#### Alladi Ramakrishnan Hall

#### Why is Serre’s problem difficult ?

#### L. Merel

##### University of Paris VII

*We will explain why (in our opinion) the difficulty to solve Serre’s problem stems*

from the same reason than the difficulty of Diophantus’ equation of lecture 1. For

that, we will use Chen’s theorem and we will make a detour through modular

jacobians, modular forms and the Birch and Swinnerton-Dyer conjecture.

Prerequisite : Basic algebraic number theory. The talk can be understood as a little

survey of the arithmetic of modular forms.

Done