Rigid local systems: arithmetic properties #1
Speaker: Hélène Esnault, Freie Universität Berlin. First of three lectures.
Abstract: Simpson (1990) conjectured that complex rigid local systems are geometric. Of course, this conjecture is deep and inaccessible. However, recently Petrov showed that it inserts in a vast arithmetic program: Simpson’s conjecture is, using Scholze’s and Liu-Zhu’s p-adic results, a consequence of the generalized Fontaine-Mazur conjecture.
If the conjecture is true, there are various consequences, all of arithmetic nature. We proved with Michael Groechenig some of them: integrality using Deligne’s theory of companions, and crystallinity using notably Faltings' p-adic Simpson correspondence.
I will try to explain a few of the aspects of these theorems. The last one is a building block of the recently uploaded proof of the André-Oort conjecture by Pila-Shankar-Tsimerman.