On the Rasmussen-Tamagawa conjecture for abelian fivefolds

Abstract

In this paper, we study the Rasmussen-Tamagawa conjecture for abelian varieties with constrained prime power torsion. Previously, Rasmussen and Tamagawa have established the conjecture under the Generalized Riemann Hypothesis for abelian varieties of any dimension over any number field, and unconditionally for those over $\mathbb{Q}$ of dimension at most three. We prove several cases of the conjecture by giving partial refinements of their techniques. Among other things, we give an unconditional proof of the conjecture for abelian fivefolds over $\mathbb{Q}$.