Jacobi's inversion problem
E702055
Jacobi's inversion problem is a fundamental question in algebraic geometry and the theory of Abelian functions, concerning the inversion of Abelian integrals and the characterization of their multi-valued inverses.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Abelian integrals | 1 |
| Jacobi's inversion problem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7978919 — 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: Jacobi's inversion problem Context triple: [Carl Gustav Jacob Jacobi, notableWork, Jacobi's inversion problem]
-
A.
Jacobi elliptic functions
Jacobi elliptic functions are a family of doubly periodic complex functions that generalize trigonometric functions and play a central role in the theory of elliptic integrals and many areas of mathematical physics.
-
B.
Weierstrass elliptic functions
Weierstrass elliptic functions are a class of doubly periodic meromorphic functions that play a central role in the theory of elliptic curves and complex analysis.
-
C.
Recherches sur les fonctions elliptiques
Recherches sur les fonctions elliptiques is a foundational mathematical treatise by Niels Henrik Abel that significantly advanced the theory of elliptic functions and laid groundwork for modern complex analysis.
-
D.
Hilbert’s twelfth problem
Hilbert’s twelfth problem is one of David Hilbert’s famous list of 23 problems, asking for a general explicit class field theory that would generate all abelian extensions of a given number field using special values of analytic functions.
-
E.
Introduction to the Theory of Algebraic Functions of One Variable
Introduction to the Theory of Algebraic Functions of One Variable is a classic monograph by Claude Chevalley that provides a rigorous, modern foundation for the theory of algebraic function fields in one variable.
- 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: Jacobi's inversion problem Target entity description: Jacobi's inversion problem is a fundamental question in algebraic geometry and the theory of Abelian functions, concerning the inversion of Abelian integrals and the characterization of their multi-valued inverses.
-
A.
Jacobi elliptic functions
Jacobi elliptic functions are a family of doubly periodic complex functions that generalize trigonometric functions and play a central role in the theory of elliptic integrals and many areas of mathematical physics.
-
B.
Weierstrass elliptic functions
Weierstrass elliptic functions are a class of doubly periodic meromorphic functions that play a central role in the theory of elliptic curves and complex analysis.
-
C.
Recherches sur les fonctions elliptiques
Recherches sur les fonctions elliptiques is a foundational mathematical treatise by Niels Henrik Abel that significantly advanced the theory of elliptic functions and laid groundwork for modern complex analysis.
-
D.
Hilbert’s twelfth problem
Hilbert’s twelfth problem is one of David Hilbert’s famous list of 23 problems, asking for a general explicit class field theory that would generate all abelian extensions of a given number field using special values of analytic functions.
-
E.
Introduction to the Theory of Algebraic Functions of One Variable
Introduction to the Theory of Algebraic Functions of One Variable is a classic monograph by Claude Chevalley that provides a rigorous, modern foundation for the theory of algebraic function fields in one variable.
- F. None of above. chosen
Statements (40)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical problem
ⓘ
problem in the theory of Abelian functions ⓘ |
| asksFor |
characterization of multi-valued inverses of Abelian integrals
ⓘ
description of arguments of Abelian functions in terms of integrals ⓘ inversion of Abelian integrals ⓘ |
| concerns |
recovery of divisor data from Abelian integrals
ⓘ
representation of points on a Jacobian by integrals of holomorphic differentials ⓘ |
| field |
algebraic geometry
ⓘ
complex analysis ⓘ number theory ⓘ theory of Abelian functions ⓘ theory of Riemann surfaces ⓘ |
| hasApplication |
construction of Abelian functions
ⓘ
explicit parametrization of algebraic curves ⓘ integrable systems ⓘ soliton theory ⓘ theory of algebraic curves ⓘ |
| hasGeneralization | inversion problems for higher-dimensional Abelian varieties ⓘ |
| historicalPeriod | 19th-century mathematics ⓘ |
| influenced |
development of Riemann surface theory
ⓘ
development of modern algebraic geometry ⓘ theory of Abelian varieties ⓘ |
| involvesConcept |
Abel's theorem
ⓘ
linked to:
Abel’s theorem
Abelian functions ⓘ Abelian integrals ⓘ
linked to:
Jacobi's inversion problem
Abel–Jacobi map ⓘ Jacobian variety ⓘ
linked to:
Jacobian varieties
Riemann surfaces ⓘ divisors on algebraic curves ⓘ multi-valued inverse functions ⓘ periods of integrals ⓘ theta functions ⓘ |
| namedAfter | Carl Gustav Jacob Jacobi ⓘ |
| relatedTo |
Abel's addition theorem
ⓘ
Abel–Jacobi theory ⓘ
linked to:
Abel–Jacobi map
Jacobi variety ⓘ
linked to:
Jacobian varieties
Riemann's theta inversion theorem ⓘ
linked to:
Riemann theta function
|
| solvedBy |
Riemann's theory of Abelian integrals
ⓘ
theory of theta functions ⓘ |
| status | classically solved ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Jacobi's inversion problem Description of subject: Jacobi's inversion problem is a fundamental question in algebraic geometry and the theory of Abelian functions, concerning the inversion of Abelian integrals and the characterization of their multi-valued inverses.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
Carl
linked to: Jacobi's inversion problem