The set \(\mathbb{P}^1(\QQ)\) of cusps#
EXAMPLES:
sage: Cusps
Set P^1(QQ) of all cusps
>>> from sage.all import *
>>> Cusps
Set P^1(QQ) of all cusps
sage: Cusp(oo)
Infinity
>>> from sage.all import *
>>> Cusp(oo)
Infinity
- class sage.modular.cusps.Cusp(a, b=None, parent=None, check=True)[source]#
Bases:
ElementA cusp.
A cusp is either a rational number or infinity, i.e., an element of the projective line over Q. A Cusp is stored as a pair (a,b), where gcd(a,b)=1 and a,b are of type Integer.
EXAMPLES:
sage: a = Cusp(2/3); b = Cusp(oo) sage: a.parent() Set P^1(QQ) of all cusps sage: a.parent() is b.parent() True
>>> from sage.all import * >>> a = Cusp(Integer(2)/Integer(3)); b = Cusp(oo) >>> a.parent() Set P^1(QQ) of all cusps >>> a.parent() is b.parent() True
- apply(g)[source]#
Return g(self), where g=[a,b,c,d] is a list of length 4, which we view as a linear fractional transformation.
EXAMPLES: Apply the identity matrix:
sage: Cusp(0).apply([1,0,0,1]) 0 sage: Cusp(0).apply([0,-1,1,0]) Infinity sage: Cusp(0).apply([1,-3,0,1]) -3
>>> from sage.all import * >>> Cusp(Integer(0)).apply([Integer(1),Integer(0),Integer(0),Integer(1)]) 0 >>> Cusp(Integer(0)).apply([Integer(0),-Integer(1),Integer(1),Integer(0)]) Infinity >>> Cusp(Integer(0)).apply([Integer(1),-Integer(3),Integer(0),Integer(1)]) -3
- denominator()[source]#
Return the denominator of the cusp a/b.
EXAMPLES:
sage: x = Cusp(6,9); x 2/3 sage: x.denominator() 3 sage: Cusp(oo).denominator() 0 sage: Cusp(-5/10).denominator() 2
>>> from sage.all import * >>> x = Cusp(Integer(6),Integer(9)); x 2/3 >>> x.denominator() 3 >>> Cusp(oo).denominator() 0 >>> Cusp(-Integer(5)/Integer(10)).denominator() 2
- galois_action(t, N)[source]#
Suppose this cusp is \(\alpha\), \(G\) a congruence subgroup of level \(N\) and \(\sigma\) is the automorphism in the Galois group of \(\QQ(\zeta_N)/\QQ\) that sends \(\zeta_N\) to \(\zeta_N^t\). Then this function computes a cusp \(\beta\) such that \(\sigma([\alpha]) = [\beta]\), where \([\alpha]\) is the equivalence class of \(\alpha\) modulo \(G\).
This code only needs as input the level and not the group since the action of Galois for a congruence group \(G\) of level \(N\) is compatible with the action of the full congruence group \(\Gamma(N)\).
INPUT:
\(t\) – integer that is coprime to N
\(N\) – positive integer (level)
OUTPUT:
a cusp
Warning
In some cases \(N\) must fit in a long long, i.e., there are cases where this algorithm isn’t fully implemented.
Note
Modular curves can have multiple non-isomorphic models over \(\QQ\). The action of Galois depends on such a model. The model over \(\QQ\) of \(X(G)\) used here is the model where the function field \(\QQ(X(G))\) is given by the functions whose Fourier expansion at \(\infty\) have their coefficients in \(\QQ\). For \(X(N):=X(\Gamma(N))\) the corresponding moduli interpretation over \(\ZZ[1/N]\) is that \(X(N)\) parametrizes pairs \((E,a)\) where \(E\) is a (generalized) elliptic curve and \(a: \ZZ / N\ZZ \times \mu_N \to E\) is a closed immersion such that the Weil pairing of \(a(1,1)\) and \(a(0,\zeta_N)\) is \(\zeta_N\). In this parameterisation the point \(z \in H\) corresponds to the pair \((E_z,a_z)\) with \(E_z=\CC/(z \ZZ+\ZZ)\) and \(a_z: \ZZ / N\ZZ \times \mu_N \to E\) given by \(a_z(1,1) = z/N\) and \(a_z(0,\zeta_N) = 1/N\). Similarly \(X_1(N):=X(\Gamma_1(N))\) parametrizes pairs \((E,a)\) where \(a: \mu_N \to E\) is a closed immersion.
