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smoothchar.py
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smoothchar.py
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r"""
Smooth characters of `p`-adic fields
Let `F` be a finite extension of `\QQ_p`. Then we may consider the group of
smooth (i.e. locally constant) group homomorphisms `F^\times \to L^\times`, for
`L` any field. Such characters are important since they can be used to
parametrise smooth representations of `\mathrm{GL}_2(\QQ_p)`, which arise as
the local components of modular forms.
This module contains classes to represent such characters when `F` is `\QQ_p`
or a quadratic extension. In the latter case, we choose a quadratic extension
`K` of `\QQ` whose completion at `p` is `F`, and use Sage's wrappers of the
Pari ``idealstar`` and ``ideallog`` methods to work in the finite group
`\mathcal{O}_K / p^c` for `c \ge 0`.
An example with characters of `\QQ_7`::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupQp
sage: K.<z> = CyclotomicField(42)
sage: G = SmoothCharacterGroupQp(7, K)
sage: G.unit_gens(2), G.exponents(2)
([3, 7], [42, 0])
The output of the last line means that the group `\QQ_7^\times / (1 + 7^2
\ZZ_7)` is isomorphic to `C_{42} \times \ZZ`, with the two factors being
generated by `3` and `7` respectively. We create a character by specifying the
images of these generators::
sage: chi = G.character(2, [z^5, 11 + z]); chi
Character of Q_7*, of level 2, mapping 3 |--> z^5, 7 |--> z + 11
sage: chi(4)
z^8
sage: chi(42)
z^10 + 11*z^9
Characters are themselves group elements, and basic arithmetic on them works::
sage: chi**3
Character of Q_7*, of level 2, mapping 3 |--> z^8 - z, 7 |--> z^3 + 33*z^2 + 363*z + 1331
sage: chi.multiplicative_order()
+Infinity
"""
from six.moves import range
import operator
from sage.structure.element import MultiplicativeGroupElement, parent
from sage.structure.parent_base import ParentWithBase
from sage.structure.sequence import Sequence
from sage.structure.richcmp import richcmp_not_equal, richcmp
from sage.rings.all import QQ, ZZ, Zmod, NumberField
from sage.rings.ring import is_Ring
from sage.misc.cachefunc import cached_method
from sage.misc.abstract_method import abstract_method
from sage.misc.misc_c import prod
from sage.categories.groups import Groups
from sage.functions.other import ceil
from sage.misc.mrange import xmrange
class SmoothCharacterGeneric(MultiplicativeGroupElement):
r"""
A smooth (i.e. locally constant) character of `F^\times`, for `F` some
finite extension of `\QQ_p`.
"""
def __init__(self, parent, c, values_on_gens):
r"""
Standard init function.
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupUnramifiedQuadratic
sage: G = SmoothCharacterGroupUnramifiedQuadratic(2, QQ)
sage: G.character(0, [17]) # indirect doctest
Character of unramified extension Q_2(s)* (s^2 + s + 1 = 0), of level 0, mapping 2 |--> 17
sage: G.character(1, [1, 17]) # indirect doctest
Character of unramified extension Q_2(s)* (s^2 + s + 1 = 0), of level 0, mapping 2 |--> 17
sage: G.character(2, [1, -1, 1, 17]) # indirect doctest
Character of unramified extension Q_2(s)* (s^2 + s + 1 = 0), of level 2, mapping s |--> 1, 2*s + 1 |--> -1, -1 |--> 1, 2 |--> 17
sage: G.character(2, [1, 1, 1, 17]) # indirect doctest
Character of unramified extension Q_2(s)* (s^2 + s + 1 = 0), of level 0, mapping 2 |--> 17
"""
MultiplicativeGroupElement.__init__(self, parent)
self._c = c
self._values_on_gens = values_on_gens
self._check_level()
def _check_level(self):
r"""
Checks that this character has the level it claims to have, and if not,
decrement the level by 1. This is called by :meth:`__init__`.
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupQp
sage: SmoothCharacterGroupQp(5, QQ).character(5, [-1, 7]) # indirect doctest
Character of Q_5*, of level 1, mapping 2 |--> -1, 5 |--> 7
"""
if self.level() == 0: return
v = self.parent().subgroup_gens(self.level())
if all([self(x) == 1 for x in v]):
new_gens = self.parent().unit_gens(self.level() - 1)
new_values = [self(x) for x in new_gens]
self._values_on_gens = Sequence(new_values, universe=self.base_ring(), immutable=True)
self._c = self._c - 1
self._check_level()
def _richcmp_(self, other, op):
r"""
Compare ``self`` and ``other``.
