Number Theory Seminar

Tuesday 7 November 2006, MR13, 2:30pm

Lattice reduction over function fields, with applications to finding rational points on curves
John Cremona (Nottingham)


     Methods for finding rational points on algebraic curves and higher-dimensional varieties based on lattice-reduction first came to attention through Elkies ANTS IV article (2000), which was based on real approximations. This was followed by a p-adic method, often referred to as "p-adic Elkies", which seems to have been thought up independently by several people, including Heath-Brown and Elkies. This method is easy to describe and implement and has been used very successfully, for example, in finding rational points on quadric intersections in P^3 (which is useful for 2- and 4-descent on elliptic curves). I will report on joint work with Nottingham student David Roberts showing that a similar method may also be applied to curves defined over F_q(T), replacing LLL-reduction of Z-lattices by an appropriate reduction theory for Fq[T]-lattices.

Last modified on 24/10/06.

Comments/corrections to Tim Dokchitser