[Paper Review] Essential dimension of abelian varieties over number fields

Patrick Brosnan, Ramesh Sreekantan|ArXiv.org|Feb 29, 2008

Algebraic Geometry and Number Theory3 citations

TL;DR

This paper proves that the essential dimension of any non-trivial abelian variety over a number field is infinite, resolving a conjecture for elliptic curves and extending it to all abelian varieties. The proof uses Galois representations on Tate modules and Bogomolov's theorem to show that certain $ε$-adic structures force infinite essential dimension via the essential dimension of $Π/\ell^n$-torsors.

ABSTRACT

We affirmatively answer a conjecture in the preprint ``Essential dimension and algebraic stacks,'' proving that the essential dimension of an abelian variety over a number field is infinite.

Motivation & Objective

• To resolve Conjecture 10.5 of [3], which posits that the essential dimension of any elliptic curve over a number field is infinite.
• To extend this result to all non-trivial abelian varieties of positive dimension over number fields.
• To establish a general criterion for infinite essential dimension based on the presence of $Π/\ell^n$-torsion and finite but nontrivial $μ_{\ell^\infty}$-torsion in field extensions.
• To provide a shorter, self-contained proof than alternative approaches using Karpenko-Merkurjev theory, while highlighting the independent interest of Theorem 3.

Proposed method

• Use of the essential dimension of the group scheme $\mathbb{Z}/\ell^n$ over field extensions, as established by Florence, to bound the essential dimension of abelian varieties from below.
• Application of Bogomolov's theorem on the Lie algebra of the Galois image to detect open subgroups in the Galois representation on $T_\ell A$.
• Construction of a Galois extension $L/k$ such that $\mathbb{Q}_\ell/\mathbb{Z}_\ell \subset A(L)$ and $1 < |\mu_{\ell^\infty}(L)| < \infty$, using Frobenius tori and Tchebotarev density.
• Leveraging the fact that $T_\ell A$ carries a Galois action and that the Frobenius torus $T_\mathfrak{p}$ at a good, non-supersingular prime has rank $\geq 2$, to ensure nontrivial torus actions.
• Use of linear algebra on the character lattice of split tori to find a rank-1 subtorus fixing a vector and acting nontrivially on the determinant.
• Reduction of the essential dimension problem to the behavior of $\mu_{\ell^\infty}(L)$ and the image of the Galois group in $\mathrm{GL}(V_\ell A)$.

Experimental results

Research questions

• RQ1 Does every non-trivial abelian variety over a number field have infinite essential dimension?
• RQ2 Can the essential dimension of an abelian variety be infinite even when its $j$-invariant is integral, thus excluding the Tate curve method?
• RQ3 What structural conditions on the Galois representation on $T_\ell A$ imply infinite essential dimension?
• RQ4 Is there a uniform criterion for infinite essential dimension based on the presence of $\mathbb{Q}_\ell/\mathbb{Z}_\ell$-torsion and finite $\mu_{\ell^\infty}$-torsion in a field extension?
• RQ5 Can the essential dimension of $A$ be bounded below by the essential dimension of $\mathbb{Z}/\ell^n$-torsors when $A$ contains such torsors?

Key findings

• The essential dimension of any non-trivial abelian variety over a number field is infinite.
• For any non-trivial abelian variety $A$ over a number field $k$, there exists an odd prime $\ell$ and a field extension $L/k$ such that $\mathbb{Q}_\ell/\mathbb{Z}_\ell \subset A(L)$ and $1 < |\mu_{\ell^\infty}(L)| < \infty$, which implies infinite essential dimension.
• The proof relies on the existence of a Frobenius torus of rank at least 2 at a good reduction prime where the reduction is not supersingular.
• The Galois representation on $T_\ell A$ contains a split torus of rank $\geq 2$ over $\mathbb{Q}_\ell$ for a positive density of primes $\ell$, enabling the construction of the required subtorus.
• The essential dimension of $\mathbb{Z}/\ell^n$ over a field $L$ with $|\mu_{\ell^\infty}(L)| = \ell^r$ grows as $\max\{1, \ell^{n-r}\}$, which tends to infinity as $n \to \infty$, thus forcing $\mathrm{ed}(A) = \infty$.
• The result holds even when the $j$-invariant is integral, thus resolving the conjecture beyond the scope of Tate curve methods.