Math Conjectures

Tate Conjecture

OPENLandmarkConjectureProposed 1963 · Standard version

Canonical statement

Let kk be a finitely generated field, X/kX/k a smooth projective variety, r≥0r\ge0, and ℓ≠char⁡k\ell\ne\operatorname{char}k a prime. The image of
cl⁡ℓ:CH⁡r(X)⊗ZQℓ⟶Heˊt2r(Xkˉ,Qℓ(r)) \operatorname{cl}_\ell: \operatorname{CH}^r(X)\otimes_{\mathbb Z}\mathbb Q_\ell \longrightarrow H_{\mathrm{\acute et}}^{2r} (X_{\bar k},\mathbb Q_\ell(r))
equals the invariant subspace
Heˊt2r(Xkˉ,Qℓ(r))Gal⁡(kˉ/k), H_{\mathrm{\acute et}}^{2r} (X_{\bar k},\mathbb Q_\ell(r))^{\operatorname{Gal}(\bar k/k)},
where CH⁡r(X)\operatorname{CH}^r(X) is the group of codimension-rr algebraic cycles modulo rational equivalence.
View source LaTeX
Let \(k\) be a finitely generated field, \(X/k\) a smooth projective variety, \(r\ge0\), and \(\ell\ne\operatorname{char}k\) a prime. The image of
\[
  \operatorname{cl}_\ell:
  \operatorname{CH}^r(X)\otimes_{\mathbb Z}\mathbb Q_\ell
    \longrightarrow
  H_{\mathrm{\acute et}}^{2r}
    (X_{\bar k},\mathbb Q_\ell(r))
\] equals the invariant subspace
\[
  H_{\mathrm{\acute et}}^{2r}
    (X_{\bar k},\mathbb Q_\ell(r))^{\operatorname{Gal}(\bar k/k)},
\] where \(\operatorname{CH}^r(X)\) is the group of codimension-\(r\) algebraic cycles modulo rational equivalence.
AG-CYCLE-003:[Tate1965Cycles][Milne1994Tate][PetersenTommasi2026HodgeTate]
Algebraic geometryNumber theory

Notes

The Tate conjecture is the arithmetic analogue of the Hodge conjecture. For a smooth projective variety XX over a finitely generated field kk and a prime ℓ≠char⁡k\ell\ne\operatorname{char}k, the ℓ\ell-adic cycle class map takes values in the Galois-invariant part of Heˊt2r(Xkˉ,Qℓ(r))H^{2r}_{\mathrm{\acute et}}(X_{\bar k},\mathbb Q_\ell(r)); the conjecture predicts that its image spans exactly this invariant subspace. Tate formulated it in 1963 and published it in his article relating algebraic cycles to poles of zeta functions [Tate1965Cycles], where the invariant subspace is tied to pole orders of zeta functions.

The divisor case r=1r=1 is known for many important classes of varieties, notably abelian varieties over finite fields, and further low-dimensional cases have been established; Milne's survey gives a detailed account of the situation over finite fields [Milne1994Tate]. Verification continues in specific families, for instance for moduli spaces of curves [PetersenTommasi2026HodgeTate].

For arbitrary codimension rr on general smooth projective varieties the conjecture is open, and no general mechanism for producing the predicted cycles is known.

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.