On the Rasmussen-Tamagawa conjecture for abelian fivefolds
Shun Ishii
TL;DR
The paper proves RT$(\mathbb{Q},5)$ unconditionally by refining Rasmussen–Tamagawa’s framework, restricting the possible invariants that can occur for abelian fivefolds and exploiting both small quadratic nonresidues and explicit quadratic residues to force contradictions for large primes $\ell$. The approach tightens the analysis of the $\ell$-adic Galois action on $A[\ell]$ and the structure of inertial and Frobenius actions via the invariants $m_{\mathbb{Q}}$ and $e_{\lambda}$, along with the special-fiber decompositions encoded by $B_d$ and $g_d$. Together, these mean that no abelian fivefold over $\mathbb{Q}$ can have the constrained torsion field $K(A[\ell^{\infty}])$ be a pro-\ell-extension of $\mathbb{Q}(\zeta_{\ell})$ for all sufficiently large \ell. The results extend the unconditional reach of RT beyond previously known cases (g ≤ 3) and illustrate new techniques that may impact related uniform and non-abelian Galois-representation problems. The paper also documents concrete nonempty instances for small \ell, illustrating the boundary between existence and nonexistence for the conjecture in dimension five.
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 $\mathbb{Q}$.