Appears in collection : Rational points on irrational varieties

When given a plane curve over Q, it is usually desirable (for computational purposes, for example) to have an equation for it with integral coefficients that is ‘small’ in a suitable sense. There are two aspects to this. One is to find an integral model with invariants that are as small as possible, or equivalently, a model that has the best possible reduction properties modulo primes. This is called minimization. The other, called reduction, is to find a unimodular transformation of the coordinates that makes the coefficients small. I will present a new algorithm that performs minimization for plane curves of any degree (this is joint work with Stephan Elsen- hans) and also explain how one can perform reduction (based on some older work of mine)