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: Barinder, M. Tip, Adam, Robert Miller, Robert Bradshaw, Chris Wuthrich

C. Parity Predictions

• People: Arijit, Anil, Adam

Computing root numbers project: notes

Main background reference: Silverman I Chapter VII (does not mention root numbers but gives background information to compute them) - reduction types of elliptic curves etc; plus Section of Silverman I Appendix C on Tate's algorithm.

Elliptic Curves over Q

root number w = \prod_p w_p * w_\infty

p is a prime of good reduction iff p does not divide the discriminant. If E has good reduction at p then w_p = +1.

p \mid \mid N means p is a prime of multiplicative reduction If E has split multiplicative reduction then w_p = -1 If E has non-split multiplicative reduction then w_p = +1

Step 1: implement w for E/Q with N square-free (already done in GP)

If p^2\mid N then p is a prime of additive reduction for E and w_p is more complicated. There are formulae to compute them, they rely on Tate's algorithm.

Elliptic Curves over general number fields

Root number classification

w = \prod_p w_p \prod_{v\div \infty} (-1)

Additive reduction w_p for p not dividing 2,3 has been done by Rohrlich (somewhat difficult) see [1] Theorem 2 which is self-contained

for p \mid 3 this has been done Kobayashi [2]

for p \mid 2 T & V Dokchitser [3] the formulae are really hard - ignore this (!!)

Would perhaps be better to do for any p, or even determine w globally (T & V Dokchitser: [4] in the introduction) There is a decision to be made as to which methods to use.

[1] D. Rohrlich, Galois Theory, elliptic curves, and root numbers, Compos. Math. 100 (1996), 311--349.

[2] S. Kobayashi, The local root number of elliptic curves with wild ramification, Math. Ann. 323 (2002), 609--623; available online

[3] http://arxiv.org/abs/math/0612054

[4] http://arxiv.org/abs/0906.1815

Working groups

Implementation:

• (Local) Armin, Charlie, Chris
• (Global) Lola, Chris, Hatice, Charlie

Reading (Kobayashi)

• Lola, AJ, Thilina, MTip

L-functions of elliptic curves over number fields

• Adam
  1. Extended the .dokchitser attribute from rationals to general number fields.
     • -The main component was producing a method to obtain the coefficients of
       • of the Dirichlet expansion of the L-series.
       -Patch has been submitted: trac #9402
  2. Discovered a bug in .count_points() attribute for elliptic curves
     • -During the course of (1) this was discovered. Apparently, .count_points()
       • caches its answer, and running a loop over different residue fields and elliptic curves results in failure.
       -No patch yet; but notified devel team: trac #9409

Implementation of root numbers

A very preliminary implementation of root numbers over number fields is attached as root_number.sage. The case of primes dividing 2 certainly has bugs at the moment. An updated version will be uploaded to ticket #9320. Also note that the implementation needs the patches #9334 (Hilbert symbol) as well as dirty_model.patch to be applied (the latter needs to be improved and will be posted to trac soon).

Finally, to work correctly the tickets #9389 and #9410 need to be addressed.

For testing against Magma, one can use magma.sage.

Fridays presentations are available as goodcop.sws and badcop.sws. The first of these also needs demo.sage.