QEtaModularCategory¶

QEtaModularCategory provides functions to check modularity conditions.

candidateLevelsCoEtaQuotient: (QEtaSpecification, PositiveInteger, NonNegativeInteger) -> List PositiveInteger: candidateLevelsCoEtaQuotient(rspec,m,t) returns an ascendingly sorted list of the nn up to 24*m*level(rspec) such that conditionCoEtaQuotient?(nn,rspec,m,t) is true.

conditionCoEtaQuotient?: (PositiveInteger, QEtaSpecification, PositiveInteger, NonNegativeInteger) -> Boolean: conditionCoEtaQuotient?(nn,rspec,m,t) returns true iff a cofactor eta-quotient can be found such that the product of the cofactor with the orbit product corresponding to the dissection (m,t) gives a modular function in level nn.

cusps: PositiveInteger -> List Cusp: cusps(nn) returns representatives for all the (inequivalent) cusps for the congruence subgroup that the respective package represents. They are sorted by their size as rational numbers with infinity being the biggest cusp. Note that cusps()=[cusp(x(1,1),x(2,1)) for x in doubleCosetRepresentatives()] where doupleCosetRepresentatives() is from the respective congruence subgroup that is connected to the package.

cuspToMatrix: (PositiveInteger, Cusp) -> Matrix Integer: For cusp=(a:c), cuspToMatrix(nn,cusp) returns a matrix gamma=[[a,b],[c,d]] corresponding to the cusp (a:c) of this domain. We assume that a/c is a normalized cusp, i.e. cusp=normalizeCusp(cusp).

etaCofactorSpace: (PositiveInteger, QEtaSpecification, PositiveInteger, NonNegativeInteger, List List Integer) -> Record(indices: List List Integer, particular: Union(Vector Integer, failed), basis: List Vector Integer): etaCofactorSpace(nn,rspec,m,t,idxs) returns a vector v and the basis of a space such that #v=#idxs and modular?(nn,members(s),rspec,m,t) is true for any s = v + reduce(_+, [z.i * basis.i for i in 1..#basis]) and any sufficiently long list z of integers. The function fails, if there is no such solution. The indices part of the result is equal to idxs and corresponds to the entries of the solution vectors.

etaCofactorSpaceSystem: (PositiveInteger, QEtaSpecification, PositiveInteger, NonNegativeInteger, List List Integer) -> Record(qmat: Matrix Fraction Integer, zrels: Vector Integer, qrhs: Vector Fraction Integer): etaCofactorSpaceSystem(nn,rspec,m,t,idxs) returns a matrix mat and a vector v such that for the integer solutions s of the equation mat*s=v it holds modular?(sspec,rspec,m,t) where sspec results from the list of the first #idxs entries of s. With idxs one can specify what indexes should be allowed in the solution space.

etaQuotientMonoidSystem: (PositiveInteger, List Cusp, List List Integer) -> Record(qmat: Matrix Fraction Integer, zrels: Vector Integer, qrhs: Vector Fraction Integer): etaQuotientMonoidSystem(nn,polesat,idxs,rhs) returns a record that can be given to zsolve$X4ti2 in order to find the generators of the monoid of (generalized) eta-quotients that are modular wrt. QMOD and have poles at most at the cusps given by polesat. The indices of the eta-functions that may appear is given by idxs.

genus: PositiveInteger -> NonNegativeInteger: genus(nn) returns the genus of the congruence subgroup connected to the package (either Gamma_0(nn) or Gamma_1(nn).

matrixEtaOrder: (PositiveInteger, List Cusp, List List Integer) -> Matrix Fraction Integer: matrixEtaOrder(nn,spitzen,idxs) returns the matrix given by [matrixEtaOrderRow(nn,cusp,idxs) for cusp in spitzen] with duplicate rows removed.

