Tate Conjecture

E680776

mathematical conjecture open problem in arithmetic geometry

The Tate Conjecture is a major open problem in arithmetic geometry that predicts a deep connection between algebraic cycles on varieties over finite fields and their Galois-invariant étale cohomology classes.

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Observed surface forms (2)

Surface form	Occurrences
Hasse–Weil conjecture	1
Tate conjecture	1

Statements (47)

Predicate	Object
instanceOf	mathematical conjecture ⓘ open problem in arithmetic geometry ⓘ
appliesTo	smooth projective varieties over finite fields ⓘ
assumes	prime l different from the characteristic of the finite field ⓘ
codimensionParameter	r ⓘ
cohomologyDegree	2r ⓘ
concerns	Galois representations NERFINISHED ⓘ algebraic cycles ⓘ varieties over finite fields ⓘ étale cohomology ⓘ
domain	smooth projective variety over a finite field ⓘ
equates	dimension of the space of Galois-invariant cohomology classes ⓘ rank of the group of algebraic cycles modulo numerical equivalence ⓘ
field	algebraic geometry ⓘ arithmetic geometry ⓘ number theory ⓘ
formulatedBy	John Tate NERFINISHED ⓘ
groupActing	absolute Galois group of the finite field ⓘ
hasConsequence	control of algebraic cycles by Galois action ⓘ description of Néron–Severi group via Galois cohomology ⓘ
hasVariant	Tate Conjecture for divisors NERFINISHED ⓘ Tate Conjecture for higher codimension cycles NERFINISHED ⓘ
implies	finiteness of the Néron–Severi group over finite fields ⓘ semisimplicity of certain Galois representations ⓘ
isAnalogOf	Hodge Conjecture NERFINISHED ⓘ Mumford–Tate Conjecture NERFINISHED ⓘ
isCentralIn	study of algebraic cycles over finite fields ⓘ theory of motives ⓘ
isRelatedTo	Beilinson Conjectures NERFINISHED ⓘ Birch and Swinnerton-Dyer Conjecture NERFINISHED ⓘ Weil Conjectures NERFINISHED ⓘ
isSpecialCaseOf	standard conjectures on algebraic cycles ⓘ
knownFor	deep connection between geometry and arithmetic of varieties over finite fields ⓘ
knownToHoldFor	K3 surfaces in some cases ⓘ abelian varieties over finite fields in many cases ⓘ divisors on abelian varieties over finite fields ⓘ
motivated	study of zeta functions of varieties over finite fields ⓘ
namedAfter	John Tate NERFINISHED ⓘ
predicts	equality between algebraic cycles and Galois-invariant cohomology classes ⓘ
relates	Galois-invariant subspace of cohomology ⓘ algebraic cycles of codimension r ⓘ l-adic étale cohomology ⓘ
status	open in general ⓘ
type	cohomological conjecture ⓘ
uses	absolute Galois group of a finite field ⓘ l-adic cohomology ⓘ
yearProposed	1963 ⓘ

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Hodge Conjecture → relatedTo → Tate Conjecture ⓘ

Birch and Swinnerton-Dyer Conjecture → relatedTo → Tate Conjecture ⓘ

this entity surface form: Hasse–Weil conjecture

John Tate → notableWork → Tate Conjecture ⓘ

this entity surface form: Tate conjecture