Grothendieck–Ogg–Shafarevich formula
E262121
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Grothendieck–Ogg–Shafarevich formula canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2394221 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Grothendieck–Ogg–Shafarevich formula Context triple: [Riemann–Hurwitz formula, relatedTo, Grothendieck–Ogg–Shafarevich formula]
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A.
Grothendieck–Riemann–Roch theorem
The Grothendieck–Riemann–Roch theorem is a fundamental result in algebraic geometry that generalizes the classical Riemann–Roch theorem by relating pushforwards in K-theory to pushforwards in cohomology via characteristic classes.
-
B.
Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
-
C.
Riemann–Roch theorem
The Riemann–Roch theorem is a fundamental result in algebraic geometry and complex analysis that relates the dimension of spaces of meromorphic sections of a line bundle on a curve to topological data such as genus and degree.
-
D.
Hirzebruch–Riemann–Roch theorem
The Hirzebruch–Riemann–Roch theorem is a fundamental result in algebraic geometry and topology that expresses the holomorphic Euler characteristic of a complex manifold in terms of characteristic classes, unifying and extending classical Riemann–Roch type formulas.
-
E.
Weil conjectures
The Weil conjectures are a set of deep statements about the zeta functions of algebraic varieties over finite fields that guided the development of modern algebraic geometry and were ultimately proved using étale cohomology.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Grothendieck–Ogg–Shafarevich formula Target entity description: 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.
-
A.
Grothendieck–Riemann–Roch theorem
The Grothendieck–Riemann–Roch theorem is a fundamental result in algebraic geometry that generalizes the classical Riemann–Roch theorem by relating pushforwards in K-theory to pushforwards in cohomology via characteristic classes.
-
B.
Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
-
C.
Riemann–Roch theorem
The Riemann–Roch theorem is a fundamental result in algebraic geometry and complex analysis that relates the dimension of spaces of meromorphic sections of a line bundle on a curve to topological data such as genus and degree.
-
D.
Hirzebruch–Riemann–Roch theorem
The Hirzebruch–Riemann–Roch theorem is a fundamental result in algebraic geometry and topology that expresses the holomorphic Euler characteristic of a complex manifold in terms of characteristic classes, unifying and extending classical Riemann–Roch type formulas.
-
E.
Weil conjectures
The Weil conjectures are a set of deep statements about the zeta functions of algebraic varieties over finite fields that guided the development of modern algebraic geometry and were ultimately proved using étale cohomology.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Grothendieck–Ogg–Shafarevich formula Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.