Upper bounds for the growth of Mordell-Weil ranks in pro-p towers of Jacobians

TL;DR

This paper investigates how Mordell-Weil ranks of Jacobians behave in pro-p towers over number fields, establishing bounds related to the Jacobian's dimension, with specific results for Fermat curves and applications to modular Jacobians.

Contribution

It provides new bounds on Mordell-Weil ranks in pro-p towers, especially showing the rank can be controlled relative to the Jacobian's dimension, including the case of Fermat curves.

Findings

Mordell-Weil rank is bounded by a constant times the Jacobian's dimension.

For Fermat curves, the constant can be made arbitrarily close to 1.

Results are applicable to modular Jacobians J(Np^m).

Abstract

We study the variation of Mordell-Weil ranks in the Jacobians of curves in a pro-p tower over a fixed number field. In particular, we show that under mild conditions the Mordell-Weil rank of a Jacobian in the tower is bounded above by a constant multiple of its dimension. In the case of the tower of Fermat curves, we show that the constant can be taken arbitrarily close to 1. The main result is used in the forthcoming paper of Guillermo Mantilla-Soler on the Mordell-Weil rank of the modular Jacobian J(Np^m).