The simplest kind of algebraic curve that is not utterly trivial is a conic curve X² – AY² – BZ² = 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 + √AY) = 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.