Basic linear algebra (vector spaces, linear maps, characteristic polynomial); group theory including the structure theorem for finitely generated abelian groups; ring theory: rings, ideals, polynomial rings; basic field theory including finite fields (but Galois theory is not essential). For example, chapters I-V of Lang's "Algebra" would be sufficient, or the following chapters of the Leiden undergraduate courses Algebra 1, 2, 3: 1–9, 11–14, 21, 22. The notes for the Leiden Algebra courses can be found here: http://websites.math.leidenuniv.nl/algebra/. We do not assume any prior knowledge of algebraic geometry.
For two weeks of the course we will also need a small amount of complex analysis: meromorphic functions, Cauchy's theorem, residues.
Aim of the course
Along various historical paths, the origins of elliptic curves can be traced back to calculus, complex analysis and algebraic geometry. Their arithmetic aspects have made elliptic curves into key objects in modern cryptography and in Wiles' proof of Fermat's last theorem. This course is an introduction to both the theoretical and the computational aspects of elliptic curves. The topics treated include a general discussion of elliptic curves and their group law, Diophantine equations in two variables, and Mordell's theorem. We will also discuss elliptic curves over finite fields and their applications.
Dr. Emma Brakkee (Leiden)
Dr. Martin Bright (Leiden)