Note that this only gets called when the
parents of ``self`` and ``other`` are identical.
INPUT:
- ``other`` -- another smooth character
- ``op`` -- a comparison operator (see :mod:`sage.structure.richcmp`)
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupQp, SmoothCharacterGroupUnramifiedQuadratic
sage: SmoothCharacterGroupQp(7, Zmod(3)).character(1, [2, 1]) == SmoothCharacterGroupQp(7, ZZ).character(1, [-1, 1])
True
sage: chi1 = SmoothCharacterGroupUnramifiedQuadratic(7, QQ).character(0, [1])
sage: chi2 = SmoothCharacterGroupQp(7, QQ).character(0, [1])
sage: chi1 == chi2
False
sage: chi2.parent()(chi1) == chi2
True
sage: chi1 == loads(dumps(chi1))
True
"""
lx = self.level()
rx = other.level()
if lx != rx:
return richcmp_not_equal(lx, rx, op)
return richcmp(self._values_on_gens, other._values_on_gens, op)
def multiplicative_order(self):
r"""
Return the order of this character as an element of the character group.
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupQp
sage: K.<z> = CyclotomicField(42)
sage: G = SmoothCharacterGroupQp(7, K)
sage: G.character(3, [z^10 - z^3, 11]).multiplicative_order()
+Infinity
sage: G.character(3, [z^10 - z^3, 1]).multiplicative_order()
42
sage: G.character(1, [z^7, z^14]).multiplicative_order()
6
sage: G.character(0, [1]).multiplicative_order()
1
"""
from sage.arith.all import lcm
from sage.rings.infinity import Infinity
if self._values_on_gens[-1].multiplicative_order() == Infinity:
return Infinity
else:
return lcm([x.multiplicative_order() for x in self._values_on_gens])
def level(self):
r"""
Return the level of this character, i.e. the smallest integer `c \ge 0`
such that it is trivial on `1 + \mathfrak{p}^c`.
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupQp
sage: SmoothCharacterGroupQp(7, QQ).character(2, [-1, 1]).level()
1
"""
return self._c
def __call__(self, x):
r"""
Evaluate the character at ``x``, which should be a nonzero element of
the number field of the parent group.
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupQp
sage: K.<z> = CyclotomicField(42)
sage: chi = SmoothCharacterGroupQp(7, K).character(3, [z^10 - z^3, 11])
sage: [chi(x) for x in [1, 2, 3, 9, 21, 1/12345678]]
[1, -z, z^10 - z^3, -z^11 - z^10 + z^8 + z^7 - z^6 - z^5 + z^3 + z^2 - 1, 11*z^10 - 11*z^3, z^7 - 1]
Non-examples::
sage: chi(QuadraticField(-1,'i').gen())
Traceback (most recent call last):
...
TypeError: no canonical coercion from Number Field in i with defining polynomial x^2 + 1 to Rational Field
sage: chi(0)
Traceback (most recent call last):
...
ValueError: cannot evaluate at zero
sage: chi(Mod(1, 12))
Traceback (most recent call last):
...
TypeError: no canonical coercion from Ring of integers modulo 12 to Rational Field
Some examples with an unramified quadratic extension, where the choice
of generators is arbitrary (but deterministic)::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupUnramifiedQuadratic
sage: K.<z> = CyclotomicField(30)
sage: G = SmoothCharacterGroupUnramifiedQuadratic(5, K)
sage: chi = G.character(2, [z**5, z**(-6), z**6, 3]); chi
Character of unramified extension Q_5(s)* (s^2 + 4*s + 2 = 0), of level 2, mapping 11*s - 10 |--> z^5, 6 |--> -z^7 - z^6 + z^3 + z^2 - 1, 5*s + 1 |--> z^6, 5 |--> 3
sage: chi(G.unit_gens(2)[0]**7 / G.unit_gens(2)[1]/5)
1/3*z^6 - 1/3*z
sage: chi(2)
-z^3
"""
v = self.parent().discrete_log(self.level(), x)
return prod([self._values_on_gens[i] ** v[i] for i in range(len(v))])
def _repr_(self):
r"""
String representation of this character.
