Achter: The p-rank strata of the moduli space of curves

The p-rank strata of the moduli space of curves
Jeff Achter
The p-rank of a curve in positive characteristic is a discrete invariant which generalizes the distinction between ordinary and supersingular elliptic curves. This invariant induces a stratification of the moduli space of (hyperelliptic) curves.

I'll discuss recent results on the geometry of these strata, with special attention to their structure at the boundary of the moduli space. This information yields new applications about the prime-to-p part of the class group of a quadratic function field with specified geometric p-rank; the existence of absolutely simple hyperelliptic Jacobians of every p-rank; and the stratification of the moduli space of curves by Newton polygon. In spite of the apparent technicality of the subject, I intend to try and draw a lot of pictures.

(Joint work with Rachel Pries)