Type: Seminar

Distribution: World

Expiry: 25 Mar 2010

Auth: billu@daumier.maths.usyd.edu.au

Speaker: Steve Donnelly Title: Solving conics over number fields Time & Place: 3-4pm, Thursday 25 March, Carslaw 535 Abstract: The simplest kind of algebraic curve that is not utterly trivial is a conic curve X^2 - A*Y^2 - B*Z^2 = 0. Here A and B lie in Q or a number field, and we are interested in finding a solution (X:Y:Z) over that field. The Hasse principle holds, which means it’s easy to decide whether a solution exists; also, from a single solution it’s easy to find all solutions i.e. a parametrization. The algorithmic problem of finding a single solution has received only belated attention: over Q, the best known algorithm was given by Denis Simon in 2005; over number fields, the usual method is to solve the norm equation N(X + sqrt{A}*Y) = B, thus reducing the problem to a (sub-exponentially!) harder one. In this talk, I’ll describe techniques used in HasRationalPoint for conics over number fields, which I’ve developed gradually over the last few years.