Module: sage.modular.abvar.cuspidal_subgroup
Cuspidal subgroups of modular abelian varieties
Author: William Stein (2007-03, 2008-02)
We compute the cuspidal subgroup of
:
sage: A = J1(13) sage: C = A.cuspidal_subgroup(); C Finite subgroup with invariants [19, 19] over QQ of Abelian variety J1(13) of dimension 2 sage: C.gens() [[(1/19, 0, 0, 9/19)], [(0, 1/19, 1/19, 18/19)]] sage: C.order() 361 sage: C.invariants() [19, 19]
We compute the cuspidal subgroup of
:
sage: A = J0(54) sage: C = A.cuspidal_subgroup(); C Finite subgroup with invariants [3, 3, 3, 3, 3, 9] over QQ of Abelian variety J0(54) of dimension 4 sage: C.gens() [[(1/3, 0, 0, 0, 0, 1/3, 0, 2/3)], [(0, 1/3, 0, 0, 0, 2/3, 0, 1/3)], [(0, 0, 1/9, 1/9, 1/9, 1/9, 1/9, 2/9)], [(0, 0, 0, 1/3, 0, 1/3, 0, 0)], [(0, 0, 0, 0, 1/3, 1/3, 0, 1/3)], [(0, 0, 0, 0, 0, 0, 1/3, 2/3)]] sage: C.order() 2187 sage: C.invariants() [3, 3, 3, 3, 3, 9]
We compute the subgroup of the cuspidal subgroup generated by rational cusps.
sage: C = J0(54).rational_cusp_subgroup(); C Finite subgroup with invariants [3, 3, 9] over QQ of Abelian variety J0(54) of dimension 4 sage: C.gens() [[(1/3, 0, 0, 1/3, 2/3, 1/3, 0, 1/3)], [(0, 0, 1/9, 1/9, 7/9, 7/9, 1/9, 8/9)], [(0, 0, 0, 0, 0, 0, 1/3, 2/3)]] sage: C.order() 81 sage: C.invariants() [3, 3, 9]
This might not give us the exact rational torsion subgroup, since it might
be bigger than order
:
sage: J0(54).rational_torsion_subgroup().multiple_of_order() 243
TESTS:
sage: C = J0(54).cuspidal_subgroup() sage: loads(dumps(C)) == C True sage: D = J0(54).rational_cusp_subgroup() sage: loads(dumps(D)) == D True
Module-level Functions
c, N, data) |
Return True if the rational number c is a rational cusp of level N. This uses remarks in Glenn Steven's Ph.D. thesis.
Input:
sage: from sage.modular.abvar.cuspidal_subgroup import is_rational_cusp_gamma0 sage: N = 27 sage: data = [n for n in range(2,N) if gcd(n,N) == 1] sage: is_rational_cusp_gamma0(Cusp(1/3), N, data) False sage: is_rational_cusp_gamma0(Cusp(1), N, data) True sage: is_rational_cusp_gamma0(Cusp(oo), N, data) True sage: is_rational_cusp_gamma0(Cusp(2/9), N, data) False
Class: CuspidalSubgroup
sage: a = J0(65)[2] sage: t = a.cuspidal_subgroup() sage: t.order() 6
Functions: lattice
self) |
Returned cached tuple of vectors that define elements of the rational homology that generate this finite subgroup.
Output:
sage: J = J0(27) sage: G = J.cuspidal_subgroup() sage: G.lattice() Free module of degree 2 and rank 2 over Integer Ring Echelon basis matrix: [1/3 0] [ 0 1/3]
Test that the result is cached:
sage: G.lattice() is G.lattice() True
Special Functions: _repr_
self) |
String representation of the cuspidal subgroup.
sage: G = J0(27).cuspidal_subgroup() sage: G._repr_() 'Finite subgroup with invariants [3, 3] over QQ of Abelian variety J0(27) of dimension 1'
Class: CuspidalSubgroup_generic
Special Functions: _compute_lattice
self, [rational_only=False]) |
Return a list of vectors that define elements of the rational homology that generate this finite subgroup.
Input:
True
, only
use rational cusps.
sage: J = J0(37) sage: C = sage.modular.abvar.cuspidal_subgroup.CuspidalSubgroup(J) sage: C._compute_lattice() Free module of degree 4 and rank 4 over Integer Ring Echelon basis matrix: [ 1 0 0 0] [ 0 1 0 0] [ 0 0 1 0] [ 0 0 0 1/3] sage: J = J0(43) sage: C = sage.modular.abvar.cuspidal_subgroup.CuspidalSubgroup(J) sage: C._compute_lattice() Free module of degree 6 and rank 6 over Integer Ring Echelon basis matrix: [ 1 0 0 0 0 0] [ 0 1/7 0 6/7 0 5/7] [ 0 0 1 0 0 0] [ 0 0 0 1 0 0] [ 0 0 0 0 1 0] [ 0 0 0 0 0 1] sage: J = J0(22) sage: C = sage.modular.abvar.cuspidal_subgroup.CuspidalSubgroup(J) sage: C._compute_lattice() Free module of degree 4 and rank 4 over Integer Ring Echelon basis matrix: [1/5 1/5 4/5 0] [ 0 1 0 0] [ 0 0 1 0] [ 0 0 0 1/5] sage: J = J1(13) sage: C = sage.modular.abvar.cuspidal_subgroup.CuspidalSubgroup(J) sage: C._compute_lattice() Free module of degree 4 and rank 4 over Integer Ring Echelon basis matrix: [ 1/19 0 0 9/19] [ 0 1/19 1/19 18/19] [ 0 0 1 0] [ 0 0 0 1]
We compute with and without the optional rational_only
option.
sage: J = J0(27); G = sage.modular.abvar.cuspidal_subgroup.CuspidalSubgroup(J) sage: G._compute_lattice() Free module of degree 2 and rank 2 over Integer Ring Echelon basis matrix: [1/3 0] [ 0 1/3] sage: G._compute_lattice(rational_only=True) Free module of degree 2 and rank 2 over Integer Ring Echelon basis matrix: [1/3 0] [ 0 1]
Class: RationalCuspidalSubgroup
sage: a = J0(65)[2] sage: t = a.rational_cusp_subgroup() sage: t.order() 6
Functions: lattice
self) |
Return lattice that defines this group.
Output: lattice
sage: G = J0(27).rational_cusp_subgroup() sage: G.lattice() Free module of degree 2 and rank 2 over Integer Ring Echelon basis matrix: [1/3 0] [ 0 1]
Test that the result is cached.
sage: G.lattice() is G.lattice() True
Special Functions: _repr_
self) |
String representation of the cuspidal subgroup.
sage: G = J0(27).rational_cusp_subgroup() sage: G._repr_() 'Finite subgroup with invariants [3] over QQ of Abelian variety J0(27) of dimension 1'