Infinite Rank of Abelian Varieties over $\mathbb Q^{\mathrm{ab}}$

Definitions and mathematical statement

Fix an algebraic closure $\overline{{\mathbb{Q}}}$ of ${\mathbb{Q}}$. The maximal abelian extension ${\mathbb{Q}}^{\mathrm{ab}}\subseteq\overline{{\mathbb{Q}}}$ is the compositum of all finite Galois extensions of ${\mathbb{Q}}$ in $\overline{{\mathbb{Q}}}$ with abelian Galois group. A nonzero abelian variety over ${\mathbb{Q}}$ means an abelian variety $A/{\mathbb{Q}}$ with $\dim A>0$. For an extension $K/{\mathbb{Q}}$, let $A(K)$ denote its abelian group of $K$-rational points, and define \[ \operatorname{rank} A(K)=\dim_{{\mathbb{Q}}}\left(A(K)\otimes_{{\mathbb{Z}}}{\mathbb{Q}}\right). \] This definition does not require $A(K)$ to be finitely generated.

The supplied question is whether \[ \text{for every abelian variety }A/{\mathbb{Q}}\text{ with }\dim A>0,\qquad \operatorname{rank} A({\mathbb{Q}}^{\mathrm{ab}})=\infty. \] Equivalently, for every such $A$ and every integer $n\geq 1$, there exist $P_1,\ldots,P_n\in A({\mathbb{Q}}^{\mathrm{ab}})$ whose images in $A({\mathbb{Q}}^{\mathrm{ab}})\otimes_{{\mathbb{Z}}}{\mathbb{Q}}$ are linearly independent over ${\mathbb{Q}}$.

The supplied record labels this question open. Its background and literature assessment instead discuss whether ${\mathbb{Q}}^{\mathrm{ab}}$ is ample. The record does not establish an equivalence between that field property and the infinite-rank assertion above; consequently, that assessment does not by itself verify the status of the stated question.

Short English statement

Does every positive-dimensional abelian variety defined over ${\mathbb{Q}}$ have infinitely many independent rational points over ${\mathbb{Q}}^{\mathrm{ab}}$, with independence measured after tensoring its group of points with ${\mathbb{Q}}$? The supplied open-status assessment concerns ampleness and requires review for its relevance to this precise question.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30001147$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The original, clean, and displayed statements all ask the infinite-rank question formulated above.
• Supplied report attribution: The Arithmetic of Fields (2009), Oberwolfach Reports, pp. 285--354, DOI 10.4171/owr/2009/05. https://doi.org/10.4171/owr/2009/05. The record gives proposed year 2009 and names no individual proposer.
• Supplied literature source: Bary-Soroker--Fehm survey. https://arxiv.org/abs/1106.1310. The supplied assessment describes this as a 2011 survey listing the ampleness question as open and reports only indirect evidence.
• Literature assessment dated exactly 2026-08-21: the record labels the status open and states that whether ${\mathbb{Q}}^{\mathrm{ab}}$ is ample remains open. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement-status label is corrected_verified, with a description of meaning-preserving semantic, typographic, and LaTeX cleanup and no unsupported source-dependent additions. This is a registry label, not independent mathematical or source verification here.