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 pari:idealstar and pari: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

class sage.modular.local_comp.smoothchar.SmoothCharacterGeneric(parent, c, values_on_gens)

Bases: MultiplicativeGroupElement

A smooth (i.e. locally constant) character of \(F^\times\), for \(F\) some finite extension of \(\QQ_p\).

galois_conjugate()

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

level()

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

multiplicative_order()

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

restrict_to_Qp()

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

class sage.modular.local_comp.smoothchar.SmoothCharacterGroupGeneric(p, base_ring)

Bases: Parent

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

alias of SmoothCharacterGeneric

base_extend(ring)

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

change_ring(ring)

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 ...>

character(level, values_on_gens)

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

compose_with_norm(chi)

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:

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:

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

discrete_log(level)

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 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 ...>

exponents(level)

The orders \(n_1, \dots, n_d\) of the generators \(x_i\) of \(F^\times / (1 + \mathfrak{p}^c)^\times\) returned by 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 ...>

ideal(level)

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 ...>

norm_character()

Return the normalised absolute value character in this group (mapping a uniformiser to \(1/q\) where \(q\) is the order of the residue field).

EXAMPLES:

sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupQp, SmoothCharacterGroupUnramifiedQuadratic
sage: SmoothCharacterGroupQp(5, QQ).norm_character()
Character of Q_5*, of level 0, mapping 5 |--> 1/5
sage: SmoothCharacterGroupUnramifiedQuadratic(2, QQ).norm_character()
Character of unramified extension Q_2(s)* (s^2 + s + 1 = 0), of level 0, mapping 2 |--> 1/4

prime()

The residue characteristic of the underlying field.

EXAMPLES:

sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupGeneric
sage: SmoothCharacterGroupGeneric(3, QQ).prime()
3

subgroup_gens(level)

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 ...>

unit_gens(level)

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 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 ...>

class sage.modular.local_comp.smoothchar.SmoothCharacterGroupQp(p, base_ring)

Bases: SmoothCharacterGroupGeneric

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

change_ring(ring)

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 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

discrete_log(level, x)

Express the class of \(x\) in \(\QQ_p^\times / (1 + p^c)^\times\) in terms of the generators returned by 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]

exponents(level)

Return the exponents of the generators returned by 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]

from_dirichlet(chi)

Given a Dirichlet character \(\chi\), return the factor at p of the adelic character \(\phi\) which satisfies \(\phi(\varpi_\ell) = \chi(\ell)\) for almost all \(\ell\), where \(\varpi_\ell\) is a uniformizer at \(\ell\).

More concretely, if we write \(\chi = \chi_p \chi_M\) as a product of characters of p-power, resp prime-to-p, conductor, then this function returns the character of \(\QQ_p^\times\) sending \(p\) to \(\chi_M(p)\) and agreeing with \(\chi_p^{-1}\) on integers that are 1 modulo M and coprime to \(p\).

EXAMPLES:

sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupQp
sage: G = SmoothCharacterGroupQp(3, CyclotomicField(6))
sage: G.from_dirichlet(DirichletGroup(9).0)
Character of Q_3*, of level 2, mapping 2 |--> -zeta6 + 1, 3 |--> 1

ideal(level)

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

number_field()

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

quadratic_chars()

Return a list of the (non-trivial) quadratic characters in this group. This will be a list of 3 characters, unless \(p = 2\) when there are 7.