EXAMPLES:
sage: Cusp(1/10).galois_action(3, 50) 1/170 sage: Cusp(oo).galois_action(3, 50) Infinity sage: c = Cusp(0).galois_action(3, 50); c 50/17 sage: Gamma0(50).reduce_cusp(c) 0
>>> from sage.all import * >>> Cusp(Integer(1)/Integer(10)).galois_action(Integer(3), Integer(50)) 1/170 >>> Cusp(oo).galois_action(Integer(3), Integer(50)) Infinity >>> c = Cusp(Integer(0)).galois_action(Integer(3), Integer(50)); c 50/17 >>> Gamma0(Integer(50)).reduce_cusp(c) 0
Here we compute the permutations of the action for t=3 on cusps for Gamma0(50).
sage: N = 50; t=3; G = Gamma0(N); C = G.cusps() sage: cl = lambda z: exists(C, lambda y:y.is_gamma0_equiv(z, N))[1] sage: for i in range(5): ....: print((i, t^i)) ....: print([cl(alpha.galois_action(t^i,N)) for alpha in C]) (0, 1) [0, 1/25, 1/10, 1/5, 3/10, 2/5, 1/2, 3/5, 7/10, 4/5, 9/10, Infinity] (1, 3) [0, 1/25, 7/10, 2/5, 1/10, 4/5, 1/2, 1/5, 9/10, 3/5, 3/10, Infinity] (2, 9) [0, 1/25, 9/10, 4/5, 7/10, 3/5, 1/2, 2/5, 3/10, 1/5, 1/10, Infinity] (3, 27) [0, 1/25, 3/10, 3/5, 9/10, 1/5, 1/2, 4/5, 1/10, 2/5, 7/10, Infinity] (4, 81) [0, 1/25, 1/10, 1/5, 3/10, 2/5, 1/2, 3/5, 7/10, 4/5, 9/10, Infinity]
>>> from sage.all import * >>> N = Integer(50); t=Integer(3); G = Gamma0(N); C = G.cusps() >>> cl = lambda z: exists(C, lambda y:y.is_gamma0_equiv(z, N))[Integer(1)] >>> for i in range(Integer(5)): ... print((i, t**i)) ... print([cl(alpha.galois_action(t**i,N)) for alpha in C]) (0, 1) [0, 1/25, 1/10, 1/5, 3/10, 2/5, 1/2, 3/5, 7/10, 4/5, 9/10, Infinity] (1, 3) [0, 1/25, 7/10, 2/5, 1/10, 4/5, 1/2, 1/5, 9/10, 3/5, 3/10, Infinity] (2, 9) [0, 1/25, 9/10, 4/5, 7/10, 3/5, 1/2, 2/5, 3/10, 1/5, 1/10, Infinity] (3, 27) [0, 1/25, 3/10, 3/5, 9/10, 1/5, 1/2, 4/5, 1/10, 2/5, 7/10, Infinity] (4, 81) [0, 1/25, 1/10, 1/5, 3/10, 2/5, 1/2, 3/5, 7/10, 4/5, 9/10, Infinity]
REFERENCES:
Section 1.3 of Glenn Stevens, “Arithmetic on Modular Curves”
There is a long comment about our algorithm in the source code for this function.
AUTHORS:
William Stein, 2009-04-18
- is_gamma0_equiv(other, N, transformation=None)[source]#
Return whether self and other are equivalent modulo the action of \(\Gamma_0(N)\) via linear fractional transformations.
INPUT:
other– CuspN– an integer (specifies the group Gamma_0(N))transformation– None (default) or either the string ‘matrix’ or ‘corner’. If ‘matrix’, it also returns a matrix in Gamma_0(N) that sends self to other. The matrix is chosen such that the lower left entry is as small as possible in absolute value. If ‘corner’ (or True for backwards compatibility), it returns only the upper left entry of such a matrix.
OUTPUT:
a boolean – True if self and other are equivalent
a matrix or an integer- returned only if transformation is ‘matrix’ or ‘corner’, respectively.
