# 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
- 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.

- -The main component was producing a method to obtain the coefficients of
- 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.

- -During the course of (1) this was discovered. Apparently, .count_points()

- Extended the .dokchitser attribute from rationals to general number fields.

## 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, #9410, and #9417 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.

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

## Parity Predictions

A brief summary of the project is available as project.pdf.

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