The London-Paris Number Theory Seminar meets twice per year, once in London and once in Paris. It is supported by a grant from the London Mathematical Society, and ANR Projet ArShiFo ANR-BLAN-0114.

London organizers: David Burns, Kevin Buzzard Fred Diamond, Yiannis Petridis, Alexei Skorobogatov, Andrei Yafaev.

Paris organizers: Pierre Charollois, David Harari, Michael Harris, Marc Hindry, Benjamin Schraen, Jacques Tilouine.

The schedule:

0900 Welcome and coffee in room 104

0930 Pierre Parent (Bordeaux): Heights on elliptic and modular curves.

1100 Sir Peter Swinnerton-Dyer (Cambridge): The effect of twisting on the 2-Selmer group.

1200--1400: lunch at L'Ardoise

1400 Tom Fisher (Cambridge): Visibility of 4-coverings of elliptic curves

1500 Coffee

1530 Jim Stankewicz (Bristol): Torsion points on CM elliptic curves over prime degree fields

Abstracts:

Parent: The determination of Galois representations mod p defined by elliptic curves over number fields can be translated in terms of rational points on modular curves. Among the most efficient tools to study rational points on curves in general are those provided by height approaches, as was illustrated by the proofs of Mordell's conjecture by Faltings and Vojta. But unfortunately these powerful methods are often non-effective, a key issue if one is interested not only in the finiteness but also in the mere existence of rational points. Modular curves are however quite special in geometric and arithmetic aspects, and that sometimes makes the question of effectivity easier. We will discuss this point of view through various recent and less recent applications of diophantine methods.

Swinnerton-Dyer: Let E be an elliptic curve whose 2-division points are rational, and let E_b be its quadratic twist by b. For any fixed E we study the distribution of the order of the 2-Selmer group of E_b as b varies. Doing this involves a sophisticated analysis of the process of 2-descent, which is joint work with Alexei Skorobogatov. If there is time, I shall also say something about the case when not all the 2-division points are rational.

Fisher: A smooth plane cubic C may naturally be viewed as a 3-covering of its Jacobian, with fibre above 0 the set of points of inflection on C. A family of 3-coverings, with constant fibre above 0, may be constructed by taking linear combinations of the equation for C and its Hessian. I will describe an analogous construction for 4-coverings of elliptic curves, and explain how this helps with the study of visibility of Tate-Shafarevich groups. This is joint work with Nils Bruin.

Stankewicz: In this talk, we answer the following question of Schuett: Let p be a prime, F a degree p number field, and E an elliptic curve over F with CM. As p grows, how does the torsion in E(F) grow? We find an answer to this question via a connection between real cyclotomic fields and torsion on CM elliptic curves. This is joint work with A. Bourdon and P. Clark.

Previous few meetings:

Eleventh meeting (Paris, 21/11/2011)

Twelfth meeting (London, 30/05/2012)

13th meeting (Paris, 22/10/2012)

14th meeting (London, 3,4/6/2013)

15th meeting (Paris, 18/11/13)

16th meeting (London, 9/6/13)

This page is maintained by Kevin Buzzard.