Images of Infinite Galois Representations

Definitions and mathematical statement

Let $\overline{{\mathbb{Q}}}$ be an algebraic closure of ${\mathbb{Q}}$, and let $G_{{\mathbb{Q}}}=\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})$ be the absolute Galois group, equipped with its Krull topology. The source uses a coefficient ring $A$ and the group $\operatorname{GL}_2(A)$ of invertible two-by-two matrices over $A$, but does not specify the class of coefficient rings or the topology on the target.

Subject to these unresolved coefficient and topology conventions, consider a continuous representation \[ \rho:G_{{\mathbb{Q}}}\longrightarrow\operatorname{GL}_2(A) \] which is irreducible, is not induced from a character of any index-two subgroup of $G_{{\mathbb{Q}}}$, and has infinite image. Define \[ \operatorname{Im}\rho=\{\rho(g):g\in G_{{\mathbb{Q}}}\}\subseteq\operatorname{GL}_2(A), \qquad |\operatorname{Im}\rho|=\infty. \] The requested problem is to determine the structure of $\operatorname{Im}\rho$ for representations satisfying these hypotheses.

Unresolved qualifications. This is a classification question whose precise scope is not supplied. The source does not define irreducibility over $A$, specify whether irreducibility is absolute or is tested after passage to a field, or specify the coefficient field or ring of the inducing characters. It also does not state whether the index-two subgroups under consideration must be open or whether their characters must be continuous. No such additional restrictions are imposed here. The intended meaning of structure---for example, abstract group structure, topological group structure, or classification as a subgroup of $\operatorname{GL}_2(A)$---is likewise unspecified. These omissions prevent a fully determined mathematical formulation without further source clarification.

Short English statement

Describe the image of an infinite, irreducible two-dimensional representation of the absolute Galois group of ${\mathbb{Q}}$, assuming it is not induced from a character of an index-two subgroup. The coefficient ring, topology, irreducibility convention, and intended classification must be clarified. The supplied partially solved label is dated curation, not an independently verified conclusion.

Sources

• Local UnsolvedMath registry, supplied complete record for permanent ID $30003881$, title Images of Infinite Galois Representations, exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. No individual proposer is identified. The supplied proposed year is 2018.
• Supplied report citation: Algebraische Zahlentheorie (2019), Oberwolfach Reports, pp. 1785--1844, DOI 10.4171/owr/2018/30. https://doi.org/10.4171/owr/2018/30. The record describes this report as documenting workshop discussions and open problems; no precise problem page is supplied.
• Supplied literature reference: Conti--Lang--Medvedovsky, Big images. https://arxiv.org/abs/1904.10519.
• Supplied literature reference: Oberwolfach report. https://doi.org/10.4171/owr/2018/30.
• Literature assessment checked at 2026-08-21: this is supplied dated curation, not independently verified. It labels the problem partially solved, states that the broad image-classification question remains unresolved, and attributes strong congruence-subgroup results under substantial hypotheses for two-dimensional pseudorepresentations and related Galois representations to Conti--Lang--Medvedovsky. The supplied background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports manual semantic, typographic, and LaTeX review. These are supplied editorial assertions; the cited sources and mathematical completeness have not been independently verified for this draft.