# Unlikely intersections study group

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

- Introduction
- Manin-Mumford conjecture following Pila and Zannier
- Unlikely intersections in families of abelian varieties (Masser-Zannier, Barroero-Capuano)
- André-Oort conjecture for
*Y(1)*^{n} (Pila)
- Ax-Lindemann-Weierstrass conjecture for
*A*_{g} (Pila-Tsimerman)
- André-Oort conjecture for
*A*_{g} (Pila-Tsimerman, Ullmo, Tsimerman)
- Faltings height of abelian varieties
- Masser-Wüstholz isogeny theorem
- André-Pink conjecture (Orr)
- Zilber-Pink conjecture for
*Y(1)*^{3} (Habegger-Pila)
- Conditional proof of the Zilber-Pink conjecture (Daw-Ren)