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Introduction to elliptic curves and
Course overview
This course is meant to introduce the students to the notion of elliptic curves over different fields and present some of their main properties. The final goal of the course is to state the Birch--Swinnerton-Dyer conjecture, one of the seven Millennium problems.
Schedule
PART 1: P. STEVENHAGEN
27/01/2025, 17:00--20:00: Analytic introduction to Elliptic curves: elliptic curves, complex tori and lattices
28/01/2025, 17:00--20:00: Group operation, isogenies and j-invariant
29/01/2025, 17:00--20:00: 2-descent
30/01/2025, 17:00--20:00: Algebraic introduction to Elliptic curves
31/01/2025, 17:00--20:00: Mordell--Weil theorem
PART 2: D. FESTI
11/02/2025, 17:00--20:00: Computing the torsion part of an elliptic curve over the rationals
13/02/2025, 17:00--20:00: Explicit examples
18/02/2025, 17:00--20:00: L-function and BSD conjecture
20/02/2025, 17:00--20:00: 2-Selmer group of an elliptic curve over the rationals
PART 3: D. FESTI
17/06/2025, 11:00--13:00: Tate modules
18/06/2025, 09:00--11:00: Weil conjectures for elliptic curves
19/06/2025, 09:00--11:00: Computation of the rational points of E:y^2=x^3-7x-6
20/06/2025, 09:00--11:00: The m-descent procedure
EXAMS: F. RUSSO
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Notes and other resources
Official presentation of the course
Stevenhagen's notes on elliptic curves
Swinnerton-Dyer: an application of computing to class field theory (1967)
Cassels: Lectures on elliptic curves (1991). Chapters 12--15.
Festi's notes on elliptic curves
Russo's exercises on the Mordell--Weil group
Table of reduction types, Kodaira's classification (Silverman's 'Arithmetic of elliptic curves, 2009, Table C.15.1, p.448)
Biographical note on Trygve Nagell (by Cassels)
Biographical note on Beppo Levi (by Schoof and Schappacher)
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