Iwasawa Theory for $p$-torsion Class Group Schemes in Characteristic $p$

Abstract: A $\mathbb{Z}_p$ tower of curves in characteristic $p$ is a sequence $C_0, C_1, C_2, \ldots$ of smooth projective curves over a perfect field of characteristic $p$ such that $C_n$ is a branched cover of $C_{n-1}$ and $C_n$ is a branched Galois $\mathbb{Z}/(p^n)$-cover of $C_0$. The genus is a well-understood invariant of algebraic curves, and the genus of $C_n$ can be seen to depend on $n$ in a simple fashion. In characteristic $p$, there are additional curve invariants like the $a$-number which are poorly understood. They describe the group-scheme structure of the $p$-torsion of the Jacobian. I will discuss work with Bryden Cais studying these invariants and suggesting that their growth is also "regular" in $\mathbb{Z}_p$ towers. This is a new kind of Iwasawa theory for function fields.

algebraic geometrynumber theoryrepresentation theory

Audience: researchers in the discipline

( paper | slides )

VIASM Arithmetic Geometry Online Seminar