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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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