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

The course webpage is https://sites.google.com/view/cmaistret/teaching#h.p_BYGoPzU848FJ.

Outline

  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?
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