Research Seminar: The Tate conjecture for K3 surfaces
This term we want to learn about K3 surfaces and in particular about (one of the) recent proofs of the Tate conjecture for K3 surfaces.
K3 surfaces play a special role in the classification of surfaces: On the one hand they have trivial canonical bundle (like elliptic curves and abelian surfaces) but on the other hand, they do not admit a group structure. As one of the sources puts it they can be defined as being proper surfaces with trivial canonical bundle plus any condition that rules out abelian surfaces.
The literature on these surfaces being vast, we have to make some choices for the seminar. However, the amount of topics also has the advantage, that many techniques from various parts of algebraic geometry have been used to study properties of K3 surfaces, which gives us the opportunity to learn several of these methods in a rather concrete application. I therefore hope that you can find a talk in the program, to share your expertise or find a talk where you can take the opportunity to talk about something you wanted to learn anyway.
|Introduction (and distribution of talks)
|Examples and basic results on K3 surfaces
|The Kuga-Satake construction
|Weil conjectures for K3-surfaces
|A cristalline interlude
|The formal Brauer group and crystalline cohomology
|Tate conjecture for K3 surfaces of finite height I
|Tate conjecture for K3 surfaces of finite height II
|The Tate conjecture for supersingular K3 surfaces I
|The Tate conjecture for supersingular K3 surfaces II
|The Tate conjecture for supersingular K3 surfaces III
|The Tate conjecture and finiteness