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupQp
sage: K.<z> = CyclotomicField(20)
sage: SmoothCharacterGroupQp(5, K).character(2, [z, z+1])._repr_()
'Character of Q_5*, of level 2, mapping 2 |--> z, 5 |--> z + 1'
Examples over field extensions::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupUnramifiedQuadratic
sage: K.<z> = CyclotomicField(15)
sage: SmoothCharacterGroupUnramifiedQuadratic(5, K).character(2, [z**5, z**3, 1, z+1])._repr_()
'Character of unramified extension Q_5(s)* (s^2 + 4*s + 2 = 0), of level 2, mapping 11*s - 10 |--> z^5, 6 |--> z^3, 5*s + 1 |--> 1, 5 |--> z + 1'
"""
gens = self.parent().unit_gens(self.level())
mapst = ", ".join( str(gens[i]) + ' |--> ' + str(self._values_on_gens[i]) for i in range(len(gens)) )
return "Character of %s, of level %s, mapping %s" % (self.parent()._field_name(), self.level(), mapst)
def _mul_(self, other):
r"""
Product of self and other.
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupQp
sage: K.<z> = CyclotomicField(20)
sage: chi1 = SmoothCharacterGroupQp(5, K).character(2, [z, z+1])
sage: chi2 = SmoothCharacterGroupQp(5, K).character(2, [z^4, 3])
sage: chi1 * chi2 # indirect doctest
Character of Q_5*, of level 1, mapping 2 |--> z^5, 5 |--> 3*z + 3
sage: chi2 * chi1 # indirect doctest
Character of Q_5*, of level 1, mapping 2 |--> z^5, 5 |--> 3*z + 3
sage: chi1 * SmoothCharacterGroupQp(5, QQ).character(2, [-1, 7]) # indirect doctest
Character of Q_5*, of level 2, mapping 2 |--> -z, 5 |--> 7*z + 7
"""
if other.level() > self.level():
return other * self
return self.parent().character(self.level(), [self(x) * other(x) for x in self.parent().unit_gens(self.level())])
def __invert__(self):
r"""
Multiplicative inverse of self.
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupUnramifiedQuadratic
sage: K.<z> = CyclotomicField(12)
sage: chi = SmoothCharacterGroupUnramifiedQuadratic(2, K).character(4, [z**4, z**3, z**9, -1, 7]); chi
Character of unramified extension Q_2(s)* (s^2 + s + 1 = 0), of level 4, mapping s |--> z^2 - 1, 2*s + 1 |--> z^3, 4*s + 1 |--> -z^3, -1 |--> -1, 2 |--> 7
sage: chi**(-1) # indirect doctest
Character of unramified extension Q_2(s)* (s^2 + s + 1 = 0), of level 4, mapping s |--> -z^2, 2*s + 1 |--> -z^3, 4*s + 1 |--> z^3, -1 |--> -1, 2 |--> 1/7
sage: SmoothCharacterGroupUnramifiedQuadratic(2, QQ).character(0, [7]) / chi # indirect doctest
Character of unramified extension Q_2(s)* (s^2 + s + 1 = 0), of level 4, mapping s |--> -z^2, 2*s + 1 |--> -z^3, 4*s + 1 |--> z^3, -1 |--> -1, 2 |--> 1
"""
return self.parent().character(self.level(), [~self(x) for x in self.parent().unit_gens(self.level())])
def restrict_to_Qp(self):
r"""
Return the restriction of this character to `\QQ_p^\times`, embedded as
a subfield of `F^\times`.
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupRamifiedQuadratic
sage: SmoothCharacterGroupRamifiedQuadratic(3, 0, QQ).character(0, [2]).restrict_to_Qp()
Character of Q_3*, of level 0, mapping 3 |--> 4
"""
G = SmoothCharacterGroupQp(self.parent().prime(), self.base_ring())
ugs = G.unit_gens(self.level())
return G.character(self.level(), [self(x) for x in ugs])
def galois_conjugate(self):
r"""
Return the composite of this character with the order `2` automorphism of
`K / \QQ_p` (assuming `K` is quadratic).
Note that this is the Galois operation on the *domain*, not on the
*codomain*.
