Meets: W 13.15-15.00 in von Neumann 1.023.
Starts: 15.4.2014.
The main goal of the semester is to understand some aspects of Faltings' proofs of some far-reaching finiteness theorems about abelian varieties over number fields, the highlight being the Tate conjecture, the Shafarevich conjecture, and the Mordell conjecture. There are a variety of references, including:
G. Faltings. Endlichkeitssätze für abelsche Varietäten über Zählkorpern. We won't actually be using this.
L. Szpiro (editor). Seminaire sur les pinceaux arithmétiques: la conjecture de Mordell. Treats everything in detail.
G. Faltings and G. Wüstholz (editors). Rational points. Lots of detail on the representation theory side of things.
G. Cornell and J. H. Silverman (editors). Arithmetic Geometry. A good reference in general.
Deligne and Szpiro have Bourbaki exposés overviewing the proof. There's also a great set of notes from a similar seminar run at Stanford. For general background on abelian varieties, see
D. Mumford. Abelian varieties.
J. S. Milne. Abelian varieties, available at www.jmilne.org/math/.
Milne also treats the finiteness theorems. For the most part we'll follow Faltings and Wüstholz, though we'll invert the order slightly.
15.04.2015: Overview (Ben).
22.04.2015: Abelian varieties over arbitrary fields (Daniele). Notes.
29.04.2015: Tate module (Gregor). Notes.
06.05.2015: Tate and Shafarevich conjectures from finiteness (Niels). Notes.
13.05.2015: Group schemes (Eva).
20.05.2015: Heights (Daniele). Notes.
27.05.2015: Ramification of p-divisible groups (Antareep).
03.06.2015: Proof of the Tate conjecture (Wouter). Notes.
10.06.2015: Ramification of finite group schemes (Fabio).
17.06.2015: Finiteness of isogeny classes (Emre). Notes.
24.06.2015: Heights within isogeny classes and the Shafarevich conjecture (Barbara).
01.07.2015: The Mordell conjecture (Rostislav).