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.

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.

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 $(0,\ldots,0,p,-1,0,\ldots,0)$, 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.

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.