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupUnramifiedQuadratic
sage: K.<w> = CyclotomicField(3)
sage: G = SmoothCharacterGroupUnramifiedQuadratic(2, K)
sage: chi = G.character(2, [w, -1,-1, 3*w])
sage: chi2 = chi.galois_conjugate(); chi2
Character of unramified extension Q_2(s)* (s^2 + s + 1 = 0), of level 2, mapping s |--> -w - 1, 2*s + 1 |--> 1, -1 |--> -1, 2 |--> 3*w
sage: chi.restrict_to_Qp() == chi2.restrict_to_Qp()
True
sage: chi * chi2 == chi.parent().compose_with_norm(chi.restrict_to_Qp())
True
"""
K,s = self.parent().number_field().objgen()
if K.absolute_degree() != 2:
raise ValueError( "Character must be defined on a quadratic extension" )
sigs = K.embeddings(K)
sig = [x for x in sigs if x(s) != s][0]
return self.parent().character(self.level(), [self(sig(x)) for x in self.parent().unit_gens(self.level())])
class SmoothCharacterGroupGeneric(ParentWithBase):
r"""
The group of smooth (i.e. locally constant) characters of a `p`-adic field,
with values in some ring `R`. This is an abstract base class and should not
be instantiated directly.
"""
Element = SmoothCharacterGeneric
def __init__(self, p, base_ring):
r"""
TESTS::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupGeneric
sage: G = SmoothCharacterGroupGeneric(3, QQ)
sage: SmoothCharacterGroupGeneric(3, "hello")
Traceback (most recent call last):
...
TypeError: base ring (=hello) must be a ring
"""
if not is_Ring(base_ring):
raise TypeError( "base ring (=%s) must be a ring" % base_ring )
ParentWithBase.__init__(self, base=base_ring, category=Groups())
if not (p in ZZ and ZZ(p).is_prime()):
raise ValueError( "p (=%s) must be a prime integer" % p )
self._p = ZZ.coerce(p)
def _element_constructor_(self, x):
r"""
Construct an element of this group from ``x`` (possibly noncanonically).
This only works if ``x`` is a character of a field containing the field of
self, whose values lie in a field that can be converted into self.
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupQp
sage: K.<i> = QuadraticField(-1)
sage: G = SmoothCharacterGroupQp(3, QQ)
sage: GK = SmoothCharacterGroupQp(3, K)
sage: chi = GK(G.character(0, [4])); chi # indirect doctest
Character of Q_3*, of level 0, mapping 3 |--> 4
sage: chi.parent() is GK
True
sage: G(GK.character(0, [7])) # indirect doctest
Character of Q_3*, of level 0, mapping 3 |--> 7
sage: G(GK.character(0, [i])) # indirect doctest
Traceback (most recent call last):
...
TypeError: unable to convert i to an element of Rational Field
"""
if x == 1:
return self.character(0, [1])
P = parent(x)
if (isinstance(P, SmoothCharacterGroupGeneric)
and P.number_field().has_coerce_map_from(self.number_field())):
return self.character(x.level(), [x(v) for v in self.unit_gens(x.level())])
else:
raise TypeError
def __eq__(self, other):
r"""
TESTS::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupQp
sage: G = SmoothCharacterGroupQp(3, QQ)
sage: G == SmoothCharacterGroupQp(3, QQ[I])
False
sage: G == 7
False
sage: G == SmoothCharacterGroupQp(7, QQ)
False
sage: G == SmoothCharacterGroupQp(3, QQ)
True
"""
return (type(self) == type(other) and
self.prime() == other.prime() and
self.number_field() == other.number_field() and
self.base_ring() == other.base_ring())
def __ne__(self, other):
"""
Check whether ``self`` is not equalt to ``other``.
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupQp
sage: G = SmoothCharacterGroupQp(3, QQ)
sage: G != SmoothCharacterGroupQp(3, QQ[I])
True
sage: G != 7
True
sage: G != SmoothCharacterGroupQp(7, QQ)
True
sage: G != SmoothCharacterGroupQp(3, QQ)
False
"""
return not (self == other)
def _coerce_map_from_(self, other):
r"""
Return True if self has a canonical coerce map from other.
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupQp
sage: K.<i> = QuadraticField(-1)
sage: G = SmoothCharacterGroupQp(3, QQ)
sage: GK = SmoothCharacterGroupQp(3, K)
sage: G.has_coerce_map_from(GK)
False
sage: GK.has_coerce_map_from(G)
True
sage: GK.coerce(G.character(0, [4]))
Character of Q_3*, of level 0, mapping 3 |--> 4
sage: G.coerce(GK.character(0, [4]))
Traceback (most recent call last):
...
