Zannier on Zoom (ZaZoom)

This seminar broadly followed Zannier’s book Some Problems of Unlikely Intersections in Arithmetic and Geometry, though we’ll use some other references. You can access it online for free through the UGA library.

The seminar met online on Fridays at 4pm Eastern.

References

1. Zannier, "Some Problems of Unlikely Intersections in Arithmetic and Geometry" (find on line via UGA Library)

2. Scanlon, "O-minimality as an approach to the Andre-Oort conjecture"

3. Pila, "O-minimality and diophantine geometry"

4. Jones, Wilkie "O-Minimality and Diophantine Geometry"

5. Lou van den Dries, "Tame Topology and o-Minimal Structures"

6. Baker, Ribet "Galois theory and torsion points on curves"

Talks

1. (Borys, 9/4) Problems on unlikely intersections: examples (Zannier, Introduction). Statement of the general Mordell-Lang-...-McQuillan theorem. Special cases: Faltings' theorem, Lang's theorem, the Manin–Mumford conjecture. Some extensions: statements of Bogomolov's conjecture and Tate-Voloch's conjecture. Proof of Lang's theorem for curves in (Zannier, chapter 1, unnumbered paragraph).

2. (Arvind, 9/11) Unlikely intersections in tori (Zannier, Chapter 1, rough notes)

3. (Zannier, 9/18) Movie day!

4. (Padma, 9/25) Simplest unlikely intersections in abelian varieties: the Manin-Mumford conjecture (Baker-Ribet)

5. (Daniel, 10/2) Galois actions on torsion points (notes)

6. (Ben Bakker, 10/9) o-minimality (slides, video)

7. (Freddy, 10/16) Galois action on CM points (following Serre’s chapter in Cassels-Frohlich)

8. (Haiyang, 10/23) Siegel’s theorem, following this paper, with context

9. (Borys, 10/30) Pila-Wilkie

10. (Nicholas, 11/6) Ax-Schanuel

11. (Mentzelos, 11/13) André-Oort for products of modular curves, I (after this paper of Pila) (notes)