T:z ↦ z+1
E656690
T:z ↦ z+1 is the standard parabolic modular transformation acting on the upper half-plane, serving as one of the fundamental generators of the modular group PSL(2,ℤ).
All labels observed (1)
| Label | Occurrences |
|---|---|
| T:z ↦ z+1 canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7338608 — 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: T:z ↦ z+1 Context triple: [PSL(2,ℤ), generatorAction, T:z ↦ z+1]
-
A.
FnZ
FnZ is a hip-hop production duo known for crafting atmospheric, hard-hitting beats for prominent rap artists.
-
B.
Ackermann function
The Ackermann function is a classic example of a computable function that grows faster than any primitive recursive function, often used in theoretical computer science to illustrate extreme computational complexity.
-
C.
Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
-
D.
Mittag-Leffler function
The Mittag-Leffler function is a complex function that generalizes the exponential function and plays a central role in fractional calculus and the theory of differential and integral equations.
-
E.
Riemann–Siegel theta function
The Riemann–Siegel theta function is a special function that appears in the study of the Riemann zeta function, used to express its values on the critical line in a form suitable for high-precision numerical computation.
- 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: T:z ↦ z+1 Target entity description: T:z ↦ z+1 is the standard parabolic modular transformation acting on the upper half-plane, serving as one of the fundamental generators of the modular group PSL(2,ℤ).
-
A.
FnZ
FnZ is a hip-hop production duo known for crafting atmospheric, hard-hitting beats for prominent rap artists.
-
B.
Ackermann function
The Ackermann function is a classic example of a computable function that grows faster than any primitive recursive function, often used in theoretical computer science to illustrate extreme computational complexity.
-
C.
Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
-
D.
Mittag-Leffler function
The Mittag-Leffler function is a complex function that generalizes the exponential function and plays a central role in fractional calculus and the theory of differential and integral equations.
-
E.
Riemann–Siegel theta function
The Riemann–Siegel theta function is a special function that appears in the study of the Riemann zeta function, used to express its values on the critical line in a form suitable for high-precision numerical computation.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
generator of PSL(2,ℤ)
ⓘ
modular transformation ⓘ parabolic element of PSL(2,ℤ) ⓘ |
| actsOn | upper half-plane ℍ ⓘ |
| actsTriviallyOn | q = e^{2πiz} coordinate ⓘ |
| belongsToGroup |
PSL(2,ℤ)
ⓘ
linked to:
modular group PSL(2,Z)
SL(2,ℤ) ⓘ |
| commutesWith | all integer translations z ↦ z + n ⓘ |
| extendsContinuouslyTo | ℍ ∪ {∞} ⓘ |
| generatesSubgroup | ⟨T⟩ ≅ ℤ ⓘ |
| hasDerivative | 1 everywhere on ℍ ⓘ |
| hasDeterminant | 1 ⓘ |
| hasEigenvalues | 1,1 as matrix in SL(2,ℤ) ⓘ |
| hasFixedPoint | ∞ ⓘ |
| hasJordanForm | [[1,1],[0,1]] ⓘ |
| hasMatrixRepresentative | [[1,1],[0,1]] ⓘ |
| hasMöbiusForm | z ↦ (1·z + 1)/(0·z + 1) ⓘ |
| hasOrder | infinite ⓘ |
| hasTrace | 2 ⓘ |
| identifiesBoundaryPoints | x = -1/2 and x = 1/2 in the standard fundamental domain ⓘ |
| isBasicExampleOf | parabolic isometry of the hyperbolic plane ⓘ |
| isBiholomorphismOf | upper half-plane ℍ ⓘ |
| isConjugateInPSL(2,ℤ)To | any other primitive parabolic element ⓘ |
| isCuspidalTranslationAt | the cusp at ∞ ⓘ |
| isDefinedBy | T(z) = z + 1 ⓘ |
| isElementOf | group of Möbius transformations ⓘ |
| isGeneratorWith | S : z ↦ -1/z ⓘ |
| isHolomorphicOn | upper half-plane ℍ ⓘ |
| isOrientationPreserving | true ⓘ |
| isParabolicAt | ∞ ⓘ |
| isRealAnalyticOn | ℍ ⓘ |
| isTranslationBy | 1 along the real axis ⓘ |
| isUnipotent | true ⓘ |
| isUpperTriangular | true ⓘ |
| isUsedIn |
construction of modular curves as quotients of ℍ
ⓘ
definition of q = e^{2πiz} for modular forms ⓘ tiling of ℍ by PSL(2,ℤ) images of a fundamental domain ⓘ |
| isUsedToDefine |
periodicity condition f(z+1) = f(z) for modular forms
ⓘ
standard fundamental domain of PSL(2,ℤ) ⓘ |
| maps | x + iy to (x+1) + iy ⓘ |
| preserves |
hyperbolic metric on ℍ
ⓘ
imaginary part of z ⓘ orientation of ℍ ⓘ |
| preservesSet |
horizontal lines in ℍ
ⓘ
ℤ-translates of any vertical geodesic ⓘ |
| satisfiesRelation | PSL(2,ℤ) = ⟨S,T | S² = 1, (ST)³ = 1⟩ ⓘ |
| stabilizes |
cusp ∞ of PSL(2,ℤ) action on ℍ ∪ {∞}
ⓘ
lattice ℤ in ℝ via translation ⓘ |
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: T:z ↦ z+1 Description of subject: T:z ↦ z+1 is the standard parabolic modular transformation acting on the upper half-plane, serving as one of the fundamental generators of the modular group PSL(2,ℤ).
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
modular group PSL(2,Z)