We consider Z/pZ covers of curves in characteristic p and the p-torsion group schemes of their Jacobians.  Such covers are the building blocks of the towers Bryden Cais spoke about last week.  Our aim is to understand the full Dieudonne structure, not just the kernel of Frobenius.

 

We will give more details on groups schemes killed by p and their Dieudonne modules, introduce the Ekedahl-Oort stratification of Ag, the moduli space of abelian varieties, and state basic questions on how it interacts with the locus of Jacobians in Ag.  (This gives an introduction to the AWS2024 courses of Karemaker and Pries.)  We end by stating a structural result on Dieudonne modules of p-covers and some of the restrictions it places on group schemes.