MODULAR-REPRESENTATIONS OF THE GALOIS GROUP OF A LOCAL FIELD, AND A GENERALIZATION OF THE SHAFAREVICH CONJECTURE

Let M-GAMMA-crys(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-GAMM-crys(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 Z: if X is a smooth proper scheme over the ring of integers of the field Q (respectively Q(square-root -1), Q(square-root -3), Q(square-root 5)), then the Hodge numbers of the complex manifold X(C) satisfy h(ij) = 0 for i not-equal j and i + j less-than-or-equal-to 3 (respectively i + j less-than-or-equal-to 2).