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).
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| T : z ↦ z + 1 | 0 |
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)
NERFINISHED
ⓘ
SL(2,ℤ) NERFINISHED ⓘ |
| 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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
PSL(2,ℤ)