GALOIS MODULI OF PERIOD $p$ GROUP SCHEMES OVER A RING OF WITT VECTORS

Abstract

Necessary and sufficient conditions (completely sufficient only when 2$ SRC=http://ej.iop.org/images/0025-5726/31/1/A01/tex_im_1042_img2.gif/>) are obtained that are satisfied by the Galois modules of the geometric points of a finite commutative period group scheme defined over a ring of Witt vectors. As an application of these results it is proved that there are no abelian schemes over the ring of integers of the fields , , , , , . The case of the field answers a conjecture of Shafarevich (at the 1962 ICM in Stockholm) that there do not exist abelian varieties or curves of genus defined over this field and having everywhere good reduction. Bibliography: 15 titles.