Tim Dokchitser (Cambridge University): Complex L-functions and the Birch and Swinnerton-Dyer conjecture

Structure of the course

• Quick review of Elliptic curves over Q and the Mordell-Weil theorem
• Elliptic curves over finite fields, heuristics for their distribution and the naive version of BSD
• L-functions of elliptic curves and the BSD-conjecture
• Root numbers and how to compute them
• Parity predictions, Goldfeld's conjecture and ranks of elliptic curves over number fields

Prerequisites

Some familiarity with basic algebraic number theory (number fields, primes), and having seen elliptic curves

Background reading

J. H. Silverman, "The arithmetic of elliptic curves", Chapters 3, 7 and 8.

Sage Reference Manual on elliptic curves: http://sagemath.org/doc/reference/plane_curves.html, up to `Isogenies'.

Computational projects

There will be many small problems and larger assignments to play with, illustrating all the concepts and conjectures from the course.

• A. Root Numbers over K for elliptic curves (implement)
  • People: Armin, Charlie, Hatice, Christ, Lola, Robert Miller, Thilina, M. Tip, Robert Bradshaw

• B. #III(E/K)_{an} function (L-functions, connection to Wuthrich)

  • People: Berinder, M. Tip, Adam, Robert Miller, Robert Bradshaw, Chris Wuthrich
• C. Parity Predictions
  • People: Arijit, Anil, Adam