Uniform bounds of base change conductors and link with the generalized Szpiro conjecture

TL;DR

This paper establishes uniform bounds for base change conductors of abelian varieties, linking them to the generalized Szpiro conjecture and simplifying its proof by reducing to the semi-stable case.

Contribution

It provides a uniform bound for the base change conductor based on the dimension and degree, connecting height differences to the conjecture.

Findings

Bound depends on abelian variety dimension and number field degree

Reduction of the generalized Szpiro conjecture to semi-stable cases

Facilitates progress towards proving the conjecture

Abstract

The difference between the Faltings height of an abelian variety $A$ defined over a number field $k$ and its stable height is measured by the so-called base change conductor. In this paper, we give a uniform bound of the base change conductor in terms of the dimension of $A$ and the degree of $k$. This allows us to reduce the generalized Szpiro conjecture to the semi-stable case.