TypeError: no canonical coercion from Group of smooth characters of Q_3* with values in Number Field in i with defining polynomial x^2 + 1 to Group of smooth characters of Q_3* with values in Rational Field
sage: G.character(0, [4]) in GK # indirect doctest
True
The coercion framework handles base extension, so we test that too::
sage: K.<i> = QuadraticField(-1)
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupUnramifiedQuadratic
sage: G = SmoothCharacterGroupUnramifiedQuadratic(3, QQ)
sage: G.character(0, [1]).base_extend(K)
Character of unramified extension Q_3(s)* (s^2 + 2*s + 2 = 0), of level 0, mapping 3 |--> 1
"""
if isinstance(other, SmoothCharacterGroupGeneric) \
and other.number_field() == self.number_field() \
and self.base_ring().has_coerce_map_from(other.base_ring()):
return True
else:
return False
def prime(self):
r"""
The residue characteristic of the underlying field.
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupGeneric
sage: SmoothCharacterGroupGeneric(3, QQ).prime()
3
"""
return self._p
@abstract_method
def change_ring(self, ring):
r"""
Return the character group of the same field, but with values in a
different coefficient ring. To be implemented by all derived classes
(since the generic base class can't know the parameters).
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupGeneric
sage: SmoothCharacterGroupGeneric(3, QQ).change_ring(ZZ)
Traceback (most recent call last):
...
NotImplementedError: <abstract method change_ring at ...>
"""
pass
def base_extend(self, ring):
r"""
Return the character group of the same field, but with values in a new
coefficient ring into which the old coefficient ring coerces. An error
will be raised if there is no coercion map from the old coefficient
ring to the new one.
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupQp
sage: G = SmoothCharacterGroupQp(3, QQ)
sage: G.base_extend(QQbar)
Group of smooth characters of Q_3* with values in Algebraic Field
sage: G.base_extend(Zmod(3))
Traceback (most recent call last):
...
TypeError: no canonical coercion from Rational Field to Ring of integers modulo 3
"""
if not ring.has_coerce_map_from(self.base_ring()) :
ring.coerce(self.base_ring().an_element())
# this is here to flush out errors
return self.change_ring(ring)
@abstract_method
def _field_name(self):
r"""
A string representing the name of the p-adic field of which this is the
character group. To be overridden by derived subclasses.
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupGeneric
sage: SmoothCharacterGroupGeneric(3, QQ)._field_name()
Traceback (most recent call last):
...
NotImplementedError: <abstract method _field_name at ...>
"""
pass
def _repr_(self):
r"""
String representation of self.
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupQp
sage: SmoothCharacterGroupQp(7, QQ)._repr_()
'Group of smooth characters of Q_7* with values in Rational Field'
"""
return "Group of smooth characters of %s with values in %s" % (self._field_name(), self.base_ring())
@abstract_method
def ideal(self, level):
r"""
Return the ``level``-th power of the maximal ideal of the ring of
integers of the p-adic field. Since we approximate by using number
field arithmetic, what is actually returned is an ideal in a number
field.
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupGeneric
sage: SmoothCharacterGroupGeneric(3, QQ).ideal(3)
Traceback (most recent call last):
...
NotImplementedError: <abstract method ideal at ...>
"""
pass
@abstract_method
def unit_gens(self, level):
r"""
A list of generators `x_1, \dots, x_d` of the abelian group `F^\times /
(1 + \mathfrak{p}^c)^\times`, where `c` is the given level, satisfying
no relations other than `x_i^{n_i} = 1` for each `i` (where the
integers `n_i` are returned by :meth:`exponents`). We adopt the
convention that the final generator `x_d` is a uniformiser (and `n_d =
0`).
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupGeneric
sage: SmoothCharacterGroupGeneric(3, QQ).unit_gens(3)
Traceback (most recent call last):
...
NotImplementedError: <abstract method unit_gens at ...>
"""
pass
@abstract_method
def exponents(self, level):
r"""
The orders `n_1, \dots, n_d` of the generators `x_i` of `F^\times / (1
+ \mathfrak{p}^c)^\times` returned by :meth:`unit_gens`.
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupGeneric
sage: SmoothCharacterGroupGeneric(3, QQ).exponents(3)
Traceback (most recent call last):
...
NotImplementedError: <abstract method exponents at ...>
"""
pass
@abstract_method
def subgroup_gens(self, level):
r"""
A set of elements of `(\mathcal{O}_F / \mathfrak{p}^c)^\times`
generating the kernel of the reduction map to `(\mathcal{O}_F /
\mathfrak{p}^{c-1})^\times`.
