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These are notes for Céline Maistret's course MA842 at BU Spring 2019.

The course webpage is


  1. Elliptic curves and their ranks

    1. Background

      1. Mordell Weil theorem (state and prove) (ANT and cohomological proof)
      2. Non-effectivity
      3. Computing the rank (descent)
    2. The Birch and Swinnerton-Dyer conjecture

      1. Heuristic via counting points omn the reduced curve
      2. \(L\)-functions
      3. BSD-1
      4. Local arithmetic invariants and BSD-2
    3. Parity of ranks

      1. Isogeny invariants of BSD 2
      2. Galois representations and local root numbers
      3. The parity conjecture
  2. Abelian surfaces

    1. Background on genus 2 curves and their Jacobians
    2. BSD in this case
    3. Computability of local arithmetic invariants
    4. 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:

  1. Vladimir Dokchitser - Lecture course
  2. Silverman - Arithmetic of Elliptic Curves
  3. Milne - Abelian Varieties?