On the Slope Filtration of d - modules over the Robba Ring

Given a p-adic representation of the Galois group of a local field, we show that its Galois cohomology can be computed using the associated 6tale (o, F)-module over the Robba ring; this is a variant of a result of Herr. We then establish analogues, for not necessarily etale (ý, F)-modules over the Robba ring, of the Euler-Poincard characteristic formula and Tate local duality for p-adic representations. These results are expected to intervene in the duality theory for Selmer groups associated to de Rham representations. We introduce the notion of families of q-modules which arises naturally from both rigid cohomology and p-adic Hodge theory. We then prove the local constancy of generic HNpolygons of families of overconvergent 0-modules and the semicontinuity of HN-polygons of families of q-modules over reduced affinoid algebras. These results are prospective for a slope theory of families of (overconvergent) 0-modules. Thesis Supervisor: Kiran Sridhara Kedlaya Title: Associate Professor of Mathematics Acknowledgments Many thanks are due to my advisor, Kiran Sridhara Kedlaya, for suggesting the topic of this thesis, for many useful discussions and crucial suggestions, for enabling me to attend many conferences, and for spending much time reviewing early drafts of this thesis. In this last regard, thanks are due as well to Jay Pottharst, and Liang Xiao. Thanks Chris Davis for many help on English. Thanks also to Chris Davis, Liang Xiao for all kinds of help and the much happy time shared. I would like to thank my wife, Qiuye Zhao, for the love she provides. Finally, thanks to my parents, Shaolu Liu and Yujie Tai, for giving me the freedom to do what I like and the support and encouragement that I always received. Financial support was provided by MIT Department of Mathematics.