EXAMPLES:

sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupQp
sage: SmoothCharacterGroupQp(7, QQ).quadratic_chars()
[Character of Q_7*, of level 0, mapping 7 |--> -1,
 Character of Q_7*, of level 1, mapping 3 |--> -1, 7 |--> -1,
 Character of Q_7*, of level 1, mapping 3 |--> -1, 7 |--> 1]
sage: SmoothCharacterGroupQp(2, QQ).quadratic_chars()
[Character of Q_2*, of level 0, mapping 2 |--> -1,
 Character of Q_2*, of level 2, mapping 3 |--> -1, 2 |--> -1,
 Character of Q_2*, of level 2, mapping 3 |--> -1, 2 |--> 1,
 Character of Q_2*, of level 3, mapping 7 |--> -1, 5 |--> -1, 2 |--> -1,
 Character of Q_2*, of level 3, mapping 7 |--> -1, 5 |--> -1, 2 |--> 1,
 Character of Q_2*, of level 3, mapping 7 |--> 1, 5 |--> -1, 2 |--> -1,
 Character of Q_2*, of level 3, mapping 7 |--> 1, 5 |--> -1, 2 |--> 1]

subgroup_gens(level)

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]

unit_gens(level)

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]

class sage.modular.local_comp.smoothchar.SmoothCharacterGroupQuadratic(p, base_ring)

Bases: SmoothCharacterGroupGeneric

The group of smooth characters of \(E^\times\), where \(E\) is a quadratic extension of \(\QQ_p\).

discrete_log(level, x, gens=None)

Express the class of \(x\) in \(F^\times / (1 + \mathfrak{p}^c)^\times\) in terms of the generators returned by self.unit_gens(level), or a custom set of generators if given.

EXAMPLES:

sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupUnramifiedQuadratic
sage: G = SmoothCharacterGroupUnramifiedQuadratic(2, QQ)
sage: G.discrete_log(0, 12)
[2]
sage: G.discrete_log(1, 12)
[0, 2]
sage: v = G.discrete_log(5, 12); v
[0, 2, 0, 1, 2]
sage: g = G.unit_gens(5); prod([g[i]**v[i] for i in [0..4]])/12 - 1 in G.ideal(5)
True
sage: G.discrete_log(3,G.number_field()([1,1]))
[2, 0, 0, 1, 0]
sage: H = SmoothCharacterGroupUnramifiedQuadratic(5, QQ)
sage: x = H.number_field()([1,1]); x
s + 1
sage: v = H.discrete_log(5, x); v
[22, 263, 379, 0]
sage: h = H.unit_gens(5); prod([h[i]**v[i] for i in [0..3]])/x - 1 in H.ideal(5)
True

sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupRamifiedQuadratic
sage: G = SmoothCharacterGroupRamifiedQuadratic(3, 1, QQ)
sage: s = G.number_field().gen()
sage: dl = G.discrete_log(4, 3 + 2*s)
sage: gs = G.unit_gens(4); gs[0]^dl[0] * gs[1]^dl[1] * gs[2]^dl[2] * gs[3]^dl[3] - (3 + 2*s) in G.ideal(4)
True

An example with a custom generating set:

sage: G.discrete_log(2, s+3, gens=[s, s+1, 2])
[1, 2, 0]

extend_character(level, chi, vals, check=True)

Return the unique character of \(F^\times\) which coincides with \(\chi\) on \(\QQ_p^\times\) and maps the generators of the quotient returned by quotient_gens() to vals.

INPUT:

• chi: a smooth character of \(\QQ_p\), where \(p\) is the residue characteristic of \(F\), with values in the base ring of self (or some other ring coercible to it)

• level: the level of the new character (which should be at least the level of chi)

• vals: a list of elements of the base ring of self (or some other ring coercible to it), specifying values on the quotients returned by quotient_gens().

A ValueError will be raised if \(x^t \ne \chi(\alpha^t)\), where \(t\) is the smallest integer such that \(\alpha^t\) is congruent modulo \(p^{\rm level}\) to an element of \(\QQ_p\).

