Time: 16:00-17:00, Thursday, November 30 2023
Venue: E4-233, Yungu Campus
Speaker: Heer Zhao, University of Duisburg-Essen
Title: Log $p$-divisible groups
Abstract:
Let K be a complete DVF of mixed characteristic (0,p), R its ring of integers, and G the absolute Galois group of K.
(a) In general it is not possible to extend a p-div (p-divisible) group with sst (semi-stable) reduction over K into a p-div group over R;
(b) a famous conjecture of Fontaine (proved by Fontaine, Laffaille, Breuil, and Kisin) states that a Z_p-representation of G with HT (Hodge-Tate) weights in {0,1} is crystalline iff it arises from a p-div group over R, and thus in general a sst Z_p-representations of G with HT weights in {0,1} does not arise from a p-div over R.
Log geometry is the natural framework to deal with degenerations (or extensions). Using Kato's theory of log p-div groups, we (joint with Alessandra Bertapelle and Shanwen Wang) make the impossible possible.
More precisely, we show that the following three categories are equivalent: (1) log p-div groups over R (endowed with the canonical log structure); (2) p-div groups with sst reduction over K; (3) sst Z_p-representations of G with HT weights in {0,1}.
If time permits, I will also briefly mention Serre-Tate theorem (joint with Matti Wurthen) and Tate theorem for log p-div groups.
