Ranks of Jacobians in towers of function fields

TL;DR

This paper combines Ulmer's rank formula and Zarhin's bounds to construct and analyze higher-dimensional, absolutely simple Jacobians over rational function fields with bounded ranks across towers.

Contribution

It introduces new examples of simple Jacobians with bounded ranks in towers of function fields, utilizing combined theoretical tools for explicit rank computation.

Findings

Constructed Jacobians with bounded ranks in towers

Computed ranks at every layer of the tower in many cases

Provided examples of absolutely simple Jacobians over $k(t)$

Abstract

Let $k$ be a field of characteristic zero and let $K=k(t)$ be the rational function field over $k$. In this paper we combine a formula of Ulmer for ranks of certain Jacobians over $K$ with strong upper bounds on endomorphisms of Jacobians due to Zarhin to give many examples of higher dimensional, absolutely simple Jacobians over $k(t)$ with bounded rank in towers $k(t^{1/p^r})$. In many cases we are able to compute the rank at every layer of the tower.