On the deformations of cyclic Galois coverings of algebraic curves

点击翻译按钮获取中文摘要

Our final aim of this research is to lift a pair (C, sigma), of a complete nonsingular curve C and its automorphism sigma of order p^n over a field of characteristic p(0), to a pair over a field of characteristic zero. For this purpose, we must construct a theory of deformations of Witt groups to tori. The n-dimensional Witt group W_n is an extension of W_<n-1> by G_a, and it contains Z/p^n as the extension of Z/p^<n-1> by Z/p. The deformations we require should preserves thus ffitrations of Witt groups. In this research, we found out that to hnadle this kind of deformations we needed a kind of vanishing theorem df extension groups of group schemes over an Artin local rings. More-over, using this vanishing theorem, we showed that we could control the deformations of W_n as an extension of W_<n-1> by G_a, the surjectivity of a specialization map, and the existence of deformations of W_n to a torus keeping the filtrations and the constant subgroup scheme Z/p^n. Using tyhese theorems, we could precisely construct the deformations of Artin-SchreierWitt exact sequences to exact sequences of kummer type. These deformed exact sequences give exactly the unified theory of the Artin-Schreier-Witt theory and the Kummer theory. In fact, we can check the unified theory by computing the first cohomology group for these deformed Witt groups. Our theory is sufficiently general, but unfortunately we could not decide the defining rings of these deformations from this direction, and for this purpose we must develop another kind of method. In fact, looking the deformations of isogenies of group schemes more precisely, we can see that our deformation of W_n is defined over the ring Z_<(p)>[mu_p^n]. Moreover, these deformations should be given from the unit groups of group rings. From this view point, we could also give some partial results.