$\lambda$-Independence and Hodge Groups of Motives

Definitions and mathematical statement

Let $M$ be a pure polarizable motive, with coefficient field $E$, Betti realization $V_B=H_B(M)$, and $\lambda$-adic realizations $V_\lambda$ carrying representations $\rho_\lambda$ of the intended base field's absolute Galois group. The base field, category of motives, and comparison identifications are not specified in the supplied record. For each finite place $\lambda$ of $E$, let \[ G_\lambda=\overline{\rho_\lambda(\operatorname{Gal}(\overline k/k))}^{\,\mathrm{Zar}} \subseteq\operatorname{GL}_{E_\lambda}(V_\lambda). \] This display uses $k$ as a placeholder for the missing field of definition, not an added universal assumption.

The source's Conjecture A asks for one algebraic subgroup \[ H\subseteq\operatorname{GL}_E(V_B), \] defined over $E$ and independent of $\lambda$, such that, after the intended comparison $V_B\otimes_E E_\lambda\cong V_\lambda$, \[ G_\lambda=H\otimes_E E_\lambda\quad\text{for every }\lambda, \qquad H^\circ=\operatorname{Hg}(M). \] The closure equality makes precise "Zariski-dense image in $H$" over a completion. The same global subgroup is requested for all places, including its component group, not merely equality of ranks or independence of the connected groups up to abstract isomorphism. The equality with $\operatorname{Hg}(M)$ requires an agreed coefficient-field realization of that Betti Hodge group.

A substantive convention is unresolved: the usual special Hodge group, generated by the Hodge circle, differs from the full Mumford--Tate group. If $\operatorname{Hg}$ means the former and the $\rho_\lambda$ are the unmodified usual realizations, the unrestricted assertion is incompatible with the Tate motive ${\mathbb{Q}}(1)$ over ${\mathbb{Q}}$: its special Hodge group is trivial, whereas its $\ell$-adic cyclotomic or inverse-cyclotomic realization has connected Zariski closure $\mathbb G_m$. Thus the report may use a different Hodge-group convention, a weight/normalization restriction, or modified Galois representations. None of these repairs is selected. This is a formulation check on the supplied wording, not a claimed resolution of the intended conjecture.

The supplied assessment dated 2026-08-21 describes a broad Mumford--Tate--Serre generalization as not known in general. It does not settle the missing conventions or justify replacing the displayed $\operatorname{Hg}$ by a different group.

Short English statement

Conjecture A asks for one algebraic group over the coefficient field whose base changes are all the Galois monodromy groups and whose connected part is the stated Hodge group. The meaning of Hodge group and the realization normalization are essential; under the usual special-Hodge convention the unrestricted wording has a Tate-motive obstruction.

Sources

• Local UnsolvedMath registry, permanent ID $30000868$, exact source code catalogue entry, proposed year 2007. The original names this "Conjecture A"; no individual proposer is supplied.
• Modulformen (2008), Oberwolfach Reports, DOI 10.4171/owr/2007/51, pp. 3013--3072. https://doi.org/10.4171/owr/2007/51.
• Supplied reference labelled "Mumford-type Mumford--Tate groups": https://doi.org/10.1016/S0022-4049(99)00174-7.
• The supplied corrected_verified note describes editorial and LaTeX cleanup without added source-dependent content. The open label and assessment checked 2026-08-21 are dated supplied triage requiring independent verification of the intended statement and current status.