Iwasawa theory of Frobenius-torsion class group schemes

For a prime p, let {C_n} be a Z_p-tower of smooth projective curves over a finite field of characteristic p. The p-part of the class group of the function field of C_n is known to depend on n in a regular way.  In characteristic p, the p-divisible group of the Jacobian of C_n is a geometric generalization of the p-part of class group.  I will discuss work with Bryden Cais, Joe Kramer-Miller, and James Upton which establishes a kind of Iwasawa theory for the Frobenius-torsion part of this "class group scheme".  In particular, invariants of the curves C_n like the a-number depend regularly on n.