Short Courses in Number Theory

Ana Caraiani: A glimpse into the Langlands programme

20th May, 2024

The goal of this lecture series is to give you a glimpse into the Langlands programme, a central topic at the intersection of algebraic number theory, algebraic geometry and representation theory. In the first lecture, we will look at a celebrated instance of the Langlands correspondence, namely the modularity of elliptic curves. I will try to give you a sense of the different meanings of modularity and of the multitude of ingredients that go into establishing such a result. In the following lectures, I will focus on the more geometric ingredients, first in the special case of the modular curve and then for higher-dimensional Shimura varieties.

Charlotte Chan: Deligne-Lusztig theory: examples and applications

20th May, 2024

Geometry and representation theory are intertwined in deep and foundational ways. One of the most important instances of this relationship was uncovered in the 1970s by Deligne and Lusztig: the representation theory of matrix groups over finite fields is encoded in the geometry of a natural 'partition' of flag varieties. Recent developments have revealed rich connections between Deligne-Lusztig varieties and geometry studied in number-theoretic contexts. In this lecture series, we give an example-based tour of these ideas, focusing on how to extract concrete information from theory.

Peter Scholze: Real local Langlands as geometric Langlands on the twistor-ℙ1

4th March, 2024

In 2014, Fargues realized that one can formulate the local Langlands correspondence over p-adic fields as a geometric Langlands correspondence on the Fargues-Fontaine curve. This raises the question of a similar realization of the local Langlands correspondence over the real numbers. The goal of these lectures is to explain a possible formulation. As part of this, we will give a new perspective on the theory of variations of twistor structure, a generalization of the theory of variations of Hodge structure. This uses the theory of analytic stacks developed in our joint work with Clausen, of which we will give a brief overview.

Han Yu: Irrational rotations and number theory

19th November, 2020

An LMS online lecture course in number theory and dynamics.

The main goal of this mini-course is to illustrate a proof of Furstenberg's ×2,×3 theorem: The ×2,×3 orbit of any irrational number on the unit interval is dense. Key results that will be needed for the proof are topological properties of irrational rotation on the unit interval. We will discuss those results and provide detailed backgrounds as well as proofs. At the end of the course, I will introduce various results and problems on digit expansions of integers. The following topics will be covered:

   1.  Irrational rotations on torus;
   2.  Diophantine approximation: Dirichlet theorem, Roth's theorem, Baker's theory of linear forms of logarithms;
   3.  Furstenberg's ×2,×3 theorem;
   4.  Results and problems on digit expansions of integers;
   5.  Furstenberg's theorem on 2-dimensional torus (if time permits).

Note: For 2., I will mostly state the results without giving proofs as they are out of the scope of this mini-course.

Tiago Fonseca: A crash course on modular forms and cohomology

11th September, 2020

An LMS online lecture course in modular forms.

This is a geometrically flavoured introduction to the theory of modular forms. We will start with a standard introduction to some basic analytic aspects concerning modular forms and to their interpretation as sections of line bundles on modular curves.

Then, our main goal will be to explain how one can attach certain 2-dimensional cohomology groups to Hecke eigenforms. In this course, we will only deal with algebraic de Rham and Betti cohomology, but this can also serve to build geometric intuition on the l-adic setting, which gives rise to the famous l-adic representations attached to modular forms.

We will finish with a discussion on the Eichler-Shimura isomorphism, periods of modular forms, and, depending on time, Manin's theorem on the critical values of L-functions of modular forms.