Cuspidal subgroups of modular abelian varieties¶
AUTHORS:
- William Stein (2007-03, 2008-02)
EXAMPLES: We compute the cuspidal subgroup of \(J_1(13)\):
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, 9/19, 9/19)], [(0, 1/19, 0, 9/19)]]
sage: C.order()
361
sage: C.invariants()
[19, 19]
We compute the cuspidal subgroup of \(J_0(54)\):
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 \(81\):
sage: J0(54).rational_torsion_subgroup().multiple_of_order()
243
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class
sage.modular.abvar.cuspidal_subgroup.CuspidalSubgroup(abvar, field_of_definition=Rational Field)¶ Bases:
sage.modular.abvar.cuspidal_subgroup.CuspidalSubgroup_genericEXAMPLES:
sage: a = J0(65)[2] sage: t = a.cuspidal_subgroup() sage: t.order() 6
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lattice()¶ Returned cached tuple of vectors that define elements of the rational homology that generate this finite subgroup.
OUTPUT:
tuple- cached
EXAMPLES:
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
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class
sage.modular.abvar.cuspidal_subgroup.CuspidalSubgroup_generic(abvar, field_of_definition=Rational Field)¶
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class
sage.modular.abvar.cuspidal_subgroup.RationalCuspSubgroup(abvar, field_of_definition=Rational Field)¶ Bases:
sage.modular.abvar.cuspidal_subgroup.CuspidalSubgroup_genericEXAMPLES:
sage: a = J0(65)[2] sage: t = a.rational_cusp_subgroup() sage: t.order() 6
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lattice()¶ Return lattice that defines this group.
OUTPUT: lattice
EXAMPLES:
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
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class
sage.modular.abvar.cuspidal_subgroup.RationalCuspidalSubgroup(abvar, field_of_definition=Rational Field)¶ Bases:
sage.modular.abvar.cuspidal_subgroup.CuspidalSubgroup_genericEXAMPLES:
sage: a = J0(65)[2] sage: t = a.rational_cuspidal_subgroup() sage: t.order() 6
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lattice()¶ Return lattice that defines this group.
OUTPUT: lattice
EXAMPLES:
sage: G = J0(27).rational_cuspidal_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
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sage.modular.abvar.cuspidal_subgroup.is_rational_cusp_gamma0(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:
c- a cuspN- a positive integerdata- the list [n for n in range(2,N) if gcd(n,N) == 1], which is passed in as a parameter purely for efficiency reasons.
EXAMPLES:
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