Faculty of Science

Eigenvarieties and Bloch–Kato Selmer Groups
Provider: Faculty of Science

Activity no.: 5584-24-07-31
	Enrollment deadline: 30/04/2024

Place

Department of Mathematical Sciences

Date and time

27.05.2024, at: 09:00 - 31.05.2024, at: 16:00

Regular seats

ECTS credits

2.50

Contact person

Nina Weisse E-mail address: weisse@math.ku.dk

Enrolment Handling/Course Organiser

Ian Kiming E-mail address: kiming@math.ku.dk

Teaching language

English partially in English

Semester/Block

Spring

Exam details

Formative assessment.

Course workload

Course workload category	Hours
Lectures	20.00
Preparation	40.00
Theory exercises	10.00

Sum	70.00

Content

This series of lectures involves three interconnected topics in arithmetic geometry. The purpose is to study cases of the twist of the BSD conjecture and the Bloch-Kato conjecture.

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.

Learning outcome

Knowledge:
• Must possess knowledge of the practice and central applied theories and methodologies of the profession and the subject area.
• Basic knowledge on p-adic Hodge theory, Bloch–Kato—Selmer Groups and p-adic Galois representations associated to automorphic forms.
• Basic knowledge on Shimura varieties and p-adic L-functions.
• Basic knowledge on Hida and Coleman theory.

Skills:
• Must be able to apply the central methodologies and tools of the subject area as well as be able to apply the skills related to work in the profession.
• Must be able to apply techniques in p-adic Hodge theory to study some problems in arithmetic geometry.
• Must be able to use Coleman theory to study some problems in Iwasawa theory.

Competences:
• Must be able to acquire new knowledge, skills and competences related to the profession in a structured context.
• Must be able to handle development-oriented situations.

Literature

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

Target group

Master and PhD students in arithmetic and algebraic Geometry, and algebraic Number Theory.

Teaching and learning methods

We will have 20h of theoretical lectures distributed between the speakers.
We will have 10h of theoretical exercices (2h each day).

Lecturers

I) Mladen Dimitrov (University of Lille): Course on the geometry of Eigenvarieties at classical points.
II) Marco Adamo Seveso (Università degli Studi di Milan): Course on the twisted p-adic BSD conjecture.
III) Denis Benois (University of Bordeaux): Introduction to p-adic Hodge theory.

Remarks

Course price: 120 €

Registraton and more information at the course webpage: www.math.ku.dk/english/calendar/events/eigenvarieties/

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