The Calabi–Yau modular-form Hasse–Witt polynomial conjecture

Let ${\sf S}$ be the affine moduli scheme of pairs $(X,\alpha)$ with universal family ${\sf X}\to{\sf S}$ and universal holomorphic form $\alpha\in H^0({\sf X},\Omega^n_{{\sf X}/{\sf S}})$. Let $t_i$ be homogeneous global regular functions of weights $k_i$, and let $t_i(q)$ denote their $q$-expansions. Hasse–Witt polynomial conjecture. For any prime $p$ not dividing $N$, there is a homogeneous polynomial $A_p\in\mathbb{F}_p[t]$ of degree $p-1$, with $\deg(t_i)=k_i$, such that

$$C(\alpha)=A_p(t)^{\frac{1}{p}}\alpha,$$

where $C$ is the Cartier operator, and

$$A_p(t_1(q),t_2(q),\ldots,t_s(q))\equiv_p1.$$

The text explains that the first assertion is known in several geometric cases, whereas the $p$-integrality and resulting congruence remain open in general.