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 $p\ne 2$. 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 $\Q $ with good reduction modulo all primes $l\ne 3$ and semi-stable reduction modulo 3 then for the Hodge numbers of $Y_C=Y\otimes _{\Q}\ C$, one has $h^2(Y_C)=h^{1,1}(Y_C)$.

2010 Mathematics Subject Classification: 11S20, 11G35, 14K15

Keywords and Phrases: p-adic semi-stable representations, Shafarevich Conjecture

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