def _coerce_map_from_(self, S):
r"""
Coercion from a parent ``S``.
There is a coercion from ``S`` if ``S`` has a coerce map to `\Q`
or if `S = \Q/m\Z` for `m` a multiple of `n`.
TESTS::
sage: G2 = QQ/(2*ZZ)
sage: G3 = QQ/(3*ZZ)
sage: G4 = QQ/(4*ZZ)
sage: G2.has_coerce_map_from(QQ)
True
sage: G2.has_coerce_map_from(ZZ)
True
sage: G2.has_coerce_map_from(ZZ['x'])
False
sage: G2.has_coerce_map_from(G3)
False
sage: G2.has_coerce_map_from(G4)
True
sage: G4.has_coerce_map_from(G2)
False
"""
if QQ.has_coerce_map_from(S):
return True
if isinstance(S, QmodnZ) and (S.n / self.n in ZZ):
return True
def __add__(self, other):
r"""
Return the subsemigroup of `\QQ` generated by this semigroup and ``other``.
INPUT:
- ``other`` -- a discrete value (semi-)group or a rational number
EXAMPLES::
sage: from sage.rings.valuation.value_group import DiscreteValueSemigroup, DiscreteValueGroup
sage: D = DiscreteValueSemigroup(1/2)
sage: D + 1/3
Additive Abelian Semigroup generated by 1/3, 1/2
sage: D + D
Additive Abelian Semigroup generated by 1/2
sage: D + 1
Additive Abelian Semigroup generated by 1/2
sage: DiscreteValueGroup(2/7) + DiscreteValueSemigroup(4/9)
Additive Abelian Semigroup generated by -2/7, 2/7, 4/9
"""
if isinstance(other, DiscreteValueSemigroup):
return DiscreteValueSemigroup(self._generators + other._generators)
if isinstance(other, DiscreteValueGroup):
return DiscreteValueSemigroup(self._generators +
(other._generator,
-other._generator))
from sage.structure.element import is_Element
if is_Element(other) and QQ.has_coerce_map_from(other.parent()):
return self + DiscreteValueSemigroup(other)
raise ValueError(
"`other` must be a DiscreteValueGroup, a DiscreteValueSemigroup or a rational number"
)
def construct_from_maass_products(ring, weight, products, is_basis = True,
provides_maass_spezialschar = False,
is_integral = False,
lazy_rank_check = True) :
r"""
Pass the return value of spanning_maass_products of a space of Siegel modular
forms of same type. This will return a space using these forms.
Whenever ``is_basis`` is False the the products are first filtered to yield a
basis.
"""
assert QQ.has_coerce_map_from(ring.base_ring()) or ring.base_ring() is ZZ, \
"%s doesn't have rational base field" % ring
dim = ring.graded_submodule(weight).dimension()
## if the products don't provide a basis, we have to choose one
if not is_basis :
## we prefer to use Maass lifts, since they are very cheap
maass_forms = []; non_maass_forms = []
for p in products :
if len(p[0]) == 1 :
maass_forms.append(p)
else :
non_maass_forms.append(p)
monomials = set()
lift_polys = []
products = []
for lifts, lift_poly in maass_forms + non_maass_forms :
red_poly = ring(ring.relations().ring()(lift_poly))._reduce_polynomial()
monomials = monomials.union(set(red_poly.monomials()))
M = matrix( QQ, len(lift_polys) + 1,
[ poly.monomial_coefficient(m)
for poly in lift_polys + [lift_poly]
for m in monomials] )
# TODO : Use linbox
try :
if magma(M).Rank() > len(lift_polys) :
break
except TypeError :
for i in xrange(10) :
Mp = matrix(Qp(random_prime(10**10), 10), M)
if Mp.rank() > len(lift_polys) : break
else :
if lazy_rank_check :
continue
elif M.rank() <= len(lift_polys) :
continue
lift_polys.append(red_poly)
products.append((lifts, red_poly))
if len(products) == dim :
break
else :
raise ValueError, "products don't provide a basis"
basis = []
if provides_maass_spezialschar :
maass_form_indices = []
for i, (lifts, lift_poly) in enumerate(products) :
e = ring(ring.relations().ring()(lift_poly))
if len(lifts) == 1 :
l = lifts[0]
e._set_fourier_expansion(
SiegelModularFormG2MaassLift(l[0], l[1], ring.fourier_expansion_precision(),
is_integral = is_integral) )
