# Liftability of mod p Hilbert modular forms of parellel weight

If $f$ is a mod $p$ Katz modular form of weight $k$ for $k \geq 2$, there is a classical modular form $g$ such that reduction of $g$ is $f$.

Is there a similar property holds for Hilbert modular forms of higher weights? Namely, let $f$ be a mod $p$ Hilbert modular form of parallel weight $k$ for some big $k$ (in the sense of Katz). Then, does there exist a classical Hilbert modular form $g$ of same weight that gives $f$ by reduction? If so, is there any bound for $k$?

Let $$\overline{\mathcal{M}}(N,\mathcal{O})$$ be the minimal compactification of the Hilbert modular scheme over $$\mathbb{Z}_p$$ for a totally real field $$F$$ and $$\omega$$ be the determinant of the co-normal sheaf ($$p\nmid N$$). It is known that $$\omega$$ is ample. We have an exact sequence of sheaf such that the first map is multiplying by $$p$$.
$$0\rightarrow \omega^{k} \rightarrow \omega^{k} \rightarrow \omega^{k}/p\rightarrow 0$$
Since $$\overline{\mathcal{M}}(N,\mathcal{O})$$ is projective an $$\omega$$ is ample, then for a large $$k$$, $$H^{1}(\overline{\mathcal{M}}(N,\mathcal{O}), \omega^{k})$$ is trivial. Moreover, the support of $$\omega^{k}/p$$ is the special fibre of $$\overline{\mathcal{M}}(N,\mathcal{O})$$, Hence by applying the global section to the above exact sequence, we find that $$M_k(N,\mathbb{Z}_p) \otimes \mathbb{F}_p$$ surjects to $$M_k(N,\mathbb{F}_p)$$.
In the case where we have a modular curve $$X_1(N)$$, we can use the same argument combined with Riemann-Roch to show that $$H^1(X_1(N),\omega^{k})$$ is trivial for a large $$k$$.
• Is it known explicitly for which $k,p$ this is true? Also is there a similar result for forms with character? I have recently posted a question here: mathoverflow.net/q/306397/21698 Jul 23, 2018 at 8:21