## Modular representations of the Galois group of a local field, and a generalization of the Shafarevich conjecture.(English. Russian original)Zbl 0733.14008

Math. USSR, Izv. 35, No. 3, 469-518 (1990); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 53, No. 6, 1135-1182 (1989).
Summary: Let $$M\Gamma^{crys}({\mathbb{Q}}_ p)$$ be the category of crystalline representations of the Galois group of the field of fractions of the ring of Witt vectors of an algebraically closed field of characteristic $$p>0.$$ The author describes the subfactors annihilated by multiplication by p of the representations from $$M\Gamma^{crys}({\mathbb{Q}}_ p)$$ arising from filtered modules of filtration length $$<p,$$ and proves a generalization of the Shafarevich conjecture that there do not exist abelian schemes over $${\mathbb{Z}}:$$ if X is a smooth proper scheme over the ring of integers of the field $${\mathbb{Q}}$$ (resp. $${\mathbb{Q}}(\sqrt{-1})$$, $${\mathbb{Q}}(\sqrt{-3})$$, $${\mathbb{Q}}(\sqrt{5}))$$, then the Hodge numbers of the complex manifold $$X_ C$$ satisfy $$h^{ij}=0$$ for $$i\neq j$$ and $$i+j\leq 3$$ (resp. $$i+j\leq 2)$$.

### MSC:

 14F30 $$p$$-adic cohomology, crystalline cohomology 11S20 Galois theory 13F35 Witt vectors and related rings
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