p-adic Gross-Zagier formula

The p-adic Gross-Zagier formula relates the derivative of a p-adic L-function to the p-adic height of an arithmetic cycle. Currently, several types of p-adic Gross-Zagier formulas exist in various settings; however, this lecture focuses on the most basic p-adic Gross-Zagier formula proved by Perrin-Riou. Many important ideas useful for subsequent generalizations are already contained in its proof.

p-adic L-functions

The course will provide an introduction to p-adic L-functions, which are p-adic avatars of complex L-functions. They arise from p-adic congruences among special values of complex L-functions. We will illustrate the constructions and applications of p-adic L-functions through some examples (Kubota--Leopoldt p-adic L-functions, Rankin-Selberg p-adic L-functions, symmetric square p-adic L-functions, p-adic L-functions for GSp_4×GL_2).

Kato's Euler systems and their applications

The goal of this lecture series is to explain the notion of Kato's Euler systems, the methodology of Kolyvagin systems, and how they are used in the arithmetic of elliptic curves.

Some numerical examples will be computed in the end.

Modular construction of Selmer classes

Congruence between modular forms is one of the most effective methods to construct Selmer classes. 

In these lectures, I will explain examples from Eisenstein congruence on GL(2) and endoscopic congruence on GSp(4).