elif len(lifts) == 2 :
(l0, l1) = tuple(lifts)
e._set_fourier_expansion(
EquivariantMonoidPowerSeries_LazyMultiplication(
SiegelModularFormG2MaassLift(l0[0], l0[1], ring.fourier_expansion_precision(),
is_integral = is_integral),
SiegelModularFormG2MaassLift(l1[0], l1[1], ring.fourier_expansion_precision(),
is_integral = is_integral) ) )
else :
e._set_fourier_expansion(
prod( SiegelModularFormG2MaassLift(l[0], l[1], ring.precision(),
is_integral = is_integral)
for l in lifts) )
basis.append(e)
if provides_maass_spezialschar and len(lifts) == 1 :
maass_form_indices.append(i)
ss = ring._submodule(basis, grading_indices = (weight,), is_heckeinvariant = True)
if provides_maass_spezialschar :
maass_coords = [ ss([0]*i + [1] + [0]*(dim-i-1))
for i in maass_form_indices ]
ss.maass_space.set_cache(
SiegelModularFormG2Submodule_maassspace(ss, map(ss, maass_coords)) )
return ss
def construct_from_maass_products(ring,
weight,
products,
is_basis=True,
provides_maass_spezialschar=False,
is_integral=False,
lazy_rank_check=True):
r"""
Pass the return value of spanning_maass_products of a space of Siegel modular
forms of same type. This will return a space using these forms.
Whenever ``is_basis`` is False the the products are first filtered to yield a
basis.
"""
assert QQ.has_coerce_map_from(ring.base_ring()) or ring.base_ring() is ZZ, \
"%s doesn't have rational base field" % ring
dim = ring.graded_submodule(weight).dimension()
## if the products don't provide a basis, we have to choose one
if not is_basis:
## we prefer to use Maass lifts, since they are very cheap
maass_forms = []
non_maass_forms = []
for p in products:
if len(p[0]) == 1:
maass_forms.append(p)
else:
non_maass_forms.append(p)
monomials = set()
lift_polys = []
products = []
for lifts, lift_poly in maass_forms + non_maass_forms:
red_poly = ring(
ring.relations().ring()(lift_poly))._reduce_polynomial()
monomials = monomials.union(set(red_poly.monomials()))
M = matrix(QQ,
len(lift_polys) + 1, [
poly.monomial_coefficient(m)
for poly in lift_polys + [lift_poly]
for m in monomials
])
# TODO : Use linbox
try:
if magma(M).Rank() > len(lift_polys):
break
except TypeError:
for i in xrange(10):
Mp = matrix(Qp(random_prime(10**10), 10), M)
if Mp.rank() > len(lift_polys): break
else:
if lazy_rank_check:
continue
elif M.rank() <= len(lift_polys):
continue
lift_polys.append(red_poly)
products.append((lifts, red_poly))
if len(products) == dim:
break
else:
raise ValueError, "products don't provide a basis"
basis = []
if provides_maass_spezialschar:
maass_form_indices = []
for i, (lifts, lift_poly) in enumerate(products):
e = ring(ring.relations().ring()(lift_poly))
if len(lifts) == 1:
l = lifts[0]
e._set_fourier_expansion(
SiegelModularFormG2MaassLift(
l[0],
l[1],
ring.fourier_expansion_precision(),
is_integral=is_integral))
elif len(lifts) == 2:
(l0, l1) = tuple(lifts)
e._set_fourier_expansion(
EquivariantMonoidPowerSeries_LazyMultiplication(
SiegelModularFormG2MaassLift(
l0[0],
l0[1],
ring.fourier_expansion_precision(),
is_integral=is_integral),
SiegelModularFormG2MaassLift(
l1[0],
l1[1],
ring.fourier_expansion_precision(),
is_integral=is_integral)))
else:
e._set_fourier_expansion(
prod(
SiegelModularFormG2MaassLift(
l[0], l[1], ring.precision(), is_integral=is_integral)
for l in lifts))
basis.append(e)
if provides_maass_spezialschar and len(lifts) == 1:
maass_form_indices.append(i)
ss = ring._submodule(basis,
grading_indices=(weight, ),
is_heckeinvariant=True)
if provides_maass_spezialschar:
maass_coords = [
ss([0] * i + [1] + [0] * (dim - i - 1)) for i in maass_form_indices
]
ss.maass_space.set_cache(
SiegelModularFormG2Submodule_maassspace(ss, map(ss, maass_coords)))
return ss
