Unlikely intersections study group

Jan-March 2018, UCL

Wednesdays 2.00 - 3.30 pm, room 505, UCL maths department (10 January - 21 March)

The aim of the study group is to understand recent progress on unlikely intersections in Shimura varieties, following the strategy of Pila and Zannier using o-minimality. The following themes will appear, each in the context of several different results:

The study group will focus on the case of Shimura varieties, in particular products of modular curves (Y(1)n) and the moduli space of principally polarised abelian varieties (Ag). As it is possible to define these moduli spaces concretely, familiarity with the general theory of Shimura varieties should not be required. We will also discuss abelian varieties (the Manin-Mumford conjecture) as a warm-up, and there will be a talk on results on families of abelian varieties (which can be interpreted as mixed Shimura varieties).

Programme

10 Jan Introduction Andrei Yafaev Notes by Gregorio
17 Jan Manin-Mumford conjecture following Pila and Zannier Michele Giacomini Notes by Gregorio
24 Jan André-Oort conjecture for Y(1)n (Pila) Domenico Valloni
31 Jan Ax-Lindemann-Weierstrass theorem for Ag (Pila-Tsimerman) Andrei Yafaev
7 Feb Heights of abelian varieties Netan Dogra
14 Feb André-Oort conjecture for Ag (Pila-Tsimerman, Ullmo, Tsimerman) Gregorio Baldi
21 Feb Masser-Wüstholz isogeny theorem Martin Orr
28 Feb Families of abelian varieties (Masser-Zannier, Barroero-Capuano) Laura Capuano
7 Mar André-Pink conjecture (Orr) Domenico Valloni
14 Mar Zilber-Pink conjecture for Y(1)3 (Habegger-Pila) tbc
21 Mar Zilber's Conjecture on Intersections with Tori tbc
The dates of the last few talks are subject to change, depending on the availability of speakers.

More detailed programme