def _image_of_abvar(self, A):
"""
Compute the image of the abelian variety `A` under this
morphism.
INPUT:
- ``A`` - an abelian variety
OUTPUT an abelian variety
EXAMPLES::
sage: t = J0(33).hecke_operator(2)
sage: t._image_of_abvar(J0(33).new_subvariety())
Abelian subvariety of dimension 1 of J0(33)
::
sage: t = J0(33).hecke_operator(3)
sage: A = J0(33)[0]
sage: B = t._image_of_abvar(A); B
Abelian subvariety of dimension 1 of J0(33)
sage: B == A
False
sage: A + B == J0(33).old_subvariety()
True
::
sage: J = J0(37) ; A, B = J.decomposition()
sage: J.projection(A)._image_of_abvar(A)
Abelian subvariety of dimension 1 of J0(37)
sage: J.projection(A)._image_of_abvar(B)
Abelian subvariety of dimension 0 of J0(37)
sage: J.projection(B)._image_of_abvar(A)
Abelian subvariety of dimension 0 of J0(37)
sage: J.projection(B)._image_of_abvar(B)
Abelian subvariety of dimension 1 of J0(37)
sage: J.projection(B)._image_of_abvar(J)
Abelian subvariety of dimension 1 of J0(37)
"""
D = self.domain()
C = self.codomain()
if A is D:
B = self.matrix()
else:
if not A.is_subvariety(D):
raise ValueError, "A must be an abelian subvariety of self."
# Write the vector space corresponding to A in terms of self's
# vector space, then take the image under self.
B = D.vector_space().coordinate_module(A.vector_space()).basis_matrix() * self.matrix()
V = (B * C.vector_space().basis_matrix()).row_module(QQ)
lattice = V.intersection(C.lattice())
base_field = C.base_field()
return abelian_variety.ModularAbelianVariety(C.groups(), lattice, base_field)
def AbelianVariety(X):
"""
Create the abelian variety corresponding to the given defining
data.
INPUT:
- ``X`` - an integer, string, newform, modsym space,
congruence subgroup or tuple of congruence subgroups
OUTPUT: a modular abelian variety
EXAMPLES::
sage: AbelianVariety(Gamma0(37))
Abelian variety J0(37) of dimension 2
sage: AbelianVariety('37a')
Newform abelian subvariety 37a of dimension 1 of J0(37)
sage: AbelianVariety(Newform('37a'))
Newform abelian subvariety 37a of dimension 1 of J0(37)
sage: AbelianVariety(ModularSymbols(37).cuspidal_submodule())
Abelian variety J0(37) of dimension 2
sage: AbelianVariety((Gamma0(37), Gamma0(11)))
Abelian variety J0(37) x J0(11) of dimension 3
sage: AbelianVariety(37)
Abelian variety J0(37) of dimension 2
sage: AbelianVariety([1,2,3])
Traceback (most recent call last):
...
TypeError: X must be an integer, string, newform, modsym space, congruence subgroup or tuple of congruence subgroups
"""
if isinstance(X, (int, long, Integer)):
X = Gamma0(X)
if is_CongruenceSubgroup(X):
X = X.modular_symbols().cuspidal_submodule()
elif isinstance(X, str):
from sage.modular.modform.constructor import Newform
f = Newform(X, names='a')
return ModularAbelianVariety_newform(f, internal_name=True)
elif isinstance(X, sage.modular.modform.element.Newform):
return ModularAbelianVariety_newform(X)
if is_ModularSymbolsSpace(X):
return abvar.ModularAbelianVariety_modsym(X)
if isinstance(X,
(tuple, list)) and all([is_CongruenceSubgroup(G)
for G in X]):
return abvar.ModularAbelianVariety(X)
raise TypeError(
"X must be an integer, string, newform, modsym space, congruence subgroup or tuple of congruence subgroups"
)