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupGeneric
sage: SmoothCharacterGroupGeneric(3, QQ).subgroup_gens(3)
Traceback (most recent call last):
...
NotImplementedError: <abstract method subgroup_gens at ...>
"""
pass
@abstract_method
def discrete_log(self, level):
r"""
Given an element `x \in F^\times` (lying in the number field `K` of
which `F` is a completion, see module docstring), express the class of
`x` in terms of the generators of `F^\times / (1 +
\mathfrak{p}^c)^\times` returned by :meth:`unit_gens`.
This should be overridden by all derived classes. The method should
first attempt to canonically coerce `x` into ``self.number_field()``,
and check that the result is not zero.
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupGeneric
sage: SmoothCharacterGroupGeneric(3, QQ).discrete_log(3)
Traceback (most recent call last):
...
NotImplementedError: <abstract method discrete_log at ...>
"""
pass
def character(self, level, values_on_gens):
r"""
Return the unique character of the given level whose values on the
generators returned by ``self.unit_gens(level)`` are
``values_on_gens``.
INPUT:
- ``level`` (integer) an integer `\ge 0`
- ``values_on_gens`` (sequence) a sequence of elements of length equal
to the length of ``self.unit_gens(level)``. The values should be
convertible (that is, possibly noncanonically) into the base ring of self; they
should all be units, and all but the last must be roots of unity (of
the orders given by ``self.exponents(level)``.
.. note::
The character returned may have level less than ``level`` in general.
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupQp
sage: K.<z> = CyclotomicField(42)
sage: G = SmoothCharacterGroupQp(7, K)
sage: G.character(2, [z^6, 8])
Character of Q_7*, of level 2, mapping 3 |--> z^6, 7 |--> 8
sage: G.character(2, [z^7, 8])
Character of Q_7*, of level 1, mapping 3 |--> z^7, 7 |--> 8
Non-examples::
sage: G.character(1, [z, 1])
Traceback (most recent call last):
...
ValueError: value on generator 3 (=z) should be a root of unity of order 6
sage: G.character(1, [1, 0])
Traceback (most recent call last):
...
ValueError: value on uniformiser 7 (=0) should be a unit
An example with a funky coefficient ring::
sage: G = SmoothCharacterGroupQp(7, Zmod(9))
sage: G.character(1, [2, 2])
Character of Q_7*, of level 1, mapping 3 |--> 2, 7 |--> 2
sage: G.character(1, [2, 3])
Traceback (most recent call last):
...
ValueError: value on uniformiser 7 (=3) should be a unit
TESTS::
sage: G.character(1, [2])
Traceback (most recent call last):
...
AssertionError: 2 images must be given
"""
S = Sequence(values_on_gens, universe=self.base_ring(), immutable=True)
assert len(S) == len(self.unit_gens(level)), "{0} images must be given".format(len(self.unit_gens(level)))
n = self.exponents(level)
for i in range(len(S)):
if n[i] != 0 and not S[i]**n[i] == 1:
raise ValueError( "value on generator %s (=%s) should be a root of unity of order %s" % (self.unit_gens(level)[i], S[i], n[i]) )
elif n[i] == 0 and not S[i].is_unit():
raise ValueError( "value on uniformiser %s (=%s) should be a unit" % (self.unit_gens(level)[i], S[i]) )
return self.element_class(self, level, S)
def _an_element_(self):
r"""
Return an element of this group. Required by the coercion machinery.
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupQp
sage: K.<z> = CyclotomicField(42)
sage: G = SmoothCharacterGroupQp(7, K)
sage: G.an_element() # indirect doctest
Character of Q_7*, of level 0, mapping 7 |--> z
"""
return self.character(0, [self.base_ring().an_element()])
def _test_unitgens(self, **options):
r"""
Test that the generators returned by ``unit_gens`` are consistent with
the exponents returned by ``exponents``.
