# 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

class sage.modular.abvar.cuspidal_subgroup.CuspidalSubgroup(abvar, field_of_definition=Rational Field)

EXAMPLES:

sage: a = J0(65)[2]
sage: t = a.cuspidal_subgroup()
sage: t.order()
6

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

class sage.modular.abvar.cuspidal_subgroup.CuspidalSubgroup_generic(abvar, field_of_definition=Rational Field)
class sage.modular.abvar.cuspidal_subgroup.RationalCuspSubgroup(abvar, field_of_definition=Rational Field)

EXAMPLES:

sage: a = J0(65)[2]
sage: t = a.rational_cusp_subgroup()
sage: t.order()
6

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

class sage.modular.abvar.cuspidal_subgroup.RationalCuspidalSubgroup(abvar, field_of_definition=Rational Field)

EXAMPLES:

sage: a = J0(65)[2]
sage: t = a.rational_cuspidal_subgroup()
sage: t.order()
6

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

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 cusp
• N - a positive integer
• data - 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