Tim Dokchitser

Counting reduction types

For curves over Q, one could ask how they behave when the equation is reduced mod p, and classify the possible behaviour. For elliptic curves there are 10 possibilities, called Kodaira types (or Kodaira families), and I will explain how to extend this classification to arbitrary genus. In every genus g there are finitely possible reduction families, and I will try to say something about how many there are. This is a purely combinatorial question that leads to inequalities, upper bounds, rigid caterpillars and other fascinating objects.