||Three conjectures will be studied:
Manin-Mumford, Andre-Oort and the Hecke Orbit conjecture.We will show that these statements (in arithmetic algebraic geometry) are very similar: all three study a set of points in an algebraic variety and we ask what the
closure is of this set.
In the first part of my talk I will discuss the basic concepts used: abelian varieties and moduli spaces. I will give examples
and some easy proofs. Some of the aspects will
be illustrated with the theory of elliptic curves. If you want to obtain a general impression about these things, the first
half of the talk will provide this.
In the second part I will discuss some more technical details, and an application. The proof of the Hecke Orbit conjecture is joint work with Ching-Li Chai, using a result by Chia-Fu Yu.