Can the Unit Root Splitting be extended?

Question

Let $X/\mathrm{Spec}\hspace{0.1em}{\mathbb{Z}_p}$ be a modular curve (with some level $N$ structure), and $p$ is prime that does not divide $N$. The special fibre $X_{\mathbb{F}_p}$ contains the ordinary locus as an open subscheme, i.e. the set of points which classify elliptic curves (with relevant level structure) having good ordinary reduction at $p$. Suppose $\pi \colon \mathcal{E} \to X$ is the universal generalized elliptic curve. The relative de Rham sheaf $H^1_{\mathrm{dR}}(\mathcal{E}/X)$ comes with a Hodge filtration $$ 0 \to \omega_{\mathcal{E}} \to H^1_{\mathrm{dr}}(\mathcal{E}/X) \to \omega_{\mathcal{E}}^{-1} \to 0 $$ and the action of the Frobenius morphism over $\mathbb{F}_p$ respects this filtration. Over the ordinary locus this filtration admits a canonical splitting called the unit root splitting which commutes with the Frobenius action.

My question is the following: can the unit root splitting be extended over some of the supersingular points? In other words, is there a splitting of the Hodge filtration over an open set that strictly contains the ordinary locus, which commutes with the Frobenius action?

Thanks in advance!

Answer 1

No, there is no such extension. This is proved in Urban's article "Nearly overconvergent modular forms" (preprint version here, official publisher version here), Prop 3.1.3:

Proposition 3.1.3. For any $\rho < 1$ and any Zariski open $V \subset X^{\ge \rho}$, the unit root splitting on $V^{ord} := U \cap X^{ord}$ does not extend to a splitting of the Hodge filtration of $H^1$ over any finite cover of $V$.

(Here $X^{\ge \rho}$ is a strict neighbourhood of the ordinary locus in the rigid-analytic generic fibre of the modular curve.)