时间： 2月25日（周四）下午 14:00—16:00
3月01日（周二）下午13:00—15:00
3月10日（周四）下午 13:00—15:00
3月14日（周一）下午 13:00—15:00

摘要：This short course will be divided into two parts:

(1) Given a smooth quasi-projective scheme X over the Witt ring W(k) of the algebraic closure of a finite field of characteristic p, I shall first explain briefly the notion "Fontaine-Faltings module" on X introduced by Fontaine and Faltings and the notions "Higgs-de Rham flow" and periodic Higgs bundles on X invented in a recent work byLan-Sheng-Zuo. I shall then discuss a p-adic correspondence between the category of crystalline representations of the etale fundamental group of the generic fibre of X and the category of periodic Higgs bundles on X. As an application we construct an absolute irreducible rank-2crystalline representation of the etalefundamental group of a so-called canonical lifted hyperbolic curve via the uniformization Higgs bundle, which should be regarded as a p-adic analogue of Hitchin-Simpson's uniformization theorem on complex hyperbolic curves via uniformization Higgs bundles. This talk is based on my joint papers with Lan, Sheng and Yang.

(2) We will also introduce the finiteness of CM Jacobians of smooth hyperelliptic curves and superelliptic curves. Coleman-Oort conjecture predicts that, there exist at most finitely many smooth curves of genus g (up to isomorphism) whose Jacobians are CM abelian varieties for g sufficiently large, or equivalently, there exists no Shimura subvariety of positive dimension contained generically in the Torelli locus of smooth curves when g is sufficiently large. In this talk, I will try to report the recent work joint with Chen and Zuo on a partial solution to this conjecture, that is, we prove that Coleman-Oort conjecture holds true for the cases of smooth hyperelliptic curves and superelliptic curves.

华东师范大学数学系
上海市核心数学与实践重点实验室