Open problem, Arithmetic and number theory, Geometry, posed 1963

Tate conjecture

Open

Let $X$ be a smooth projective variety over a field $k$ finitely generated over its prime field, and fix a prime $\ell \ne \operatorname{char}(k)$ . Is the cycle class map $\bigoplus_{r} \left(\operatorname{CH}^r(X) \otimes \mathbb{Q}_\ell\right) \to \bigoplus_r H^{2r}_{\mathrm{et}}(X_{\bar{k}}, \mathbb{Q}_\ell(r))^{\operatorname{Gal}(\bar{k}/k)}$ from algebraic cycles with $\mathbb{Q}_\ell$ -coefficients to Galois-invariant $\ell$ -adic cohomology classes surjective?

Research frontier as of 2026

As of 2026, the Tate conjecture is proven for divisors on abelian varieties over finite fields (Tate 1966) and number fields (Faltings 1983), and for divisors on K3 surfaces over finite fields of characteristic $p \ne 2$ (Nygaard–Ogus, Madapusi Pera, Charles, Maulik 2012–2014) and in characteristic zero (André 1996, Tankeev). It remains open for higher-codimension cycles on general abelian varieties and for most smooth projective varieties beyond dimension 2.

Best known results

Proven for divisors on abelian varieties over finite fields (Tate 1966) and over number fields (Faltings 1983, via the finiteness theorems underlying the Mordell conjecture).
Proven for divisors on K3 surfaces over finite fields of odd characteristic (Nygaard, Ogus, Charles, Madapusi Pera, Maulik 2012–2014) and over number fields and $\mathbb{C}$ (André 1996, using the Kuga–Satake correspondence).

Tools and where they stop

Tool	Achieved	Where it stops
Finiteness theorems for isogeny classes and the Faltings height	Proves the Tate conjecture for abelian varieties over finitely generated fields by bounding heights of isogenous abelian varieties in a fixed isogeny class	Relies on the group structure and moduli theory specific to abelian varieties, with no known analogue for higher-codimension cycles on general varieties
Kuga–Satake correspondence and crystalline Torelli theorems	Transfers the Tate conjecture for K3 surfaces to the already-known case for abelian varieties by embedding the K3 lattice into the cohomology of an auxiliary abelian variety	Specific to the rank-22 weight-2 Hodge structure of K3 surfaces; no analogous auxiliary abelian variety construction is known for general higher-weight cohomology

Open questions

Does the Tate conjecture hold for divisors on arbitrary smooth projective surfaces over finite fields, beyond abelian and K3 surfaces?
Is the Tate conjecture true for higher-codimension algebraic cycles on abelian varieties of dimension $\ge 4$ ?

References

John Tate (1965). Algebraic cycles and poles of zeta functions
John Tate (1966). Endomorphisms of abelian varieties over finite fields · DOI:10.1007/BF01404549
Gerd Faltings (1983). Endlichkeitssätze für abelsche Varietäten über Zahlkörpern · DOI:10.1007/BF01388432
Keerthi Madapusi Pera (2015). The Tate conjecture for K3 surfaces in odd characteristic · DOI:10.1007/s00222-014-0557-5