QEtaUserSupport(C, MODG)ΒΆ

qetaus.spad line 112 [edit on github]

QEtaUserSupport provides support functions for creating, manipulating, and finding eta-quotients and q-Pochhammer quotient relations and their dissection. modularity of an element of a specification ring.

minimalRootOfUnity: (List QEtaSpecificationExpressionMonomial, List Matrix Integer) -> PositiveInteger

minimalRootOfUnity(spexmons,gammas) returns the lcm of minimalRootOfUnity(spexmon,gammas) over all spexmon in spexmons.

minimalRootOfUnity: (QEtaSpecification, QEtaSpecification, PositiveInteger, NonNegativeInteger) -> PositiveInteger

minimalRootOfUnity(sspec,rspec,m,t) returns minimalRootOfUnity(sspec,rspec,m,t,cuspMatrices()$MODG).

minimalRootOfUnity: (QEtaSpecification, QEtaSpecification, PositiveInteger, NonNegativeInteger, List Matrix Integer) -> PositiveInteger

minimalRootOfUnity(sspec,rspec,m,t,gammas) returns the smallest positive integer n such that the $q$-expansion of the modular function corresponding to $F_{s,r,m,t}(gammatau)$ at all cusps of MODG given by gammas can be represented in $mathbb{Q}[xi]((q))$ where $xi$ is a primitive $n$-th root of unity. For the definition of $F_{s,r,m,t}(gammatau)$ see eqref{eq:F_{s,r,m,t}(tau)} and eqref{eq:F_{s,r,m,t}(gamma*tau)} in qeta.tex.

minimalRootOfUnity: (QEtaSpecificationExpression C, List Matrix Integer) -> PositiveInteger

minimalRootOfUnity(spex,gammas) returns minimalRootOfUnity(support(spex),gammas).

minimalRootOfUnity: (QEtaSpecificationExpressionMonomial, List Matrix Integer) -> PositiveInteger

minimalRootOfUnity(spexmon,gammas) returns minimalRootOfUnity(sspec,rspec,m,t,gammas), if spexmon is a modular function that can be represented by a dissection and a corresponding cofactor of the form (sspec,rspec,m,t)

minimalRootOfUnity: List QEtaSpecificationExpressionMonomial -> PositiveInteger

minimalRootOfUnity(spexmons) returns the lcm of minimalRootOfUnity(spexmon) over all spexmon in spexmons.

minimalRootOfUnity: QEtaSpecificationExpression C -> PositiveInteger

minimalRootOfUnity(spex) returns minimalRootOfUnity(support(spex)).

minimalRootOfUnity: QEtaSpecificationExpressionMonomial -> PositiveInteger

minimalRootOfUnity(spexmon) returns minimalRootOfUnity(sspec,rspec,m,t), if spexmon is a modular function that can be represented by a dissection and a corresponding cofactor of the form (sspec,rspec,m,t)

modular?: QEtaSpecificationExpression C -> Boolean

modular?(x) returns true if modular?(m)$MODG is true for every monomial m of etaExpression(x), i.e. x represents a modular function with respect to MODG

modular?: QEtaSpecificationExpressionMonomial -> Boolean

modular?(x) returns true if x represents a modular function wrt MODG. It corresponds to checking whether eqref{eq:F_s-r-m-t(tau)} is a modular function according to Theorem~ref{thm:RaduConditions} and Theorem~ref{thm:condition-co-eta-quotient-gamma1} in qeta.tex. Currently, if this function returns true, it is guaranteed that x is modular? wrt. MODG. If this functions returns false, x can still be modular, but this function is currently unable to detect it. If qSpecificatioon(x)=1 and gfSpecification(x)=1, then it is equivalent to modular?(etaSpecification(x)). gfSpecification is different from 1, the dissections of the generating function must appear with all other dissections belonging to the same orbit.

modularFunctionInfinity: QEtaSpecificationExpression C -> ModularFunctionQSeriesInfinity C

modularEtaQuotientInfinity(spex) returns the series expansion of the modular function given by the specification spex at the cusp infinity. It basically forms the C-linear combination of all the modular functions corresponding to its terms.

modularFunctionInfinity: QEtaSpecificationExpressionMonomial -> ModularFunctionQSeriesInfinity C

modularFunctionInfinity(spexmon) returns the series expansion of the modular function given by the specification spexmon at the cusp infinity. It checks whether the specification is indeed modular for the group respective by MODG.

qetaGrades: (List QEtaSpecificationExpressionMonomial, List Matrix Integer) -> List List Integer

qetaGrades(spexmons,gammas) returns qetaGrades(spexmon,gammas) for every spexmon in spexmons.

qetaGrades: (QEtaSpecificationExpression C, List Matrix Integer) -> List List Integer

qetaGrades(spex,gammas) returns qetaGrades(support(spex),gammas).

qetaGrades: (QEtaSpecificationExpressionMonomial, List Matrix Integer) -> List Integer

qetaGrades(spexmon,gammas) returns the grades of the modular function represented by spexmon at the cusps of MODG given through gammas. Not every specification is allowed, but only those that correspond to a dissection of a q-Pochhammer quotient, that is made modular by completing the orbit for the dissection and multiplying by a respective eta-quotient. In short, failure?(specificationMonomial(spexmon)$YMEQ(MODG)) must return false.

qetaGrades: List QEtaSpecificationExpressionMonomial -> List List Integer

qetaGrades(x) returns qetaGrades(x,cuspMatrices()$MODG)

qetaGrades: QEtaSpecificationExpression C -> List List Integer

qetaGrades(x) returns qetaGrades(x,cuspMatrices()$MODG)

qetaGrades: QEtaSpecificationExpressionMonomial -> List Integer

qetaGrades(x) returns qetaGrades(x,cuspMatrices()$MODG)