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupUnramifiedQuadratic
sage: SmoothCharacterGroupUnramifiedQuadratic(2, Zmod(8))._test_unitgens()
"""
T = self._tester(**options)
for c in range(6):
gens = self.unit_gens(c)
exps = self.exponents(c)
T.assert_(exps[-1] == 0)
T.assert_(all([u != 0 for u in exps[:-1]]))
T.assert_(all([u.parent() is self.number_field() for u in gens]))
I = self.ideal(c)
for i in range(len(exps[:-1])):
g = gens[i]
for m in range(1, exps[i]):
if (g - 1 in I):
T.fail("For generator g=%s, g^%s = %s = 1 mod I, but order should be %s" % (gens[i], m, g, exps[i]))
g = g * gens[i]
# reduce g mod I
if hasattr(I, "small_residue"):
g = I.small_residue(g)
else: # I is an ideal of ZZ
g = g % (I.gen())
if not (g - 1 in I):
T.fail("For generator g=%s, g^%s = %s, which is not 1 mod I" % (gens[i], exps[i], g))
I = self.prime() if self.number_field() == QQ else self.ideal(1)
T.assert_(gens[-1].valuation(I) == 1)
# This implicitly tests that the gens really are gens!
_ = self.discrete_log(c, -1)
def _test_subgroupgens(self, **options):
r"""
Test that the values returned by :meth:`~subgroup_gens` are valid.
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupQp
sage: SmoothCharacterGroupQp(2, CC)._test_subgroupgens()
"""
T = self._tester(**options)
for c in range(1, 6):
sgs = self.subgroup_gens(c)
I2 = self.ideal(c-1)
T.assert_(all([x-1 in I2 for x in sgs]), "Kernel gens at level %s not in kernel!" % c)
# now find the exponent of the kernel
n1 = prod(self.exponents(c)[:-1])
n2 = prod(self.exponents(c-1)[:-1])
n = n1 // n2
# if c > 1, n will be a prime here, so that logs below gets calculated correctly
logs = []
for idx in xmrange(len(sgs)*[n]):
y = prod( map(operator.pow, sgs, idx) )
L = tuple(self.discrete_log(c, y))
if L not in logs:
logs.append(L)
T.assert_(n2 * len(logs) == n1, "Kernel gens at level %s don't generate everything!" % c)
def compose_with_norm(self, chi):
r"""
Calculate the character of `K^\times` given by `\chi \circ \mathrm{Norm}_{K/\QQ_p}`.
Here `K` should be a quadratic extension and `\chi` a character of `\QQ_p^\times`.
EXAMPLES:
When `K` is the unramified quadratic extension, the level of the new character is the same as the old::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupQp, SmoothCharacterGroupRamifiedQuadratic, SmoothCharacterGroupUnramifiedQuadratic
sage: K.<w> = CyclotomicField(6)
sage: G = SmoothCharacterGroupQp(3, K)
sage: chi = G.character(2, [w, 5])
sage: H = SmoothCharacterGroupUnramifiedQuadratic(3, K)
sage: H.compose_with_norm(chi)
Character of unramified extension Q_3(s)* (s^2 + 2*s + 2 = 0), of level 2, mapping -2*s |--> -1, 4 |--> -w, 3*s + 1 |--> w - 1, 3 |--> 25
In ramified cases, the level of the new character may be larger:
.. link
::
sage: H = SmoothCharacterGroupRamifiedQuadratic(3, 0, K)
sage: H.compose_with_norm(chi)
Character of ramified extension Q_3(s)* (s^2 - 3 = 0), of level 3, mapping 2 |--> w - 1, s + 1 |--> -w, s |--> -5
On the other hand, since norm is not surjective, the result can even be trivial:
.. link
::
sage: chi = G.character(1, [-1, -1]); chi
Character of Q_3*, of level 1, mapping 2 |--> -1, 3 |--> -1
sage: H.compose_with_norm(chi)
Character of ramified extension Q_3(s)* (s^2 - 3 = 0), of level 0, mapping s |--> 1
"""
if chi.parent().number_field() != QQ: raise ValueError
if self.number_field().absolute_degree() != 2: raise ValueError
n = chi.level()
P = chi.parent().prime() ** n
m = self.number_field()(P).valuation(self.ideal(1))
return self.character(m, [chi(x.norm(QQ)) for x in self.unit_gens(m)])
class SmoothCharacterGroupQp(SmoothCharacterGroupGeneric):
r"""
The group of smooth characters of `\QQ_p^\times`, with values in some fixed
base ring.
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupQp
sage: G = SmoothCharacterGroupQp(7, QQ); G
Group of smooth characters of Q_7* with values in Rational Field
sage: TestSuite(G).run()
sage: G == loads(dumps(G))
True
"""
def unit_gens(self, level):
r"""
Return a set of generators `x_1, \dots, x_d` for `\QQ_p^\times / (1 +
p^c \ZZ_p)^\times`. These must be independent in the sense that there
are no relations between them other than relations of the form
`x_i^{n_i} = 1`. They need not, however, be in Smith normal form.
