Beilinson conjectures

E685698

conjectural framework in arithmetic geometry mathematical conjecture

Beilinson conjectures are a set of deep conjectures in arithmetic geometry that relate special values of L-functions to algebraic K-theory and motivic cohomology, generalizing phenomena seen in cases like the Birch and Swinnerton-Dyer conjecture.

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Statements (50)

Predicate	Object
instanceOf	conjectural framework in arithmetic geometry ⓘ mathematical conjecture ⓘ
appliesTo	L-functions of algebraic varieties ⓘ L-functions of motives ⓘ L-functions of number fields ⓘ
concerns	critical values of L-functions ⓘ non-critical values of L-functions at integers ⓘ rational structures on cohomology ⓘ special values at negative integers ⓘ special values at positive integers ⓘ
field	algebraic K-theory ⓘ arithmetic geometry ⓘ motivic cohomology ⓘ number theory ⓘ
formulatedBy	Alexander Beilinson NERFINISHED ⓘ
formulatedIn	1980s ⓘ
generalizes	Birch and Swinnerton-Dyer conjecture NERFINISHED ⓘ Dirichlet’s class number formula NERFINISHED ⓘ class number formula ⓘ
hasPart	Beilinson conjecture on motivic cohomology NERFINISHED ⓘ Beilinson conjecture on regulators NERFINISHED ⓘ Beilinson conjecture on special values of L-functions NERFINISHED ⓘ
implies	parts of the Bloch–Kato conjecture in certain cases ⓘ
influenced	development of motivic cohomology ⓘ formulation of the Bloch–Kato Tamagawa number conjecture ⓘ
mainTheme	relation between special values of L-functions and algebraic K-theory ⓘ relation between special values of L-functions and motivic cohomology ⓘ
namedAfter	Alexander Beilinson NERFINISHED ⓘ
predicts	leading Taylor coefficients of L-functions at integers ⓘ order of vanishing of L-functions at integers ⓘ rationality properties of normalized L-values ⓘ relations between special L-values and determinants of regulator pairings ⓘ
relatedTo	Bloch–Kato conjecture NERFINISHED ⓘ Bloch’s conjecture on Chow groups NERFINISHED ⓘ Deligne conjecture on critical values of L-functions ⓘ Equivariant Tamagawa number conjecture NERFINISHED ⓘ Lichtenbaum conjectures ⓘ Tamagawa number conjecture NERFINISHED ⓘ
relatesConcept	Deligne cohomology NERFINISHED ⓘ absolute Hodge cohomology ⓘ algebraic K-groups ⓘ higher regulators ⓘ motivic cohomology groups ⓘ regulator maps ⓘ
status	open ⓘ
typicalDomain	motives over number fields ⓘ smooth projective varieties over number fields ⓘ
usesConcept	Chow groups ⓘ Ext-groups in the category of mixed motives ⓘ higher Chow groups ⓘ

Referenced by (2)

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

Birch and Swinnerton-Dyer Conjecture → relatedTo → Beilinson conjectures ⓘ

Alexander Beilinson → knownFor → Beilinson conjectures ⓘ