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

Abrashkin, V. A.

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

Keywords:

crystalline representations of the Galois group; Witt vectors; characteristic p; Shafarevich conjecture; Hodge numbers