These are notes for Céline Maistret's course MA842 at BU Spring 2019.
The course webpage is
Elliptic curves and their ranks
- Mordell Weil theorem (state and prove) (ANT and cohomological proof)
- Computing the rank (descent)
The Birch and Swinnerton-Dyer conjecture
- Heuristic via counting points omn the reduced curve
- Local arithmetic invariants and BSD-2
Parity of ranks
- Isogeny invariants of BSD 2
- Galois representations and local root numbers
- The parity conjecture
- Background on genus 2 curves and their Jacobians
- BSD in this case
- Computability of local arithmetic invariants
- Parity conjecture
Evaluation, none, when not around will give exercise/project, if you come regularly and do a computation you pass.
Main references that we will be following:
- Vladimir Dokchitser - Lecture course
- Silverman - Arithmetic of Elliptic Curves
- Milne - Abelian Varieties?