# Surjectivity of reduction for Hilbert modular forms

Fix a totally real field $K$, a level $\mathfrak{n}$, a (parallel) weight $k\geq 2$ and a primitive ray class character $\chi$ modulo $\mathfrak{n}$.

Then one can form the space $S_k(\mathfrak{n},\chi)$ of Hilbert cusp forms (as in Shimura). This is the full Hecke module so can be considered as $h_K^{+}$ tuples of modular forms that transform for various “Gamma_0” style groups.

Consider the module of forms which have $\mathfrak{p}$-integral Shimura coefficients. Then there is a well defined mod $\mathfrak{p}$ reduction map into the space of mod $\mathfrak{p}$ HMF’s of weight $k$, level $\mathfrak{n}$ and character $\bar{\chi}$ (a la Katz).

Is it known explicitly when this map is surjective, i.e. is there an explicit analogue of Carayol’s lemma? (as in Edixhoven’s Serre’s conjecture chapter in “Modular Forms and FLT”). Could the answer be $k\geq 2$ as in the rational case?

If the result is only known for large enough $k$ then that is fine too, but would prefer to know an explicit absolute bound.