Workshop

This talk is intended as an introduction to the introductory workshop. It will introduce some of the main players -- special cohomology classes, Euler systems, p-adic L-functions -- largely by focusing on the special case of cyclotomic units and Dirichlet characters.

I will give an introduction to the theory of special algebraic cycles, emphasizing their relation to arithmetic theta functions. The parallels between the number field (Shimura variety) and function field (moduli of shtukas) situations will be highlighted, and if time permits I will mention some new perspectives from derived algebraic geometry.

The Langlands program intersects the theory of motives and algebraic cycles in many different ways. I will talk about some of these, giving a broad overview of the subject, including some recent developments and open problems.

Heegner points were introduced in the ‘70s by Birch, inspired by earlier work of Heegner. They are points on rational elliptic curves defined over ring class fields of imaginary quadratic fields and they are a key ingredient to prove much of what is nowadays known about the Birch and Swinnerton-Dyer conjecture. In these lectures, we will recall their construction and the main results involving them. In particular, we will focus on Kolyvagin’s work, which provides a bound on Selmer groups of elliptic curves. If time permits we will discuss Iwasawa theoretic analogues of this statement (leading to one divisibility in Perrin-Riou’s main conjecture) and possible generalisations of the theory of Heegner points.

Perrin-Riou's big logarithm map plays an important role in Iwasawa theory. It sends systems of local Galois cohomology clases to p-adic analytic functions. In particular, it allows us to convert an Euler systems to a p-adic L-function, which is often the first step towards studying Iwasawa main conjectures. I will first review the original construction of this map developed by Perrin-Riou and its generalizations. I will then discuss how this map is used in studying Iwasawa theory in several concrete settings.

This talk is intended as an introduction to the introductory workshop. It will introduce some of the main players -- special cohomology classes, Euler systems, p-adic L-functions -- largely by focusing on the special case of cyclotomic units and Dirichlet characters.

I will give an introduction to p-adic L-functions. I will explain the conjecture by Coates and Perrin-Riou on the existence of p-adic L-functions and their definition of the modified Euler factors at p for p-adic interpolations. I will also talk about some approaches for constructing p-adic L-functions with some examples.

Euler systems are one of the most powerful tools for proving new cases of the Birch--Swinnerton-Dyer and Bloch--Kato conjectures. In my lectures, I will give an introduction to the theory of Euler systems, and give some examples of how to construct and where to find them.

Rigid meromorphic cocycles are classes in the first cohomology of SL_2(Z[1/p]) acting on the non-zero rigid meromorphic functions on the Drinfeld p-adic upper half plane. Their values at real quadratic points are conjectured to be analogues of singular moduli for real quadratic fields. In this talk, I will discuss the relation between real quadratic singular moduli and derivatives of p-adic families of modular forms. These results can be fit into an emerging p-adic Kudla program.