matrixEtaOrderElement: (PositiveInteger, Cusp, List Integer) -> Fraction Integer: z:=matrixEtaOrderElement(nn,cusp,idx). The parameter corresponds to the index of a (generalized) eta-quotient, i.e. it can be of the form idx=[delta,g] or of the form idx=[delta] with g in the range 0..delta. Then z is the coefficient of r_{delta,g} in cite[Thm.~4]{Robins_GeneralizedDedekindEtaProducts_1994}, except for the case nn=4, gcd(denom(cusp),nn)=2 (or Gamma1(4)-equivalent) where we return half of this coefficient. If idx=[delta] then z=1/2*matrixEtaOrderElement(nn,cusp,[delta,0])/2. and thus corresponds to the coefficient of $r_delta$ in eqref{eq:order-rbar-non-adjusted} in Theorem~ref{thm:matrixEtaOrderRobins}. Note that Robins refers to an expansion in the uniformizing variable q^(gcd(c,nn)/nn) (where cusp=a/c). That is equal to q^(1/width(c)$CongruenceSubgroupGamma1(nn)) except for the case nn=4 and gcd(nn,c)=2. In that case the width of the cusp 1/2 is 1 and not 2=4/gcd(4.2). For this case, we deviate from Robins’ coefficients and adapt to an expansion in q rather than an expansion in q^(1/2).

matrixEtaOrderFull: (PositiveInteger, List Cusp, List List Integer) -> Matrix Fraction Integer: matrixEtaOrder(nn,spitzen,idxs) returns a matrix given by [matrixEtaOrderRow(nn,cusp,idxs) for cusp in spitzen].

matrixEtaOrderRow: (PositiveInteger, Cusp, List List Integer) -> List Fraction Integer: matrixEtaOrderRow(nn, cusp, idxs) returns [matrixEtaOrderElement(nn,cusp,idx) for idx in idxs].

matrixModular: (PositiveInteger, List List Integer) -> Matrix Fraction Integer: matrixModular(nn, idxs) returns a matrix mat mat*v=0 for an (extended) exponent vector where the exponents correspond to idxs, i.e. #v=#idxs+k where k corresponds to additional (slack) variables due to the modularity conditions. The rows corresponding to slack variables have been normalized so that the entry of the slack variable (there is only one) is 1.

minimalLevelCoEtaQuotient: (QEtaSpecification, PositiveInteger, NonNegativeInteger) -> PositiveInteger: minimalLevelCoEtaQuotient(rspec, m, t) returns the smallest nn of candidateLevalsCoEtaQuotient(rspec,m,t).

modular?: (QEtaSpecification, QEtaSpecification, PositiveInteger, NonNegativeInteger) -> Boolean: modular?(sspec,rspec,m,t) returns true iff (sspec,rspec,m,t) specifies a modular function in level(sspec).

modular?: QEtaSpecification -> Boolean: modular?(rspec) returns true iff spec specifies a modular function in level(rspec).

modularOrbit: (QEtaSpecification, PositiveInteger, NonNegativeInteger) -> List NonNegativeInteger: modularOrbit(rspec, m, t) (for $Gamma_0(N) where N is level(rspec)) computes the elements of $modularOrbit{r,m,t)$ as defined in qeta.tex, cite[Def.~42]{Radu_RamanujanKolberg_2015} and cite[Lemma 4.35]{Radu_PhD_2010} were r is pureExponents(rspec) and rspec is expected to specify a pure eta-quotient. For $Gamma_1(N)$, modularOrbit(rspec, m, t) returns [t].

width: (PositiveInteger, Cusp) -> PositiveInteger: width(nn, cusp) returns the width of the cusp=(a:c) for G(nn) where G is the congruence subgroup of SL_2(ZZ) connected to this package.

width: (PositiveInteger, Matrix Integer) -> PositiveInteger: width(nn, gamma) returns the width of the cusp=(a:c) for G(nn) where G is the congruence subgroup of SL_2(ZZ) connected to this package and gamma = matrix [[a,b],[c,d]].