7.3 (Manning) — Number fields as trace fields

Definitions and mathematical statement

A number field $k$ is totally real if every embedding $k\hookrightarrow{\mathbb{C}}$ has image in ${\mathbb{R}}$. For an oriented complete hyperbolic $3$-manifold $M=\mathbb H^3/\Gamma$, let $\widetilde\Gamma$ be the inverse image of its holonomy group in $\operatorname{SL}(2,{\mathbb{C}})$. Its ordinary trace field is \[ t(M)={\mathbb{Q}}\left(\operatorname{tr}\gamma:\gamma\in\widetilde\Gamma\right). \] The invariant trace field is the generally different field \[ k(M)={\mathbb{Q}}\left(\operatorname{tr}(\gamma^2):\gamma\in\widetilde\Gamma\right). \]

Manning, relaying an old Neumann--Reid question, asks whether for every number field $k$ that is not totally real there is a hyperbolic $3$-manifold with trace field $k$. With ordinary trace-field terminology, the target is $t(M)\cong k$. The source statement does not explicitly impose finite volume or say whether equality is of abstract fields or of specified subfields of ${\mathbb{C}}$.

The supplied summary instead identifies the question with Neumann's conjecture for the invariant trace field of each non-real concrete number field. A concrete field includes a specified embedding in ${\mathbb{C}}$, so this is not identical wording. Ordinary versus invariant trace field, abstract versus concrete realization, and finite-volume hypotheses must therefore be resolved before selecting a single formulation.

The summary reports realization of non-real multiquadratic fields, experimental support and an unrestricted arithmetic-orbifold realization assertion for non-totally-real fields. These are retained as supplied literature claims. In particular, the last assertion needs checking: arithmetic Kleinian-lattice trace fields have restrictive complex-place hypotheses, so its unrestricted scope is not adopted as a theorem. Orbifold realization and manifold realization must also be distinguished.

Short English statement

Can every number field that is not totally real occur as a hyperbolic $3$-manifold trace field? The source says trace field, while its summary switches to invariant trace fields and concrete embeddings.

Sources

• Manning, attributed item 7.3, described as an old Neumann--Reid question, in Delp, Hoffoss and Manning, Problems In Groups, Geometry, and Three-Manifolds (2015), supplied PDF page 6. https://arxiv.org/pdf/1512.04620#page=6; original-status link https://arxiv.org/abs/1512.04620. Recorded access 2026-07-29; extraction source-tex checked against rendered PDF text, as reported in the record.
• The supplied summary cites An experimental investigation of Neumann's conjecture, https://arxiv.org/abs/1609.08719, mentions discussion with Reid, Neumann's realizing-arithmetic-invariants lectures (Columbia volume/conference, 2006--2009), and a Reid--Neumann multiquadratic-field observation. Complete lecture and partial-result citations are absent; no author list is reconstructed for the arXiv item.
• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $10900051$, exact source code catalogue entry; supplied partially_solved/PARTIAL-PROGRESS status.