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)
Abstract:
Methods for finding rational points on algebraic curves and
higherdimensional varieties based on latticereduction first came to
attention through Elkies ANTS IV article (2000), which was based on real
approximations. This was followed by a padic method, often referred to
as "padic Elkies", which seems to have been thought up independently by
several people, including HeathBrown 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 4descent 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
LLLreduction of Zlattices by an appropriate reduction theory for
Fq[T]lattices.



