250086 VO From Elliptic Curves to Fermat's Last Theorem (2024S)

Elliptic Curves represent a paradigma for the confluence of various mathematical theories and techniques: Algebraic Geometry, Group Theory, Complex Analysis, Number Theory, Algebraic Topology, and Differential Geometry, with numerous applications in Coding Theory and Cryptography.

The objective of the course is to illustrate this multi-faceted situation by comparing the different approaches: the study of equations, double periodic functions, tori, and Riemann surfaces of genus 1. Special interest will then be laid on number theoretic results, looking at Q-rational points on elliptic curves (Mordell's theorem), modularity (Shimura-Taniyama conjecture) and the relation to Fermat's last theorem, respectively, the ABC-conjecture. These last items can only be described roughly as the required methods would go far beyond the scope of the course. However, we very much hope to be able to give a reasonable account on the phenomena and results which have been observed.