Title of the course: Elliptic curves
Instructor: Dr. Dino Festi
Institution: Università degli Studi di Milano
Dates: 25-30 July 2022
Prerequisites: Group, ring  and field theory, Galois theory, basics of complex analysis, basics of algebraic geometry.
Level: Advanced undergraduate, master student
Abstract: In this course we introduce the notion of elliptic curves. These geometric objects are fascinating for their extremely rich algebraic and geometric structure, and during the last decades they have gained a global interest thanks to their application to cryptography.
We will present elliptic curves as double covers of the projective line and we will characterize them by means of the $j$-function, thus presenting the first example of a modular curve.
We will focus on elliptic curves over the field of rational numbers, introducing their $L$-series.
We will then state the modularity theorem and the Birch–Swinnerton-Dyer conjecture.
For a more thorough enjoyment, we suggest the participants also attend Veniani’s course on modular forms.
Language: EN
Textbook: Silverman’s “The arithmetic of elliptic curves” and my own notes, which will be available online. The plan is to roughly cover sections 10, 2, 3, 6, 7, in this order, of the notes attached.