Computational Algebra Seminar: Donnelly -- Solving conics over number fields

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.