QEtaQuotientSpecifications4ti2 QMOD¶
qetaqmspecs.spad line 234 [edit on github]
- QMOD: QEtaModularCategory 
QEtaQuotientSpecificationss4ti2 computes a monoid basis of (specifications for) (generalized) eta-quotients with a pole at most at infinity.
- etaQuotientInfinitySpecifications: (PositiveInteger, List List Integer, Integer) -> List QEtaSpecification
- etaQuotientInfinitySpecifications(nn,idxs,grd)returns specifications for generalized eta-quotients with a pole of pole-order- grd>0at infinity and no pole at any other cusp.
- etaQuotientMonoidInfinitySpecifications: (PositiveInteger, List List Integer) -> List QEtaSpecification
- etaQuotientMonoidInfinitySpecifications(nn,idxs)returns monoidGenerators etaQuotientMonoidInfinityZSpecifications(- nn,- idxs).
- etaQuotientMonoidInfinitySpecifications: (PositiveInteger, List List Integer, String) -> List QEtaSpecification
- etaQuotientMonoidInfinitySpecifications(nn,idxs,basedir)reads the file “etaQuotientMonoidInfinitySpecifications.input” in directory given by dir:=concat[- basedir,- "/",string(- nn)] if it exists and interprets it as specifications of level- nnor returns etaQuotientMonoidInfinitySpecifications(- nn,- idxs). The return value will be saved in the above mentione file if it was not already there. WARNING: The function blindly trusts the contents of an existing file in the respective place. The user is responsible for reading from or storing in a directory that corresponds to the- idxsand QMOD parameters.
- etaQuotientMonoidInfinityZSpecifications: (PositiveInteger, List List Integer) -> Record(zinhom: List QEtaSpecification, zhom: List QEtaSpecification, zfree: List QEtaSpecification)
- etaQuotientMonoidInfinityZSpecifications(nn,idxs)returns etaQuotientMonoidZSpecifications(- nn,- idxs,[infinity()]).
- etaQuotientMonoidNoPolesSolutions: (PositiveInteger, List List Integer, List Cusp) -> Record(zinhom: List Vector Integer, zhom: List Vector Integer, zfree: List Vector Integer)
- etaQuotientMonoidNoPolesSolutions(nn,idxs,nopolesat)returns etaQuotientMonoidSolutions(- nn,- idxs,[],- nopolesat).
- etaQuotientMonoidNoPolesZSpecifications: (PositiveInteger, List List Integer, List Cusp) -> Record(zinhom: List QEtaSpecification, zhom: List QEtaSpecification, zfree: List QEtaSpecification)
- etaQuotientMonoidNoPolesZSpecifications(nn,idxs,nopolesat)returns etaQuotientMonoidZSpecifications(- nn,- idxs,[],- nopolesat), but sorted via sortByComponentGrade(- x,polesat) for polesat- :=[- cfor- cin cusps(- nn)$QMOD | not member?(- c,- nopolesat)].
- etaQuotientMonoidSolutions: (PositiveInteger, List List Integer, List Cusp) -> Record(zinhom: List Vector Integer, zhom: List Vector Integer, zfree: List Vector Integer)
- etaQuotientMonoidSolutions(nn,idxs,polesat)returns etaQuotientMonoidSolutions(- nn,- idxs,[],nopolesat) = etaQuotientMonoidNoPolesSolutions(- nn,- idxs,nopolesat) for nopolesat- :=[- cfor- cin cusps(- nn)$QMOD | not member?(- c,- polesat)].
- etaQuotientMonoidSolutions: (PositiveInteger, List List Integer, List Cusp, List Cusp) -> Record(zinhom: List Vector Integer, zhom: List Vector Integer, zfree: List Vector Integer)
- etaQuotientMonoidSolutions(nn,idxs,polesat,nopolesat)returns etaQuotientMonoidSolutions(- nn,- idxs,- polesat,rels,ords,- nopolesat) for rels:=[- -1for- cin- polesat] and ords:=[0 for- cin- polesat].