EXAMPLES:
sage: x = Cusp(2,3) sage: y = Cusp(4,5) sage: x.is_gamma0_equiv(y, 2) True sage: _, ga = x.is_gamma0_equiv(y, 2, 'matrix'); ga [-1 2] [-2 3] sage: x.is_gamma0_equiv(y, 3) False sage: x.is_gamma0_equiv(y, 3, 'matrix') (False, None) sage: Cusp(1/2).is_gamma0_equiv(1/3,11,'corner') (True, 19) sage: Cusp(1,0) Infinity sage: z = Cusp(1,0) sage: x.is_gamma0_equiv(z, 3, 'matrix') ( [-1 1] True, [-3 2] )
>>> from sage.all import * >>> x = Cusp(Integer(2),Integer(3)) >>> y = Cusp(Integer(4),Integer(5)) >>> x.is_gamma0_equiv(y, Integer(2)) True >>> _, ga = x.is_gamma0_equiv(y, Integer(2), 'matrix'); ga [-1 2] [-2 3] >>> x.is_gamma0_equiv(y, Integer(3)) False >>> x.is_gamma0_equiv(y, Integer(3), 'matrix') (False, None) >>> Cusp(Integer(1)/Integer(2)).is_gamma0_equiv(Integer(1)/Integer(3),Integer(11),'corner') (True, 19) >>> Cusp(Integer(1),Integer(0)) Infinity >>> z = Cusp(Integer(1),Integer(0)) >>> x.is_gamma0_equiv(z, Integer(3), 'matrix') ( [-1 1] True, [-3 2] )
ALGORITHM: See Proposition 2.2.3 of Cremona’s book ‘Algorithms for Modular Elliptic Curves’, or Prop 2.27 of Stein’s Ph.D. thesis.
- is_gamma1_equiv(other, N)[source]#
Return whether self and other are equivalent modulo the action of Gamma_1(N) via linear fractional transformations.
INPUT:
other– CuspN– an integer (specifies the group Gamma_1(N))
OUTPUT:
bool– True if self and other are equivalentint– 0, 1 or -1, gives further information about the equivalence: If the two cusps are u1/v1 and u2/v2, then they are equivalent if and only if v1 = v2 (mod N) and u1 = u2 (mod gcd(v1,N)) or v1 = -v2 (mod N) and u1 = -u2 (mod gcd(v1,N)) The sign is +1 for the first and -1 for the second. If the two cusps are not equivalent then 0 is returned.
EXAMPLES:
sage: x = Cusp(2,3) sage: y = Cusp(4,5) sage: x.is_gamma1_equiv(y,2) (True, 1) sage: x.is_gamma1_equiv(y,3) (False, 0) sage: z = Cusp(QQ(x) + 10) sage: x.is_gamma1_equiv(z,10) (True, 1) sage: z = Cusp(1,0) sage: x.is_gamma1_equiv(z, 3) (True, -1) sage: Cusp(0).is_gamma1_equiv(oo, 1) (True, 1) sage: Cusp(0).is_gamma1_equiv(oo, 3) (False, 0)
>>> from sage.all import * >>> x = Cusp(Integer(2),Integer(3)) >>> y = Cusp(Integer(4),Integer(5)) >>> x.is_gamma1_equiv(y,Integer(2)) (True, 1) >>> x.is_gamma1_equiv(y,Integer(3)) (False, 0) >>> z = Cusp(QQ(x) + Integer(10)) >>> x.is_gamma1_equiv(z,Integer(10)) (True, 1) >>> z = Cusp(Integer(1),Integer(0)) >>> x.is_gamma1_equiv(z, Integer(3)) (True, -1) >>> Cusp(Integer(0)).is_gamma1_equiv(oo, Integer(1)) (True, 1) >>> Cusp(Integer(0)).is_gamma1_equiv(oo, Integer(3)) (False, 0)
- is_gamma_h_equiv(other, G)[source]#
Return a pair (b, t), where b is True or False as self and other are equivalent under the action of G, and t is 1 or -1, as described below.
Two cusps \(u1/v1\) and \(u2/v2\) are equivalent modulo Gamma_H(N) if and only if \(v1 = h*v2 (\mathrm{mod} N)\) and \(u1 = h^{(-1)}*u2 (\mathrm{mod} gcd(v1,N))\) or \(v1 = -h*v2 (mod N)\) and \(u1 = -h^{(-1)}*u2 (\mathrm{mod} gcd(v1,N))\) for some \(h \in H\). Then t is 1 or -1 as c and c’ fall into the first or second case, respectively.
