Even Mod-p Galois Representations in Langlands Reciprocity

Definitions and mathematical statement

Let $p$ be a prime, let $\overline{{\mathbb{Q}}}$ be an algebraic closure of ${\mathbb{Q}}$, and write \[ G_{{\mathbb{Q}}}=\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}}). \] Consider a homomorphism \[ \rho:G_{{\mathbb{Q}}}\longrightarrow\operatorname{GL}_2({\mathbb{F}}_p). \] The representation $\rho$ is absolutely irreducible if the induced action on $\overline{{\mathbb{F}}}_p^{\,2}$, obtained by extending scalars from ${\mathbb{F}}_p$, has no nonzero proper invariant subspace. Choose an embedding $\overline{{\mathbb{Q}}}\hookrightarrow{\mathbb{C}}$, and let $c\in G_{{\mathbb{Q}}}$ be the resulting complex conjugation. The representation is even if \[ \det\left(\rho(c)\right)=1\in{\mathbb{F}}_p^{\times}. \] This condition is independent of the choice of complex conjugation because such elements are conjugate in $G_{{\mathbb{Q}}}$. For $p=2$, the determinant condition is automatic; the supplied record does not specify whether this characteristic is intended to be included. In the usual continuous interpretation of a mod-$p$ Galois representation, $G_{{\mathbb{Q}}}$ has its profinite topology and $\operatorname{GL}_2({\mathbb{F}}_p)$ has the discrete topology. Continuity is not explicit in the supplied statement and remains a qualification to verify.

Source question. For absolutely irreducible even representations $\rho:G_{{\mathbb{Q}}}\to\operatorname{GL}_2({\mathbb{F}}_p)$, what does the arithmetic Langlands reciprocity programme predict, and what does it establish?

This is a research question rather than a specified conjectural assertion. The supplied record identifies no particular automorphic objects, correspondence, compatibility conditions, or claimed existence theorem. It also supplies no additional ramification or local hypotheses. Consequently, the intended reciprocity statement and its quantifiers cannot be made more specific from this record alone. The original wording describes the displayed representations as examples ("such as"), whereas the normalized statement makes them the explicit subject; any intended broader class remains unspecified.

Short English statement

The question asks what arithmetic Langlands reciprocity predicts or proves for absolutely irreducible two-dimensional representations of $G_{{\mathbb{Q}}}$ over ${\mathbb{F}}_p$ whose determinant at complex conjugation is $1$. A precise proposed correspondence is not supplied. Continuity, the intended treatment of $p=2$, and the scope beyond the displayed representations require clarification from the source.

Sources

• Local UnsolvedMath registry: permanent ID $30001835$, source problem code catalogue entry, title Even Mod-p Galois Representations in Langlands Reciprocity; collection: Oberwolfach Reports Open Problems. No individual proposer is supplied. The registry records proposed year $2011$.
• Supplied report citation: Explicit Methods in Number Theory (2012), Oberwolfach Reports, pp. 1971--2044, DOI 10.4171/owr/2011/35. Supplied source URL: https://doi.org/10.4171/owr/2011/35. The background attributes the question to workshop open-problem discussions documented in Oberwolfach Reports. No problem-specific page or named contributor is provided.
• Supplied literature source, labelled "original OWR report": https://doi.org/10.4171/owr/2011/35.
• Literature assessment dated 2026-08-21: the supplied curation labels the question open and reports that no general theorem resolving this broad arithmetic Langlands question was found. This is supplied dated curation, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This verification claim is supplied provenance and has not been independently checked here. The citation's $2012$ title date and the registry's $2011$ proposed year are retained as supplied.