Tamagawa numbers

E685697

arithmetic invariant invariant of algebraic groups invariant of elliptic curves number theoretic invariant

Tamagawa numbers are arithmetic invariants attached to algebraic groups or elliptic curves that measure certain volume or local factor contributions in number theory, notably appearing in the Birch and Swinnerton-Dyer conjecture.

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Statements (49)

Predicate	Object
instanceOf	arithmetic invariant ⓘ invariant of algebraic groups ⓘ invariant of elliptic curves ⓘ number theoretic invariant ⓘ
appearsAsFactorIn	BSD formula denominator ⓘ volume computations for arithmetic quotients ⓘ
appearsIn	Birch and Swinnerton-Dyer formula NERFINISHED ⓘ formula for the leading term of the L-function of an elliptic curve at s = 1 ⓘ mass formulae for algebraic groups ⓘ
associatedWith	algebraic group over a number field ⓘ elliptic curve over a number field ⓘ
conjecturallyRelatedTo	finiteness of Tate–Shafarevich groups ⓘ rank of elliptic curves ⓘ
context	Langlands program NERFINISHED ⓘ arithmetic of reductive groups ⓘ automorphic forms ⓘ
definedOver	number fields ⓘ
definedUsing	adelic quotient of an algebraic group ⓘ product of local measures ⓘ
describedAs	volume of adelic points modulo rational points ⓘ
field	algebraic geometry ⓘ arithmetic geometry ⓘ number theory ⓘ
historicalDevelopment	introduced in the mid-20th century ⓘ
namedAfter	Toshio Tamagawa NERFINISHED ⓘ
property	factorizes as a product of local contributions ⓘ invariant under isomorphism of algebraic groups over a number field ⓘ
relatedTo	Galois cohomology ⓘ Haar measure ⓘ Tamagawa measure NERFINISHED ⓘ Weil’s adelic formalism NERFINISHED ⓘ adelic points ⓘ cohomology of algebraic groups ⓘ local factors of L-functions ⓘ rational points ⓘ
specialCase	Tamagawa number of a semisimple algebraic group NERFINISHED ⓘ Tamagawa number of a torus ⓘ Tamagawa number of an elliptic curve ⓘ
studiedBy	André Weil NERFINISHED ⓘ Goro Shimura NERFINISHED ⓘ John Tate NERFINISHED ⓘ Yutaka Taniyama NERFINISHED ⓘ
takesValuesIn	positive rational numbers ⓘ
usedIn	Birch and Swinnerton-Dyer conjecture NERFINISHED ⓘ Tamagawa measure theory NERFINISHED ⓘ Weil conjectures for algebraic groups NERFINISHED ⓘ arithmetic of elliptic curves ⓘ rational points on abelian varieties ⓘ study of L-functions ⓘ

Referenced by (1)

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

Birch and Swinnerton-Dyer Conjecture → relatesConcept → Tamagawa numbers ⓘ