The raison d'être of p-adic Hodge theory are the comparison theorems between p-adic étale cohomology of a smooth proper variety over a p-adic field and other p-adic cohomologies of differential nature (Hodge, de Rham, crystalline...). Faltings' approach for this theory is up to now the most successful. He obtained all comparison theorems from a basic one, namely his main comparison theorem. In the first part of my lecture, I will explain this theorem and in the second part, I will discuss a relative version roughly sketched by Faltings in the appendix of his 2002 Astérisque article. The actual proof requires more work (joint with Michel Gros): as one says, the devil is in the details...

Ahmed Abbess教授，法国国家科学研究中心和巴黎高级科学研究所研究部主任，主要研究特征数p>0或p进位的完美域簇上层的几何及上同调性质，用于算术和代数几何。

Abbess教授与Takeshi Saito（东京）就分歧理论方面的合作取得重大突破。最近他的工作重点是p进位霍奇理论，并与Michel Gros共同研究了基于法尔廷斯方法的p进位辛普森对应。

法国国家科学研究中心(CNRS)成立于1939年，是法国最大的科学技术研究机构，也是欧洲最大的基础研究机构之一。