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 overall strategy and the Pila-Wilkie counting theorem. We will not prove the Pila-Wilkie theorem, and sketch only enough model theory to state what is needed.
• Functional transcendence results and how they are used (the geometric part of the proofs).
• Large Galois orbits and associated number theory.

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 (A_g). 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).

Draft programme

1. Introduction
2. Manin-Mumford conjecture following Pila and Zannier
3. Unlikely intersections in families of abelian varieties (Masser-Zannier, Barroero-Capuano)
4. André-Oort conjecture for Y(1)ⁿ (Pila)
5. Ax-Lindemann-Weierstrass conjecture for A_g (Pila-Tsimerman)
6. André-Oort conjecture for A_g (Pila-Tsimerman, Ullmo, Tsimerman)
7. Faltings height of abelian varieties
8. Masser-Wüstholz isogeny theorem
9. André-Pink conjecture (Orr)
10. Zilber-Pink conjecture for Y(1)³ (Habegger-Pila)
11. Conditional proof of the Zilber-Pink conjecture (Daw-Ren)