"""
Abelian varieties attached to newforms
TESTS::
sage: A = AbelianVariety('23a')
sage: loads(dumps(A)) == A
True
"""
from sage.databases.cremona import cremona_letter_code
from sage.rings.all import QQ, ZZ
from sage.modular.modform.element import Newform
from sage.modular.arithgroup.all import is_Gamma0, is_Gamma1, is_GammaH
from abvar import ModularAbelianVariety_modsym_abstract
import homspace
class ModularAbelianVariety_newform(ModularAbelianVariety_modsym_abstract):
"""
A modular abelian variety attached to a specific newform.
"""
def __init__(self, f, internal_name=False):
"""
Create the modular abelian variety `A_f` attached to the
newform `f`.
INPUT:
f -- a newform
EXAMPLES::
sage: f = CuspForms(37).newforms('a')[0]
sage: f.abelian_variety()
Newform abelian subvariety 37a of dimension 1 of J0(37)
"""
if not isinstance(f, Newform):
raise TypeError, "f must be a newform"
self.__f = f
self._is_hecke_stable = True
K = f.qexp().base_ring()
if K == QQ:
variable_name = None
else:
variable_name = K.variable_name()
self.__named_newforms = { variable_name: self.__f }
if not internal_name:
self.__named_newforms[None] = self.__f
ModularAbelianVariety_modsym_abstract.__init__(self, (f.group(),), QQ,
is_simple=True, newform_level = (f.level(), f.group()),
isogeny_number=f.number(), number=0)
def _modular_symbols(self,sign=0):
"""
EXAMPLES::
sage: f = CuspForms(52).newforms('a')[0]
sage: A = f.abelian_variety()
sage: A._modular_symbols()
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 15 for Gamma_0(52) of weight 2 with sign 0 over Rational Field
sage: A._modular_symbols(1)
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 10 for Gamma_0(52) of weight 2 with sign 1 over Rational Field
sage: A._modular_symbols(-1)
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 5 for Gamma_0(52) of weight 2 with sign -1 over Rational Field
"""
return self.__f.modular_symbols(sign=sign)
def newform(self, names=None):
r"""
Return the newform that this modular abelian variety is attached to.
EXAMPLES::
sage: f = Newform('37a')
sage: A = f.abelian_variety()
sage: A.newform()
q - 2*q^2 - 3*q^3 + 2*q^4 - 2*q^5 + O(q^6)
sage: A.newform() is f
True
If the a variable name has not been specified, we must specify one::
sage: A = AbelianVariety('67b')
sage: A.newform()
Traceback (most recent call last):
...
TypeError: You must specify the name of the generator.
sage: A.newform('alpha')
q + alpha*q^2 + (-alpha - 3)*q^3 + (-3*alpha - 3)*q^4 - 3*q^5 + O(q^6)
If the eigenform is actually over `\QQ` then we don't have to specify
the name::
sage: A = AbelianVariety('67a')
sage: A.newform()
q + 2*q^2 - 2*q^3 + 2*q^4 + 2*q^5 + O(q^6)
"""
try:
return self.__named_newforms[names]
except KeyError:
self.__named_newforms[names] = Newform(self.__f.parent().change_ring(QQ), self.__f.modular_symbols(1), names=names, check=False)
return self.__named_newforms[names]
def label(self):
"""
Return canonical label that defines this newform modular
abelian variety.
OUTPUT:
string
EXAMPLES::
sage: A = AbelianVariety('43b')
sage: A.label()
'43b'
"""
G = self.__f.group()
if is_Gamma0(G):
group = ''
elif is_Gamma1(G):
group = 'G1'
elif is_GammaH(G):
group = 'GH[' + ','.join([str(z) for z in G._generators_for_H()]) + ']'
return '%s%s%s'%(self.level(), cremona_letter_code(self.factor_number()), group)
def factor_number(self):
"""
Return factor number.
OUTPUT:
int
EXAMPLES::
sage: A = AbelianVariety('43b')
sage: A.factor_number()
1
"""
try:
return self.__factor_number
except AttributeError:
self.__factor_number = self.__f.number()
return self.__factor_number
def _repr_(self):
"""
String representation of this modular abelian variety.
EXAMPLES::
sage: AbelianVariety('37a')._repr_()
'Newform abelian subvariety 37a of dimension 1 of J0(37)'
"""
return "Newform abelian subvariety %s of dimension %s of %s" % (
self.newform_label(), self.dimension(), self._ambient_repr())
def endomorphism_ring(self):
"""
Return the endomorphism ring of this newform abelian variety.
EXAMPLES::
sage: A = AbelianVariety('23a')
sage: E = A.endomorphism_ring(); E
Endomorphism ring of Newform abelian subvariety 23a of dimension 2 of J0(23)
We display the matrices of these two basis matrices::
sage: E.0.matrix()
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
sage: E.1.matrix()
[ 0 1 -1 0]
[ 0 1 -1 1]
[-1 2 -2 1]
[-1 1 0 -1]
The result is cached::
sage: E is A.endomorphism_ring()
True
"""
try:
return self.__endomorphism_ring
except AttributeError:
pass
E = homspace.EndomorphismSubring(self)
self.__endomorphism_ring = E
return self.__endomorphism_ring
def _calculate_endomorphism_generators(self):
"""
EXAMPLES::
sage: A = AbelianVariety('43b')
sage: B = A.endomorphism_ring(); B # indirect doctest
Endomorphism ring of Newform abelian subvariety 43b of dimension 2 of J0(43)
sage: [b.matrix() for b in B.gens()]
[
[1 0 0 0] [ 0 1 0 0]
[0 1 0 0] [ 1 -2 0 0]
[0 0 1 0] [ 1 0 -2 -1]
[0 0 0 1], [ 0 1 -1 0]
]
"""
M = self.modular_symbols()
bound = M.sturm_bound()
d = self.dimension()
T1list = self.hecke_operator(1).matrix().list()
EndVecZ = ZZ**(len(T1list))
V = EndVecZ.submodule([T1list])
n = 2
while V.dimension() < d:
W = EndVecZ.submodule([((self.hecke_operator(n).matrix())**i).list()
for i in range(1,d+1)])
V = V+W
n += 1
return V.saturation().basis()
