Algebraisk Geometri

Algebra og Geometri Seminar

Prof. Christian Liedtke (Technische Universität München)

Seminar Rom Delta

Title: Crystalline Galois Representations arising from K3 Surfaces

Let K be a p-adic field, let X be a K3 surface over K, and assume that X has potential semi-stable reduction. Then, we show that the following are equivalent:  
1) the l-adic Galois representation on H^2(\bar{X},Q_l) is unramified for one l different from p 
2) the l-adic Galois representation on H^2(\bar{X},Q_l) is unramified for all l different from p 
3) the p-adic Galois representation on H^2(\bar{X},Q_p) is crystalline 
4) the surface has good reduction after an unramified extension of K
This is an analog of the classical Serre-Tate theorem for Abelian varieties. We also show by counter-examples that neither 1), nor 2), nor 3) implies that X has good reduction of X over K. However, in this case X admits a proper model over O_K, whose special fiber X_0 has at worst canonical singularities. Now, if the Galois-representation on H^2(\bar{X},Q_p) is crystalline, then functors of Fontaine and Kisin provide us with an F-crystal over W(k) that looks like the crystalline cohomology of some smooth K3 surface - in fact, we will show that this is the crystalline cohomology of the minimal resolution of singularities of X_0. In my talk, I will introduce all the above notions and functors (which will not give me much time to give proofs). Part of this is joint with Matsumoto, part of this is joint with Chiarellotto and Lazda.