Tate conjecture

Tate conjecture
George Bergman · GFDL 1.2 · source
Name	Tate conjecture
Field	Algebraic geometry, Number theory
Introduced	1963
Introduced by	John Tate
Related conjectures	Hodge conjecture, Birch and Swinnerton-Dyer conjecture, Standard conjectures on algebraic cycles, Langlands program

Tate conjecture The Tate conjecture posits a precise relationship between algebraic cycles on varieties over finite fields and Galois-invariant classes in ℓ-adic cohomology, linking ideas from Algebraic geometry, Number theory, Arithmetic geometry, Étale cohomology, and the Langlands program. Formulated by John Tate in the 1960s, it stands alongside the Hodge conjecture and the Birch and Swinnerton-Dyer conjecture as a central open problem guiding research on motives, Shimura varieties, and the Weil conjectures. The conjecture has deep implications for the structure of zeta functions, L-functions, and the arithmetic of Abelian varietys and K3 surfaces.

Statement

For a smooth projective variety X over a finite field F_q and a prime ℓ different from the characteristic, the conjecture asserts that the ℓ-adic cycle class map induces a surjection from the group of algebraic cycles on X modulo numerical equivalence (tensored with Q_ℓ) onto the subspace of ℓ-adic cohomology fixed by the action of the absolute Galois group of F_q. In concrete terms, the rank of the group of codimension r algebraic cycles modulo numerical equivalence equals the multiplicity of q^r as a reciprocal root of the numerator of the zeta function ζ(X, s). The formulation ties together the algebraic cycle groups, Étale cohomology, and the behavior of Frobenius as an element of the Galois group of F_q.

Background and motivation

Tate proposed the conjecture after work on the Weil conjectures and the development of Étale cohomology by Alexander Grothendieck and collaborators, and following investigations by André Weil and Bernard Dwork. Motivations include understanding the extent to which cohomological invariants are "explained" by algebraic geometry as opposed to transcendental phenomena, analogous to the role of the Hodge decomposition for complex varieties and the role of Mordell–Weil theorem for rational points on Elliptic curves. The conjecture also interacts with the formulation of Grothendieck's standard conjectures on algebraic cycles and with conjectural descriptions of motives pursued by Pierre Deligne, Alexander Beilinson, and James Milne.

Known cases and results

Proved cases include the Tate conjecture for divisors (codimension 1) on abelian varieties and K3 surfaces in many instances. Notable results were obtained by Tate himself for abelian varieties over finite fields, and by Tetsuji Shioda and Tatsuyuki Shioda on elliptic surfaces and singular K3 surfaces; breakthroughs for certain K3 surfaces used work of Jean-Pierre Serre, Peter Scholze, and Christopher Skinner in related contexts. Results for surfaces over finite fields were advanced by Jean-Louis Colliot-Thélène and J. S. Milne; the case of varieties dominated by products of curves follows from work on the Tate module and Honda-Tate theorem. For varieties over number fields, the Tate conjecture is related to the Hodge conjecture via reduction modulo primes and to modularity results like those established by Andrew Wiles and Richard Taylor for elliptic curves, which influence special cases. Counterparts over finite fields often leverage the Weil conjectures proved by Pierre Deligne.

Techniques and approaches

Approaches include the study of Frobenius eigenvalues using the Weil conjectures and ℓ-adic representations of the Galois group, deformation and lifting techniques to relate varieties in characteristic p to characteristic 0 using Crystalline cohomology and p-adic Hodge theory developed by Jean-Marc Fontaine and Gerd Faltings. Other methods invoke automorphic forms and the Langlands correspondence to analyze Galois representations, drawing on work by Robert Langlands, Michael Harris, and Richard Taylor. Techniques from the theory of Motives, as advanced by Yves André and Uwe Jannsen, and inputs from the study of Brauer groups and the Tate–Shafarevich group for Abelian varietys have provided structural tools. Recent innovations use perfectoid spaces and techniques of Peter Scholze to control cohomology and special fibers.

Consequences and connections

If true, the Tate conjecture would yield finiteness results for algebraic cycles modulo numerical equivalence and imply parts of the Standard conjectures on algebraic cycles proposed by Grothendieck. It would determine ranks of Néron–Severi groups of varieties over finite fields and relate these ranks to orders of poles of zeta functions, impacting the study of zeta functions and special value formulas akin to predictions from the Birch and Swinnerton-Dyer conjecture. The conjecture connects to the Hodge conjecture through comparison isomorphisms and to the Langlands program via the arithmetic of Galois representations of Automorphic representations. It has implications for rational points on varieties, the structure of Motives, and constraints on possible Frobenius eigenvalues informed by the Honda-Tate theorem.

Open problems and conjectural refinements

Major open problems include proving the Tate conjecture in codimension greater than 1 for general varieties, establishing uniformity statements for families of varieties such as K3 surfaces and higher-dimensional Calabi–Yau manifolds, and relating the conjecture to precise formulations of the Tamagawa number conjecture and refined conjectures in the Bloch–Kato conjecture framework. Refinements consider integral and p-adic versions, compatibility with expected properties of the category of motives proposed by Grothendieck and developed by Jannsen, and explicit links with the Langlands reciprocity for Galois representations arising from algebraic cycles. Progress on these fronts often interweaves advances in p-adic Hodge theory, modularity theorems, and geometric methods coming from Shimura varieties and Perfectoid space techniques.

Category:Algebraic geometry