CongruenceSubgroup¶

CongruenceSubgroup is the category of congruence subgroups $Gamma$ of $L2Z such as $Gamma_0(N) and $Gamma_1(N)$.

cuspOf: Matrix Integer -> Cusp: cuspOf(mat) returns the cusp represented by the transformation matrix mat. It returns cusp(mat(1,1),mat(2,1)). cuspOf is a left-inverse of cuspToMatrix, i.e. cuspOf(cuspToMatrix(c))=c.

cusps: () -> List Cusp: If $Gamma$ denotes this domain, then cusps() returns representatives for all the (inequivalent) cusps for GAMMA 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()].

cuspToMatrix: Cusp -> Matrix Integer: For cusp=(a:c), cuspToMatrix(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).

dimensionOfCuspForms: Integer -> NonNegativeInteger: dimensionOfCuspForms(w) computes the dimension of $S_w(Gamma)$. See https://www.wstein.org/books/modform/modform/dimension_formulas.html

dimensionOfEisensteinSubspace: Integer -> NonNegativeInteger: dimensionOfEisensteinSubspace(w) computes the dimension of $E_w(Gamma)$, the Eisenstein subspace of modular forms. See https://www.wstein.org/books/modform/modform/dimension_formulas.html

dimensionOfModularForms: Integer -> NonNegativeInteger: dimensionOfModularForms(w) computes the dimension of $M_w(Gamma)$. See https://www.wstein.org/books/modform/modform/dimension_formulas.html

doubleCosetRepresentatives: () -> List Matrix Integer: doubleCosetRepresentatives() returns a list of double coset representatives of $Gamma backslash SL_2(ZZ) / SL_2(ZZ)_infty$ where $SL_2(ZZ)_infty$ are matrices of the form [[1,h],[0,1]] (or [[-1,h],[0,-1]] with h being an integer. Note that cusps()=[cusp(x(1,1),x(2,1)) for x in doubleCosetRepresentatives()].

equivalentCusps?: (Cusp, Cusp) -> Boolean: equivalentCusps?(cusp1, cusp2) returns true iff the cusp cusp1=(a:c) is equivalent to cusp2=(u:w) modulo the action of this domain.

genus: () -> NonNegativeInteger: genus() returns the genus of the congruence subgroup $Gamma$. For $Gamma_0(nn)$, it corresponds to the series https://oeis.org/A001617 . cite[p.~25]{Shimura_ArithmeticTheory_1994} and Proposition 1.40. See also https://www.wstein.org/books/modform/modform/dimension_formulas.html See Section “Modular Formas for $Gamma_1(N)$” in cite{Stein_ModularForms_2007}. level: () -> PP level() returns the level of the congruence subgroup.

index: () -> PositiveInteger: If $Gamma$ denotes this domain, then index() computes the index of $Gamma$ in SL2Z.

member?: Matrix Integer -> Boolean: member?(mat) returns true if mat is an element of $Gamma$. It is assumed that determinant(mat)=1.

normalizeCusp: Cusp -> Cusp: normalizeCusp(cusp) normalizes the cusp to another (equivalent modulo $Gamma$ cusp (u:w) such that if cusp is equivalent to infinity, then infinity() is returned; if cusp is equivalent to (0:1), (0:1) is returned. Otherwise the numerator and denominator are positive and small and coprime.

nu2: () -> Integer: nu2() returns the number of $Gamma$ inequivalent elliptic points of order 2. See cite[p.~25]{Shimura_ArithmeticTheory_1994} and Proposition 1.40 and mu_{1,3}(n) of https://www.wstein.org/books/modform/modform/dimension_formulas.html.

nu3: () -> Integer: nu3() returns the number of $Gamma$ inequivalent elliptic points of order 3. See See cite[p.~25]{Shimura_ArithmeticTheory_1994} and Proposition 1.40 and mu_{1,3}(n) of https://www.wstein.org/books/modform/modform/dimension_formulas.html.

numberOfCusps: () -> PositiveInteger: numberOfCusps() returns the number of cusps of $Gamma$. cite[p.~25]{Shimura_ArithmeticTheory_1994} and Proposition 1.40 and also https://www.wstein.org/books/modform/modform/dimension_formulas.html#id2.

projectiveIndex: () -> PositiveInteger: projectivIndex() computes the index of the image of $Gamma$ in PSL2Z.

rightCosetRepresentatives: () -> List Matrix Integer: rightCosetRepresentatives() returns a list of right coset representatives of $Gamma backslash SL_2(ZZ)$.

width: Cusp -> PositiveInteger: If GAMMA denotes this domain, then width(cusp) returns the width of the cusp=(a:c) of GAMMA.

width: Matrix Integer -> PositiveInteger: width(gamma)=width(cuspOf(gamma)).