# Ash–Stevens for Hilbert modular forms

In the theory of mod-$$p$$ modular forms, I learned a while ago about an interesting result that I think is technically due to Serre and Tate, though the proof was first published by Jochnowitz in greater generality, and later on a different proof for $$\ell \geq 5$$ was accomplished in the work of Ash and Stevens: namely, the fact that for a rational prime $$\ell$$, the systems of Hecke eigenvalues of mod-$$\ell$$ modular forms of (say) level 1 coming from modular forms of weight $$k \geq 2$$ can all be produced (up to some theta twist) as mod-$$\ell$$ reductions of systems of Hecke eigenvalues of modular forms of weight $$2 \leq k \leq \ell + 1$$.

Is there a generalization of this fact to Hilbert modular forms ? I imagine that it will be a little bit more technically complicated: for example we would need to probably replace the condition $$k \geq 2$$ with the condition that the mod-$$\ell$$ Hilbert modular form in question is liftable to characteristic zero (as far as I am concerned this is fine). By Googling, I found the long paper of Andreatta and Goren, in which a similar thing is proved for forms that are ordinary (Hilbert modular forms: mod $$p$$ and $$p$$-adic aspects, Corollary 18.15). Is this result false as stated without the ordinary hypothesis ? Is there a salvage ?

Is there a fundamental reason why we cannot use the approach of Ash–Stevens to study this problem ?

• You might want to look at various works of Fred Diamond over the last decade or so (maybe starting with the Buzzard--Diamond--Jarvis paper). 4 hours ago