The Cohen-Lenstra heuristic revisited; or: what does a random abelian group look like?

Alex Bartel (University of Warwick)

Wednesday 19th February, 2014 16:00-17:00 Maths 204


To each quadratic number field, one attaches an abelian group, called the ideal class group of the field. The study of these groups goes back to Gauss, but is also a central topic of algebraic number theory in the 20th century. The behaviour of these groups is still largely mysterious. The Cohen-Lenstra heuristic predicts their asymptotic behaviour, by postulating that ideal class groups behave like random abelian groups, the main achievement of the heuristic being to explain exactly what a "random abelian group" is supposed to be. I will explain what the Cohen-Lenstra heuristic says, why it has short-comings, and how to fix them. The number theoretic prerequisites in the talk will be minimal, but it will help if you remember what a Galois extension of Q is.

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