INPUT:
other– CuspG– a congruence subgroup Gamma_H(N)
OUTPUT:
bool– True if self and other are equivalentint– -1, 0, 1; extra info
EXAMPLES:
sage: # needs sage.libs.pari sage: x = Cusp(2,3) sage: y = Cusp(4,5) sage: x.is_gamma_h_equiv(y,GammaH(13,[2])) (True, 1) sage: x.is_gamma_h_equiv(y,GammaH(13,[5])) (False, 0) sage: x.is_gamma_h_equiv(y,GammaH(5,[])) (False, 0) sage: x.is_gamma_h_equiv(y,GammaH(23,[4])) (True, -1)
>>> from sage.all import * >>> # needs sage.libs.pari >>> x = Cusp(Integer(2),Integer(3)) >>> y = Cusp(Integer(4),Integer(5)) >>> x.is_gamma_h_equiv(y,GammaH(Integer(13),[Integer(2)])) (True, 1) >>> x.is_gamma_h_equiv(y,GammaH(Integer(13),[Integer(5)])) (False, 0) >>> x.is_gamma_h_equiv(y,GammaH(Integer(5),[])) (False, 0) >>> x.is_gamma_h_equiv(y,GammaH(Integer(23),[Integer(4)])) (True, -1)
Enumerating the cusps for a space of modular symbols uses this function.
sage: # needs sage.libs.pari sage: G = GammaH(25,[6]); M = G.modular_symbols(); M Modular Symbols space of dimension 11 for Congruence Subgroup Gamma_H(25) with H generated by [6] of weight 2 with sign 0 over Rational Field sage: M.cusps() [8/25, 1/3, 6/25, 1/4, 1/15, -7/15, 7/15, 4/15, 1/20, 3/20, 7/20, 9/20] sage: len(M.cusps()) 12
>>> from sage.all import * >>> # needs sage.libs.pari >>> G = GammaH(Integer(25),[Integer(6)]); M = G.modular_symbols(); M Modular Symbols space of dimension 11 for Congruence Subgroup Gamma_H(25) with H generated by [6] of weight 2 with sign 0 over Rational Field >>> M.cusps() [8/25, 1/3, 6/25, 1/4, 1/15, -7/15, 7/15, 4/15, 1/20, 3/20, 7/20, 9/20] >>> len(M.cusps()) 12
This is always one more than the associated space of weight 2 Eisenstein series.
sage: # needs sage.libs.pari sage: G.dimension_eis(2) 11 sage: M.cuspidal_subspace() Modular Symbols subspace of dimension 0 of Modular Symbols space of dimension 11 for Congruence Subgroup Gamma_H(25) with H generated by [6] of weight 2 with sign 0 over Rational Field sage: G.dimension_cusp_forms(2) 0
>>> from sage.all import * >>> # needs sage.libs.pari >>> G.dimension_eis(Integer(2)) 11 >>> M.cuspidal_subspace() Modular Symbols subspace of dimension 0 of Modular Symbols space of dimension 11 for Congruence Subgroup Gamma_H(25) with H generated by [6] of weight 2 with sign 0 over Rational Field >>> G.dimension_cusp_forms(Integer(2)) 0
- is_infinity()[source]#
Returns True if this is the cusp infinity.
EXAMPLES:
sage: Cusp(3/5).is_infinity() False sage: Cusp(1,0).is_infinity() True sage: Cusp(0,1).is_infinity() False
>>> from sage.all import * >>> Cusp(Integer(3)/Integer(5)).is_infinity() False >>> Cusp(Integer(1),Integer(0)).is_infinity() True >>> Cusp(Integer(0),Integer(1)).is_infinity() False
- numerator()[source]#
Return the numerator of the cusp a/b.
EXAMPLES:
sage: x = Cusp(6,9); x 2/3 sage: x.numerator() 2 sage: Cusp(oo).numerator() 1 sage: Cusp(-5/10).numerator() -1
>>> from sage.all import * >>> x = Cusp(Integer(6),Integer(9)); x 2/3 >>> x.numerator() 2 >>> Cusp(oo).numerator() 1 >>> Cusp(-Integer(5)/Integer(10)).numerator() -1