Heegner points and the original  Gross--Zagier formula

Mordell--Weil Theorem, Modularity of elliptic curves, Heegner points,
and (orginal) Gross--Zagier formula


 

I will introduce various examples of Shimura varieties, and explain what some important classes of their compactifications are like.  I will begin with complex coordinates, but when good theories of their integral models are available, I will also explain what they are like.  The lectures will be for people who are not already familiar with these topics---for most of them, some willingness to see matrices larger than 2x2 ones should suffice.  (I hope to allow simple factors of all possible types A, B, C, D, and E to show up if time permits.  Nevertheless, it is not necessary to know beforehand what this means.)

The lecture series will start with a  brief introduction to rigid analytic geometry. I will then introduce modular curves from various viewpoints (complex analytic, algebraic, and p-adic analytic) and use them to give a geometric definition of  p-adic and overconvergent modular forms and Hecke operators. I will next show how to use the p-adic geometry of the modular curves towards p-adic analytic continuation of overconvergent modular forms. Finally, I will demonstrate an application of these results to modularity of certain Galois representations which can itself be used to prove certain cases of the Artin conjecture. If time allows, I would explain briefly how these ideas extend to higher dimensions by illustrating the easier case of Hilbert modular surfaces.

Gan--Gross--Prasad conejcture 

Gan--Gross--Prasad conjeture for unitary groups, relative trace
formula and fundamental lemma, arithmetic analogues.

I will introduce various examples of Shimura varieties, and explain what some important classes of their compactifications are like.  I will begin with complex coordinates, but when good theories of their integral models are available, I will also explain what they are like.  The lectures will be for people who are not already familiar with these topics---for most of them, some willingness to see matrices larger than 2x2 ones should suffice.  (I hope to allow simple factors of all possible types A, B, C, D, and E to show up if time permits.  Nevertheless, it is not necessary to know beforehand what this means.)

These lectures will focus on the mod p representation theory of a split p-adic reductive group G, using  GL(2) as a running example. We hope to emphasize the differences between the mod p and complex representations of G while keeping in mind that the theory is partly motivated by the mod p and complex local Langlands programs.

We will start with remarks regarding finite reductive groups. We will then compare the homological properties of certain categories of mod p and complex representations of G (and the associated pro-p-Iwahori Hecke algebra). In particular, in the complex setting, the theory of coefficient systems on the Bruhat-Tits building by Schneider and Stuhler gives a way to construct explicit projective resolutions. We will explore what remains from this theory in the mod p setting. This will help us describe the first step in the construction of Colmez' functor yielding the mod p local Langlands correspondence for GL(2,Q_p).

We discuss some basic techniques for studying the arithmetic aspects of the cohomology of arithmetic groups. In contrast to the lectures of Speh, we shall focus on cohomology with coefficients over the integers or over finite fields rather than over the real numbers. 

 We give an introductory discussion of the theory of p-adic representations of the field of p-adic numbers.   Topics will include:  Tate's p-divisible groups paper, the field of norms, the formalism of period rings, and (phi,Gamma)-modules

This series of lectures, which build on Jared Weinstein's talks, will be a light introduction to the theory of phi-gamma modules and their interactions with p-adic Hodge theory. We will discuss Fontaine's equivalence of categories, give examples of phi-gamma modules and present Berger's fundamental results which link phi-gamma modules and p-adic Hodge theory. Depending on time, we may say a few words about the applications of phi-gamma modules to Galois cohomology and to the p-adic Langlands correspondence for GL_2(Q_p).