I have a question regarding non-cuspidal Hilbert modular forms. If one starts with a non-parallel weight for example, it is easy to prove that there are no Eisenstein series of any level, or as is generally stated, all forms are cuspidal. My question is what happens with mod p Hilbert modular forms? Are there (non-zero) non-cuspidal mod p Hilbert modular forms of non-parallel weight? (say at least when one or all the weights are greater than 1).


For classical modular forms, if the weight is greater than 1, the mod p modular forms are exactly the reduction of global modular forms, so the naive answer would be that there are none, but I am not too familiar with mod p Hilbert modular forms...



Eisenstein mod p Hilbert modular forms
The partial Hasse invariants $h\_1,\ldots,h\_d$ are mod $p$ Hilbert modular forms of non-parallel weight whose $q$-expansion at each cusp is equal to 1. The forms $h\_1-1,\ldots,h\_d-1$ generate the kernel of the $q$-expansion map over $\mathbb{F}\_p$. Technically, these forms are of weight $(0,\ldots,0,p,-1,0,\ldots,0)$ or $(0,\ldots,0,p-1,0,\ldots,0)$, at least when $p$ is unramified, but you can always multiply them by some large parallel weight form to get something of everywhere positive weight. As you remarked, they have no characteristic 0 lift on account being non-cuspidal and having non-parallel weight.


If you want a more detailed account of these guys, and mod $p$ Hilbert modular forms in general, I recommend Goren's *Lectures on Hilbert Modular Varieties and Modular Forms*, especially chapter 5, and Andreatta-Goren's *Hilbert Modular Forms: mod p and p-adic aspects*, available on Goren's [website](http://www.math.mcgill.ca/goren/publications.html).


