing p for quadratic totally real fields, extending some results to higher degrees.

Contribution

It constructs Hecke operators on mod p^m Hilbert modular classes and proves unramifiedness of associated Galois representations for quadratic fields, with partial results for higher degrees.

Findings

Galois representations are unramified at p for quadratic fields.

Construction of Hecke operators on mod p^m Hilbert modular classes.

Partial results and conjectures for higher degree fields.

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 ${\bf 1}$ are unramified at $p$ when $[F:\mathbb Q]=2$. Some partial and some conjectural results are obtained when $[F:\mathbb Q]>2$.