= 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 [[attachment: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
[[attachment: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, #9410, and #9417 need to be addressed.

For testing against Magma, one can use [[attachment:magma.sage]].

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

''UPDATE:'' A (hopefully) working implementation is attached to ticket #9320 now.

== Parity Predictions ==
A brief summary of the project is available as [[attachment:project.pdf]].

The worksheet I used for my presentation is located at http://www.sagenb.org/home/pub/2234