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupQp
sage: SmoothCharacterGroupQp(7, QQ).unit_gens(3)
[3, 7]
sage: SmoothCharacterGroupQp(2, QQ).unit_gens(4)
[15, 5, 2]
"""
if level == 0:
return [QQ(self.prime())]
else:
return [QQ(x) for x in Zmod(self.prime()**level).unit_gens()] + [QQ(self.prime())]
def exponents(self, level):
r"""
Return the exponents of the generators returned by :meth:`unit_gens`.
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupQp
sage: SmoothCharacterGroupQp(7, QQ).exponents(3)
[294, 0]
sage: SmoothCharacterGroupQp(2, QQ).exponents(4)
[2, 4, 0]
"""
if level == 0: return [0]
return [x.multiplicative_order() for x in Zmod(self.prime()**level).unit_gens()] + [0]
def change_ring(self, ring):
r"""
Return the group of characters of the same field but with values in a
different ring. This need not have anything to do with the original
base ring, and in particular there won't generally be a coercion map
from self to the new group -- use
:meth:`~SmoothCharacterGroupGeneric.base_extend` if you want this.
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupQp
sage: SmoothCharacterGroupQp(7, Zmod(3)).change_ring(CC)
Group of smooth characters of Q_7* with values in Complex Field with 53 bits of precision
"""
return SmoothCharacterGroupQp(self.prime(), ring)
def number_field(self):
r"""
Return the number field used for calculations (a dense subfield of the
local field of which this is the character group). In this case, this
is always the rational field.
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupQp
sage: SmoothCharacterGroupQp(7, Zmod(3)).number_field()
Rational Field
"""
return QQ
def ideal(self, level):
r"""
Return the ``level``-th power of the maximal ideal. Since we
approximate by using rational arithmetic, what is actually returned is
an ideal of `\ZZ`.
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupQp
sage: SmoothCharacterGroupQp(7, Zmod(3)).ideal(2)
Principal ideal (49) of Integer Ring
"""
return ZZ.ideal(self.prime() ** level)
def _field_name(self):
r"""
Return a string representation of the field unit group of which this is
the character group.
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupQp
sage: SmoothCharacterGroupQp(7, Zmod(3))._field_name()
'Q_7*'
"""
return "Q_%s*" % self.prime()
def discrete_log(self, level, x):
r"""
Express the class of `x` in `\QQ_p^\times / (1 + p^c)^\times` in terms
of the generators returned by :meth:`unit_gens`.
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupQp
sage: G = SmoothCharacterGroupQp(7, QQ)
sage: G.discrete_log(0, 14)
[1]
sage: G.discrete_log(1, 14)
[2, 1]
sage: G.discrete_log(5, 14)
[9308, 1]
"""
x = self.number_field().coerce(x)
if x == 0: raise ValueError( "cannot evaluate at zero" )
s = x.valuation(self.prime())
return Zmod(self.prime()**level)(x / self.prime()**s).generalised_log() + [s]
def subgroup_gens(self, level):
r"""
Return a list of generators for the kernel of the map `(\ZZ_p / p^c)^\times
\to (\ZZ_p / p^{c-1})^\times`.
INPUT:
- ``c`` (integer) an integer `\ge 1`
EXAMPLES::
sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupQp
sage: G = SmoothCharacterGroupQp(7, QQ)
sage: G.subgroup_gens(1)
[3]
sage: G.subgroup_gens(2)
[8]
sage: G = SmoothCharacterGroupQp(2, QQ)
sage: G.subgroup_gens(1)
[]
sage: G.subgroup_gens(2)
[3]
sage: G.subgroup_gens(3)
[5]
"""
if level == 0:
raise ValueError
elif level == 1:
return self.unit_gens(level)[:-1]
else:
return [1 + self.prime()**(level - 1)]
class SmoothCharacterGroupUnramifiedQuadratic(SmoothCharacterGroupGeneric):
r"""
The group of smooth characters of `\QQ_{p^2}^\times`, where `\QQ_{p^2}` is
the unique unramified quadratic extension of `\QQ_p`. We represent
`\QQ_{p^2}^\times` internally as the completion at the prime above `p` of a