## Eisenstein mod p Hilbert modular forms

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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...

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## 1 Answer

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.

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Just a technical question (while I read Goren's book) does this work for $p=2$ as well? Because if so you would get a form of weight (0,...,0,1,0,...,0) which does not satisfy the usual parity condition (but you might have a form of weight (0,...,0,2,0,..,0) modulo 2) – A. Pacetti Oct 25 2011 at 0:18
$p=2$ works fine. The "usual parity condition" plays no role, as far as I know, in the elementary "definitions and first properties" part of the theory; for example I can take a certain non-algebraic grossencharacter on $GL(1)$ of a CM extension of $F$ and induce it up to get a classical char 0 Hilbert modular form of e.g. weight $(1,2)$ on a real quadratic field. The parity condition only comes in when trying to do arithmetic -- e.g. it's the parity condition you need to get Satake parameters all defined over a number field or to attach a Galois representation. – Kevin Buzzard Oct 25 2011 at 8:07

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[NB even for $p>2$ the partial Hasse invariants don't satisfy the parity condition in general!] – Kevin Buzzard Oct 25 2011 at 8:08
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