EXAMPLES:

We extend an unramified character of \(\QQ_3^\times\) to the unramified quadratic extension in various ways.

sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupQp, SmoothCharacterGroupUnramifiedQuadratic
sage: chi = SmoothCharacterGroupQp(5, QQ).character(0, [7]); chi
Character of Q_5*, of level 0, mapping 5 |--> 7
sage: G = SmoothCharacterGroupUnramifiedQuadratic(5, QQ)
sage: G.extend_character(1, chi, [-1])
Character of unramified extension Q_5(s)* (s^2 + 4*s + 2 = 0), of level 1, mapping s |--> -1, 5 |--> 7
sage: G.extend_character(2, chi, [-1])
Character of unramified extension Q_5(s)* (s^2 + 4*s + 2 = 0), of level 1, mapping s |--> -1, 5 |--> 7
sage: G.extend_character(3, chi, [1])
Character of unramified extension Q_5(s)* (s^2 + 4*s + 2 = 0), of level 0, mapping 5 |--> 7
sage: K.<z> = CyclotomicField(6); G.base_extend(K).extend_character(1, chi, [z])
Character of unramified extension Q_5(s)* (s^2 + 4*s + 2 = 0), of level 1, mapping s |--> -z + 1, 5 |--> 7

We extend the nontrivial quadratic character:

sage: chi = SmoothCharacterGroupQp(5, QQ).character(1, [-1, 7])
sage: K.<z> = CyclotomicField(24); G.base_extend(K).extend_character(1, chi, [z^6])
Character of unramified extension Q_5(s)* (s^2 + 4*s + 2 = 0), of level 1, mapping s |--> -z^6, 5 |--> 7

Extensions of higher level:

sage: K.<z> = CyclotomicField(20); rho = G.base_extend(K).extend_character(2, chi, [z]); rho
Character of unramified extension Q_5(s)* (s^2 + 4*s + 2 = 0), of level 2, mapping 11*s - 10 |--> z^5, 6 |--> 1, 5*s + 1 |--> z^4, 5 |--> 7
sage: rho(3)
-1

Examples where it doesn’t work:

sage: G.extend_character(1, chi, [1])
Traceback (most recent call last):
...
ValueError: Invalid values for extension

sage: G = SmoothCharacterGroupQp(2, QQ); H = SmoothCharacterGroupUnramifiedQuadratic(2, QQ)
sage: chi = G.character(3, [1, -1, 7])
sage: H.extend_character(2, chi, [-1])
Traceback (most recent call last):
...
ValueError: Level of extended character cannot be smaller than level of character of Qp

quotient_gens(n)

Return a list of elements of \(E\) which are a generating set for the quotient \(E^\times / \QQ_p^\times\), consisting of elements which are “minimal” in the sense of [LW12].

In the examples we implement here, this quotient is almost always cyclic: the exceptions are the unramified quadratic extension of \(\QQ_2\) for \(n \ge 3\), and the extension \(\QQ_3(\sqrt{-3})\) for \(n \ge 4\).

EXAMPLES:

sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupUnramifiedQuadratic
sage: G = SmoothCharacterGroupUnramifiedQuadratic(7,QQ)
sage: G.quotient_gens(1)
[2*s - 2]
sage: G.quotient_gens(2)
[15*s + 21]
sage: G.quotient_gens(3)
[-75*s + 33]

A ramified case:

sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupRamifiedQuadratic
sage: G = SmoothCharacterGroupRamifiedQuadratic(7, 0, QQ)
sage: G.quotient_gens(3)
[22*s + 21]

An example where the quotient group is not cyclic:

sage: G = SmoothCharacterGroupUnramifiedQuadratic(2,QQ)
sage: G.quotient_gens(1)
[s + 1]
sage: G.quotient_gens(2)
[-s + 2]
sage: G.quotient_gens(3)
[-17*s - 14, 3*s - 2]

class sage.modular.local_comp.smoothchar.SmoothCharacterGroupRamifiedQuadratic(prime, flag, base_ring, names='s')

Bases: SmoothCharacterGroupQuadratic

The group of smooth characters of \(K^\times\), where \(K\) is a ramified quadratic extension of \(\QQ_p\), and \(p \ne 2\).

change_ring(ring)

Return the character group of the same field, but with values in a different coefficient 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 base_extend() if you want this.

