Projective varieties with bad semi-stable reduction at 3 only

Suppose $F = W(k)[1/p]$ where $W(k)$ is the ring of Witt vectors with coefficients in algebraically closed field $k$ of characteristic $p\ne2$. We construct 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 $\mathbb{Q}$ with good reduction modulo all primes $l\ne3$ and semi-stable reduction modulo $3$ then for the Hodge numbers of $Y_{\mathbb{C}}=Y\otimes_{\mathbb{Q}}\mathbb{C}$, one has $h^2(Y_{\mathbb{C}})=h^{1,1}(Y_{\mathbb{C}})$.