A Base Change Version of Rasmussen-Tamagawa Conjecture

TL;DR

This paper proves a uniform version of the Shafarevich Conjecture and applies it to confirm the Rasmussen-Tamagawa Conjecture for certain abelian varieties over number fields with specific reduction properties.

Contribution

It introduces a uniform version of the Shafarevich Conjecture and verifies the Rasmussen-Tamagawa Conjecture for abelian varieties with potential good reduction everywhere.

Findings

Proved a uniform version of the Shafarevich Conjecture.

Confirmed the Rasmussen-Tamagawa Conjecture for a class of abelian varieties.

Established reduction properties over quadratic extensions.

Abstract

We prove a certain uniform version of the Shafarevich Conjecture. As a corollary, we prove the Rasmussen-Tamagawa Conjecture for a particular class of abelian varieties $A$ defined over a number $K$ of dimension $g$ having everywhere potential good reduction, in particular, for any finite place $v$ of $K$ the localization $A_v:=A\times_{\mathrm{Spec}(K)}\mathrm{Spec}(K_v)$ has either good reduction or {\it totally bad reduction} (connected component $\tilde{\mathcal{A}}_v^0$ of the special fibre $\tilde{\mathcal{A}}_v$ of the N\'eron model $\mathcal{A}_v$ at $v$ is an affine group scheme over the residue field $k_v$ at $v$) and has good reduction over a quadratic extension of $K_v$.