Grothendieck–Ogg–Shafarevich formula

E262121

mathematical theorem result in arithmetic geometry

The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.

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All labels observed (1)

Label	Occurrences
Grothendieck–Ogg–Shafarevich formula canonical	1

Statements (44)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in arithmetic geometry ⓘ
appearsIn	SGA 7 ⓘ
appliesTo	smooth projective curves over finite fields ⓘ ℓ-adic sheaves ⓘ
assumes	ℓ different from the characteristic of the base finite field ⓘ
characterizes	Euler–Poincaré characteristic of ℓ-adic cohomology ⓘ
concerns	relationship between global and local arithmetic invariants ⓘ wild and tame ramification ⓘ
context	curves over finite fields ⓘ ℓ-adic cohomology ⓘ
describes	Euler characteristic of ℓ-adic sheaves on curves over finite fields ⓘ
field	algebraic geometry ⓘ arithmetic geometry ⓘ number theory ⓘ
formalism	derived functor cohomology ⓘ ℓ-adic sheaf theory ⓘ
generalizes	classical conductor–discriminant relations ⓘ
hasCodomain	integers (Euler characteristic values) ⓘ
hasDomain	curves over finite fields ⓘ
holdsFor	constructible ℓ-adic sheaves ⓘ lisse ℓ-adic sheaves on open subsets of curves ⓘ
involves	Galois representations ⓘ local monodromy ⓘ ramification filtration ⓘ étale cohomology ⓘ
isPartOf	Grothendieck’s theory of ℓ-adic sheaves ⓘ
isRelatedTo	Hasse–Weil zeta function ⓘ Riemann–Hurwitz formula ⓘ Weil conjectures ⓘ
language	Grothendieck topology ⓘ surface form: étale topology
namedAfter	Alexander Grothendieck ⓘ André Ogg ⓘ Igor Shafarevich ⓘ
relates	global Euler characteristic to sum of local conductors ⓘ
relatesTo	Artin conductor ⓘ Swan conductor ⓘ local invariants of ℓ-adic sheaves ⓘ ramification data ⓘ
typeOf	Euler–Poincaré characteristic formula ⓘ
usedFor	computing Euler characteristics of ℓ-adic sheaves ⓘ studying ramification of Galois representations attached to sheaves ⓘ
usedIn	study of local factors of zeta functions of curves ⓘ theory of L-functions over function fields ⓘ

Referenced by (1)

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

Riemann–Hurwitz formula → relatedTo → Grothendieck–Ogg–Shafarevich formula ⓘ