Math Conjectures

Mumford–Tate Conjecture for Abelian Varieties

OPENLandmarkConjectureProposed c. 1966 · Full conjecture

Canonical statement

Let AA be an abelian variety over a number field KK, fix an embedding K↪CK\hookrightarrow\mathbb C, and put VB=H1(A(C),Q)V_B=H_1(A(\mathbb C),\mathbb Q). Let MT⁡(A)⊆GL⁡(VB)\operatorname{MT}(A)\subseteq\operatorname{GL}(V_B) be the Mumford–Tate group. For a prime ℓ\ell, put Vℓ=Tℓ(A)⊗ZℓQℓV_\ell=T_\ell(A)\otimes_{\mathbb Z_\ell}\mathbb Q_\ell, and let GℓG_\ell be the Zariski closure of the image of Gal⁡(K‾/K)\operatorname{Gal}(\overline K/K) in GL⁡(Vℓ)\operatorname{GL}(V_\ell). Under Betti–étale comparison,
Gℓ∘=MT⁡(A)⊗QQℓ G_\ell^\circ=\operatorname{MT}(A)\otimes_{\mathbb Q}\mathbb Q_\ell
for every prime ℓ\ell.
View source LaTeX
Let \(A\) be an abelian variety over a number field \(K\), fix an embedding \(K\hookrightarrow\mathbb C\), and put \(V_B=H_1(A(\mathbb C),\mathbb Q)\). Let \(\operatorname{MT}(A)\subseteq\operatorname{GL}(V_B)\) be the Mumford–Tate group. For a prime \(\ell\), put \(V_\ell=T_\ell(A)\otimes_{\mathbb Z_\ell}\mathbb Q_\ell\), and let \(G_\ell\) be the Zariski closure of the image of \(\operatorname{Gal}(\overline K/K)\) in \(\operatorname{GL}(V_\ell)\). Under Betti–étale comparison,
\[
  G_\ell^\circ=\operatorname{MT}(A)\otimes_{\mathbb Q}\mathbb Q_\ell
\]
for every prime \(\ell\).
AG-HODGE-023:[Pink1998MumfordTate][CantoralFarfan2017MumfordTateSurvey][Commelin2019MumfordTateProducts]
Algebraic geometryNumber theory

Notes

For an abelian variety AA over a number field, the Mumford–Tate group records the linear symmetries of its rational Hodge structure, whereas the ℓ\ell-adic monodromy group is the Zariski closure of the Galois action on the Tate module. After Betti–étale comparison, the conjecture says that the connected ℓ\ell-adic group is exactly the scalar extension of the Mumford–Tate group for every prime ℓ\ell. Pink's work gives a foundational treatment of this comparison and its group-theoretic constraints [Pink1998MumfordTate].

One inclusion follows from the theory of absolute Hodge cycles, and equality is known for CM abelian varieties and many low-dimensional or constrained endomorphism types. The known landscape and its links with the Hodge and Tate conjectures are surveyed by Cantoral Farfán [CantoralFarfan2017MumfordTateSurvey]; compatibility under products supplies further nontrivial cases [Commelin2019MumfordTateProducts].

The remaining problem is the reverse inclusion for arbitrary abelian varieties, especially in high dimension with general endomorphism structure. The connected component is essential: finite components can reflect the field of definition rather than the Hodge structure, so the entry does not assert equality of the full disconnected groups.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.