Course: Eigenvarieties and Bloch–Kato Selmer Groups

I) The first topic (animated by Dimitrov) is on the geometry of Eigenvarieties at classical points. The smoothness at such points is a crucial input in the proof of many cases of the Bloch--Kato Conjecture, the Iwasawa Main Conjecture and Perrin-Riou’s Conjectures. Far more fascinating is the study of the geometry at singular points, especially at the intersection between irreducible components of the eigenvariety, as such classical points are related to trivial zeros of adjoint p-adic L-functions. In this series of lectures, Dimitrov will sketch the construction of the eigencurve, and provide the necessary p-adic background to understand how to apply such p-adic techniques in the study of the Bloch–Kato—Selmer Groups.

II) The second topic is on the ''twisted p-adic BSD conjecture'' and will be animated by Seveso:
Kolyvagin introduced in his study of Heegner points the notion of an Euler system, a collection of compatible elements of Galois cohomology groups, in the aim to give bounds of Selmer groups, and also to show the finiteness of some Tate—Shafarevich groups. Using p-adic techniques such method has been extended to study cases of the twist of the BSD conjecture. In this series of lectures, Seveso will provide the necessary p-adic background to understand how to apply such p-adic techniques in certain cases.

III) The third topic is an introduction to p-adic Hodge theory (animated by Benois). p-adic Hodge theory provides a way to study p-adic Galois representations of characteristic 0 local fields with residual characteristic p, and also provides faithful functors to categories of linear algebraic objects that are easier to study.

In this series of lectures, Benois will introduce the so-called Fontaine period rings such as B_dR, B_st, B_cris, and the formalism of B-admissible representations introduced by Fontaine. Moreover, he will provide the necessary p-adic background and p-adic techniques needed in the first and second topic.

Joël Bellaïche: The Eigenbook: Eigenvarieties, families of Galois representations, p-adic L-functions.
Denis Benois: An introduction to p-adic Hodge theory

We kindly ask the participants to arrange their own accommodation.

We recommend Hotel 9 Små Hjem, which is pleasant and inexpensive and offers rooms with a kitchen. Other inexpensive alternatives are CabInn, which has several locations in Copenhagen: the Hotel City (close to Tivoli), Hotel Scandinavia (Frederiksberg, close to the lakes), and Hotel Express (Frederiksberg) are the most convenient locations; the latter two are 2.5-3 km from the math department. Somewhat more expensive – and still recommended – options are Hotel Nora and  Ibsen's Hotel.

An additional option is to combine a stay at the CabInn Metro Hotel with a pass for Copenhagen public transportation (efficient and reliable). See information about tickets & prices.