Mumford–Tate conjecture for abelian varieties

Determine whether, for every abelian variety over a number field and every prime [?] after a suitable finite extension of the ground field, the Zariski closure of the [?]-adic Galois image equals the [?]-adic points of the Mumford–Tate group; equivalently, establish the unresolved part of the Mumford–Tate–Serre conjecture asserting that the [?]-adic Galois image is open in the corresponding Mumford–Tate group.

Background

The paper formulates the Mumford–Tate–Serre conjecture for an abelian variety A over a number field K. It predicts the existence of an algebraic group MT(A) over Q, determined only by the complex geometry of A, such that after a finite extension K'/K the image of every ℓ-adic Galois representation is contained in MT(A)(Qℓ) and is open there.

The paper notes that the containment statement and several structural assertions are known, and that the ℓ-adic image is open in the group of Qℓ-points of its own Zariski closure. The unresolved issue is whether that Zariski closure is the full Mumford–Tate group, which would establish the predicted independence of ℓ and maximality of the Galois image subject to the geometric constraints encoded by Hodge classes.

References

Even though the full conjecture remains open, various parts of it and several special cases have been proven: there is a precise description of the group $\operatorname{MT}(A)$ (which does indeed depend only on $A(C)$, and which by work of Deligne does not depend on the embedding of $K$ in $C$); part (2a) has also been proven by Deligne, and for part (2b) one does at least know that $\rho_{\ell\infty}(#1{K'})$ is open (in the $\ell$-adic topology) in the group of $Q_\ell$-points of its Zariski closure. What we don't know is whether this Zariski closure is all of $\operatorname{MT}(A){Q\ell}$.

— An introduction to algebraic groups  (2610.21926 - Lombardo, 15 Oct 2026) in Section 'The Mumford–Tate conjecture and independence of ℓ', immediately after the Mumford–Tate–Serre conjecture