SatoEigenValueProblemΒΆ
satoevp.spad line 104 [edit on github]
For given $N$, the package SatoEigenValueProblem finds $gamma$ and $tau$ such that $gamma tau = N tau$.
- gammaCandidate: (PositiveInteger, Integer, Integer, Integer, Integer) -> Union(Matrix Integer, failed)
- gammaCandidate(nn,det,v,c,d)computes for a given $(- N,- det,- v,- c,- d)$ a possible matrix [[a,- b],[- c,- d]] with a=v-d*nn,- b=(a*d-- det)- /cand fails, if- bis not an integer.
- gammaCandidateWurm: (PositiveInteger, Integer, Integer, Integer, Integer) -> Union(Matrix Integer, failed)
- gammaCandidateWurm(nn,det,v,c,d)computes for a given $(- N,- det,- v,- c,- d)$ a possible matrix [[a,- b],[- c,- d]] with a=v-d*nn,- b=(a*d-- det)- /c. It fails, if both- cand- dare even or if- bis not an integer or- det=1and either a or- dis even or if- det=2and (- cis even or any of a,- b,- dis odd).
- gammaTauQCandidateAux: (PositiveInteger, Integer, Integer, Integer, Integer, (PositiveInteger, Integer, Integer, Integer, Integer) -> Union(Matrix Integer, failed)) -> Union(Record(fgamma: Matrix Integer, ftau: AlgebraicNumber, fq: Complex Float, fnn: PositiveInteger, fdiscr: Integer), failed)
- gammaTauQCandidate( - nn,det,- v,- c,- d,gcand) computes for a given $(- N,det,- v,- c,- d)$ a possible matrix $gamma=[[a,- b],[- c,- d]]$ in SL2Z and a $tauinmathbb{- H}$ such that $gamma tau =- Ntau$. It fails if the candidate matrix would have a non-integer entry in position (1,2) or if v^2>=4*nn*det.
- gammaTauQCandidates: (PositiveInteger, Integer, Integer) -> List Record(fgamma: Matrix Integer, ftau: AlgebraicNumber, fq: Complex Float, fnn: PositiveInteger, fdiscr: Integer)
- gammaTauQCandidates(nn,c,d)returns gammaTauQCandidatesAux(- nn,1,- c,- d,gammaCandidate).
- gammaTauQCandidatesAux: (PositiveInteger, Integer, Integer, Integer, (PositiveInteger, Integer, Integer, Integer, Integer) -> Union(Matrix Integer, failed)) -> List Record(fgamma: Matrix Integer, ftau: AlgebraicNumber, fq: Complex Float, fnn: PositiveInteger, fdiscr: Integer)
- gammaTauQCandidatesAux(nn,det,c,d,f)collects non-failed gammaTauQCandidateAux(- nn,- det,- v,- c,- d,- f) for the finitely many- vwith v^2<4*nn*det.
- gammaTauQCandidatesWurm: (PositiveInteger, Integer, Integer) -> List Record(fgamma: Matrix Integer, ftau: AlgebraicNumber, fq: Complex Float, fnn: PositiveInteger, fdiscr: Integer)
- gammaTauQCandidatesWurm(nn,c,d)returns gammaTauQCandidatesAux(- nn,2,- c,- d,gammaCandidateWurm) if- dis even and gammaTauQCandidatesAux(- nn,1,- c,- d,gammaCandidateWurm) if- dis odd. It returns 0 if the conditions are not fulfilled.
- satoTriples: (PositiveInteger, PositiveInteger) -> XHashTable(List Integer, Record(fnn: PositiveInteger, fgamma: Matrix Integer, ftau: AlgebraicNumber, ftaufd: AlgebraicNumber))
- satoTriples(nn,k)returns a number of Sato triples (- nn,gamma,tau) such that moebiusTransform(gamma,tau)=nn*tau) and for t=moebiusTransform(moebiusAN(toFundamentalDomain(tau,- k),tau)) the value expPiI(2/k*t) is real. Currently,- k=1corresponds to SL2Z and- k=2to the Radu-Wurm group- W.