- etaQuotientMonoidSolutions: (PositiveInteger, List List Integer, List Cusp, List Integer, List Integer, List Cusp) -> Record(zinhom: List Vector Integer, zhom: List Vector Integer, zfree: List Vector Integer)
- etaQuotientMonoidSolutions(nn,idxs,polesat,rels,ords,nopolesat)returns specifications of (generalized) eta-quotients that (together with 1) generate the (multiplicative) monoid of all (generalized) eta-quotients (corresponding to the indices given by- idxs) that are modular functions- wrt. QMOD and have no poles at the cusps given by- nopolesatand for all cusps in- polesatthe expansion order fulfills the relation at the respective cusp. #polesat>=#rels and #polesat>=#ords. Missing values are assumed to be 0. Entries in- relscan be 1,0,- -1and stand for- >=, =, and- <=, respectively. The indices of the eta-functions that may appear is given by- idxs. It returns zsolve(etaQuotientMonoidSystem(- nn,- idxs,- polesat,- rels,- ords,- nopolesat)$QMOD)$- X4ti2. Note that the union of poleat and- nopolesatcan be a proper subset of all cusps given by cusps(- nn).
- etaQuotientMonoidZSpecifications: (PositiveInteger, List List Integer, List Cusp) -> Record(zinhom: List QEtaSpecification, zhom: List QEtaSpecification, zfree: List QEtaSpecification)
- etaQuotientMonoidZSpecifications(nn,idxs,polesat)returns z=etaQuotientMonoidZSpecifications(- nn,- idxs,[],nopolesat) for nopolesat- :=[- cfor- cin cusps(- nn)$QMOD | not member?(- c,- polesat)], but sorted via sortByComponentGrade(- z,- polesat).
- etaQuotientMonoidZSpecifications: (PositiveInteger, List List Integer, List Cusp, List Cusp) -> Record(zinhom: List QEtaSpecification, zhom: List QEtaSpecification, zfree: List QEtaSpecification)
- etaQuotientMonoidZSpecifications(nn,idxs,polesat,nopolesat)returns etaQuotientMonoidZSpecifications(- nn,- idxs,- polesat,rels,ords,- nopolesat) for rels:=[- -1for- cin- polesat] and ords:=[0 for- cin- polesat].
- etaQuotientMonoidZSpecifications: (PositiveInteger, List List Integer, List Cusp, List Integer, List Integer, List Cusp) -> Record(zinhom: List QEtaSpecification, zhom: List QEtaSpecification, zfree: List QEtaSpecification)
- etaQuotientMonoidZSpecifications(nn,idxs,polesat,rels,ords,nopolesat)returns specifications of (generalized) eta-quotients that (together with 1) generate the (multiplicative) monoid of all (generalized) eta-quotients (corresponding to the indices given by- idxs) that are modular functions- wrt. QMOD and have no poles at the cusps given by- nopolesatand for all cusps in- polesatthe expansion order fulfills the relation at the respective cusp. #polesat>=#rels and #polesat>=#ords. Missing values are assumed to be 0. Entries in- relscan be 1,0,- -1and stand for- >=, =, and- <=, respectively. The indices of the eta-functions that may appear is given by- idxs. Note that the union of poleat and- nopolesatcan be a proper subset of all cusps given by cusps(- nn).
- etaQuotientZSpecifications: (PositiveInteger, List List Integer, List Cusp, Integer) -> Record(zinhom: List QEtaSpecification, zhom: List QEtaSpecification, zfree: List QEtaSpecification)
- etaQuotientZSpecifications(nn,idxs,polesat,w)returns specifications of (generalized) eta-quotients- f(corresponding to the indices given by- idxs) that are modular- wrt. QMOD of weight- wand have poles at most at the cusps given by- polesat, i.e.- f(matrix[[a,- b],[- c,- d]]*tau)=(c*tau+d)^w*f(tau).
- withProperPoles: (Record(zinhom: List QEtaSpecification, zhom: List QEtaSpecification, zfree: List QEtaSpecification), List Cusp) -> List QEtaSpecification
- withProperPoles(zspecs,polesat)tries to compute specifications from ZSPECS that have a proper pole at all cusps in- polesat. The result is sorted via sortByComponentGrade(res,- polesat) from QEtaQuotientSortPackage(QMOD). The empty list is returned if no such specification can be found. It computes monomials with non-negative exponents for the elements in zhom and integer exponents for the elements in zfree such that the product is a specification that has a proper pole at any cusp of- polesat. We need at least one such element for the- tin the multi-samba algorithm.