Lectures on Fontaine-Laffaille theory and ramification of crystalline representations

Hattori Shin (Tokyo City University)

August 2018,  21 (Tue) 16:00~17:30, 22~24 (Wen,Thur,Fri) 15:00~16:30 아산이학관 111호   

AbstractLet K be a complete discrete valuation field of mixed characteristic (0,p) with perfect residue field and G_K its absolute Galois group. For a variety X over K, etale cohomology groups with coefficients in Q_p yield p-adic G_K-representations, which can be considered as linearizations of geometric information of X. p-adic Hodge theory enables us not only to understand etale cohomology groups via (variants of) de Rham cohomology groups, but also to classify good p-adic G_K-representations by certain semi-linear algebraic data.

On the other hand, we often encounter G_K-representations with Z_p- or p-power torsion coefficients, which are more subtle than those with Q_p-coefficients. Integral p-adic Hodge theory is a tool to study them in a compatible way with p-adic Hodge theory, and typically it is very powerful for investigating deformation of G_K-representations. The Fontaine-Laffaille theory is the most primitive form of integral p-adic Hodge theory established in 1980s. Since then integral p-adic Hodge theory has been highly developed in various ways, while the Fontaine-Laffaille theory is still useful for arithmetic study of Galois representations.

In this series of lectures, I will talk about what the Fontaine-Laffaille theory is like, present a sketch of proofs of its main theorems and explain its classical application to ramification bound for torsion etale cohomology groups due to Abrashkin and Fontaine.