A semi-stable case of the Shafarevich Conjecture

Suppose $F$ is the quotient field of the ring of Witt vectors with coefficients in an algebraically closed field $k$ of odd characteristic $p$. We construct an integral theory of $p$-adic semi-stable representations of the absolute Galois group of $F$ with Hodge–Tate weights from $[0,p)$. This modification of Breuil's theory results in the following application in the spirit of the Shafarevich Conjecture. If $Y$ is a projective algebraic variety over rational numbers with good reduction away from $3$ and semi-stable reduction modulo $3$, then for the Hodge numbers of the complexification $Y_C$ of $Y$ it holds $h^2(Y_C)=h^{1,1}(Y_C)$.