Elliptic Curves

The aim of this course is to discover the beautiful theory of elliptic curves. Elliptic curves are objects at the crossroads between geometry, analysis, algebra and number theory. They constitute one of the key ingredient in the proof of Fermat’s Last Theorem for instance, and famous open conjectures -for example the Birch and Swinnerton-Dyer conjecture- focus on these special curves. Studying compact Riemann surfaces, lattice theory and periodic functions, rational points and diophantine problems, projective and affine geometry of curves, schemes, higher Galois theory, modular forms and L functions, abelian varieties, local fields, global fields, finite fields, modern cryptography, each time these curves show up at a central place. 

As these objects really appear as a corner stone in the modern mathematical landscape, we offer a course presenting in details their various definitions and basic properties and focus on some modern applications. At times, properties from algebraic geometry, from complex analysis, from algebra, from number theory, are referred to.