Level-One Serre Modularity over Imaginary Quadratic Fields

Definitions and mathematical statement

Let $E$ be an imaginary quadratic field, meaning a degree-two extension of ${\mathbb{Q}}$ having no real embedding. Write $\mathcal O_E$ for its ring of integers and $\operatorname{Cl}(\mathcal O_E)$ for its ideal class group. The class-number-one condition is \[ \#\operatorname{Cl}(\mathcal O_E)=1. \] Fix an algebraic closure $\overline E$, and put $G_E=\operatorname{Gal}(\overline E/E)$. For a prime $p$, let ${\mathbb{F}}_p$ be the field with $p$ elements; $\operatorname{GL}_2({\mathbb{F}}_p)$ is the group of invertible two-by-two matrices over that field. The representations in the supplied question have the form \[ \rho:G_E\longrightarrow\operatorname{GL}_2({\mathbb{F}}_p). \] The source denotes their level by $N(\rho)$ and imposes $N(\rho)=1$, but does not define this invariant.

Source question. For every imaginary quadratic field $E$ of class number one and every prime $p$, is Serre's conjecture true for representations $\rho:G_E\to\operatorname{GL}_2({\mathbb{F}}_p)$ of level $N(\rho)=1$, subject to the hypotheses of the intended version of that conjecture?

Unresolved qualifications. The supplied record does not state the intended version of Serre's conjecture over imaginary quadratic fields. In particular, it does not specify the continuity or irreducibility hypotheses on $\rho$, any restrictions on $p$, the precise definition of $N(\rho)$, or the modularity conclusion, including its automorphic or cohomological realization and its weight and level conditions. A common possible interpretation of level one is trivial prime-to-$p$ Artin conductor, but the supplied record does not establish that convention; it is therefore not adopted here as a definition. The coefficient field remains exactly ${\mathbb{F}}_p$, as supplied. The question above preserves the source target, but these omissions prevent a fully specified quantified modularity assertion.

Short English statement

The source asks whether the intended form of Serre's modularity conjecture holds for level-one two-dimensional representations over ${\mathbb{F}}_p$ of absolute Galois groups of imaginary quadratic fields of class number one. The representation hypotheses, level convention, and exact modularity conclusion require clarification. The supplied registry labels the problem open; that label is not independently verified here.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001836$, title Level-One Serre Modularity over Imaginary Quadratic Fields, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2011 and no named proposer.
• Supplied source citation: Explicit Methods in Number Theory (2012), Oberwolfach Reports, pp. 1971--2044, DOI 10.4171/owr/2011/35. Source URL and literature-source label "original OWR report": https://doi.org/10.4171/owr/2011/35. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature-source label "Torrey on Serre conjecture over imaginary quadratic fields": https://doi.org/10.1016/j.jnt.2011.09.010. No fuller bibliographic description is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that Serre modularity over imaginary quadratic fields remains only partially established and that the level-one class-number-one assertion is not known in the stated generality. This is supplied dated curation, not independently verified. The background describes it as literature triage derived from the supplied OWR curation and calls for independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that its mathematical content was checked against the cited official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This is supplied verification metadata; no independent source consultation or verification is asserted here.