QEtaModularLambdaAuxiliaryPackageΒΆ
qetamodlambdatool.spad line 91 [edit on github]
QEtaModularLambdaAuxiliaryPackage provides functions for working with the modular lambda function.
- computeReducedCusps: List Record(fdelta: PositiveInteger, fgamma: Matrix Integer, fred: Matrix Integer, ftriang: Matrix Integer) -> List Record(fdelta: PositiveInteger, fgamma: Matrix Integer, fred: Matrix Integer, ftriang: Matrix Integer)
- computeReducedCusps(cmats)removes entries that have an equivalent entry. Earlier entries survive.
- cuspDeltaMatrices: (PositiveInteger, PositiveInteger, Cusp) -> Record(fdelta: PositiveInteger, fgamma: Matrix Integer, fred: Matrix Integer, ftriang: Matrix Integer)
- cuspDeltaMatrices(delta,nn,x)returns cuspDeltaMatrices(- delta,- nn,cuspToMatrix(- x)$GAMMAN(- nn)).
- cuspDeltaMatrices: (PositiveInteger, PositiveInteger, Matrix Integer) -> Record(fdelta: PositiveInteger, fgamma: Matrix Integer, fred: Matrix Integer, ftriang: Matrix Integer)
- cuspDeltaMatrices(delta,nn,gamma)returns a record of 3 matrices, namely- gammaand splitMatrix(- gamma,- delta)$QETAAUX, i.e. the respective matrix $bargamma$ and $- T=- \mt{- u}{- v}{0}{- w}$ such that the expansion $- f(deltagammatau)$ is $- f(bargammabartau)$ where $bartau=frac{- utau+v}{- w}$. From the possible different options to select the entries $(1,2)$ and $(2,2)$ of $bargamma$, we prefer one that gives $- v=0$.
- equivalentCuspsByMatrices?: (Record(fdelta: PositiveInteger, fgamma: Matrix Integer, fred: Matrix Integer, ftriang: Matrix Integer), Record(fdelta: PositiveInteger, fgamma: Matrix Integer, fred: Matrix Integer, ftriang: Matrix Integer)) -> Boolean
- Let $ - h$ be a modular function for $ $Gamma(2N)$ such that $- h(- t)- =r(lambda(- t))$ for some rational function $- r$ and the modular lambda function $lambda$. If for two cusps $- x$ and $- y$ we have $gamma_x equiv gamma_y pmod{2}$ and $bargamma_x equiv bargamma_y pmod{2}$ (the first two entries in the CUSPDELTAMATRICES record and the third (triangular) matrix is equal, then we can conclude by the transformation formula for the modular lambda function that the $- q$-expansion of $- h(gamma_xtau)- =h(gamma_ytau)$ and $- h(deltagamma_xtau)- =h(deltagamma_ytau)$. So it is enough to only consider one of these cusps. Since further down, also the Jacobi theta functions are involved, we would require then even more conditions. Namely, that $- S_3(gamma_x)- =S_3(gamma_y)$ and $- S_3(bargamma_x)- =S_3(bargamma_y)$. Those equalities, however, are superfluous, since they follow immeditately from the conditions above and the definition of $- S_3$ at https://fungrim.org/entry/9bda2f/.
- hasEquivalentCuspByMatrices?: (Record(fdelta: PositiveInteger, fgamma: Matrix Integer, fred: Matrix Integer, ftriang: Matrix Integer), List Record(fdelta: PositiveInteger, fgamma: Matrix Integer, fred: Matrix Integer, ftriang: Matrix Integer)) -> Boolean
- Check whether the cusp has an equivalent counterpart, i.e. whether the expansions of $ - h$ (and $- H/F$) has the same expansion at both cusps.