## Projective varieties with bad semi-stable reduction at 3 only.(English)Zbl 1362.11095

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

### MSC:

 11S20 Galois theory 11G35 Varieties over global fields 14K15 Arithmetic ground fields for abelian varieties
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