Sun–Yang–Zuo conjecture on finiteness of crystalline local systems

Let $(X,S)$ be a log pair over $W(\mathbb F_q)$ with $p^f\mid q$. A $\mathbb Z_{p^f}$-crystalline local system is a crystalline local system with coefficients in $\mathbb Z_{p^f}$, and $\operatorname{GL}_r(\mathbb Z_{p^f})$-crystalline local systems are the corresponding rank-$r$ objects.

Sun–Yang–Zuo conjecture.

1. If $\mathbb L_1$ and $\mathbb L_2$ are two $\mathbb Z_{p^f}$-crystalline local systems over $(X,S)/W(\mathbb F_q)$ and

$$\mathbb L_1\simeq\mathbb L_2\pmod p,$$

then $\mathbb L_1\simeq\mathbb L_2$. 2. The number of isomorphism classes of $\operatorname{GL}_r(\mathbb Z_{p^f})$-crystalline local systems over $(X,S)_{W(\mathbb F_{q^h})}$, as $h$ ranges through the positive integers, is finite.

The first part is presented as being in the spirit of the Fontaine–Mazur conjecture, while the second is described as analogous to a theorem of Litt. It asserts both rigidity under reduction modulo $p$ and finiteness across finite unramified extensions.