EXAMPLES:

sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupRamifiedQuadratic
sage: SmoothCharacterGroupRamifiedQuadratic(7, 1, Zmod(3), names='foo').change_ring(CC)
Group of smooth characters of ramified extension Q_7(foo)* (foo^2 - 35 = 0) with values in Complex Field with 53 bits of precision

exponents(c)

Return the orders of the independent generators of the unit group returned by unit_gens().

EXAMPLES:

sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupRamifiedQuadratic
sage: G = SmoothCharacterGroupRamifiedQuadratic(5, 0, QQ)
sage: G.exponents(0)
(0,)
sage: G.exponents(1)
(4, 0)
sage: G.exponents(8)
(500, 625, 0)

ideal(c)

Return the ideal \(p^c\) of self.number_field(). The result is cached, since we use the methods idealstar() and ideallog() which cache a Pari bid structure.

EXAMPLES:

sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupRamifiedQuadratic
sage: G = SmoothCharacterGroupRamifiedQuadratic(5, 1, QQ, 'a'); I = G.ideal(3); I
Fractional ideal (25, 5*a)
sage: I is G.ideal(3)
True

number_field()

Return a number field of which this is the completion at \(p\).

EXAMPLES:

sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupRamifiedQuadratic
sage: SmoothCharacterGroupRamifiedQuadratic(7, 0, QQ, 'a').number_field()
Number Field in a with defining polynomial x^2 - 7
sage: SmoothCharacterGroupRamifiedQuadratic(5, 1, QQ, 'b').number_field()
Number Field in b with defining polynomial x^2 - 10
sage: SmoothCharacterGroupRamifiedQuadratic(7, 1, Zmod(6), 'c').number_field()
Number Field in c with defining polynomial x^2 - 35

subgroup_gens(level)

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 SmoothCharacterGroupRamifiedQuadratic
sage: G = SmoothCharacterGroupRamifiedQuadratic(3, 1, QQ)
sage: G.subgroup_gens(2)
[s + 1]

unit_gens(c)

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 exponents()). We adopt the convention that the final generator \(x_d\) is a uniformiser.

EXAMPLES:

sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupRamifiedQuadratic
sage: G = SmoothCharacterGroupRamifiedQuadratic(5, 0, QQ)
sage: G.unit_gens(0)
[s]
sage: G.unit_gens(1)
[2, s]
sage: G.unit_gens(8)
[2, s + 1, s]

class sage.modular.local_comp.smoothchar.SmoothCharacterGroupUnramifiedQuadratic(prime, base_ring, names='s')

Bases: SmoothCharacterGroupQuadratic

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 quadratic number field, defined by (the obvious lift to \(\ZZ\) of) the Conway polynomial modulo \(p\) of degree 2.

EXAMPLES:

sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupUnramifiedQuadratic
sage: G = SmoothCharacterGroupUnramifiedQuadratic(3, QQ); G
Group of smooth characters of unramified extension Q_3(s)* (s^2 + 2*s + 2 = 0) with values in Rational Field
sage: G.unit_gens(3)
[-11*s, 4, 3*s + 1, 3]
sage: TestSuite(G).run()
sage: TestSuite(SmoothCharacterGroupUnramifiedQuadratic(2, QQ)).run()

change_ring(ring)

Return the character group of the same field, but with values in a different coefficient 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 base_extend() if you want this.

EXAMPLES:

sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupUnramifiedQuadratic
sage: SmoothCharacterGroupUnramifiedQuadratic(7, Zmod(3), names='foo').change_ring(CC)
Group of smooth characters of unramified extension Q_7(foo)* (foo^2 + 6*foo + 3 = 0) with values in Complex Field with 53 bits of precision

exponents(c)

The orders \(n_1, \dots, n_d\) of the generators \(x_i\) of \(F^\times / (1 + \mathfrak{p}^c)^\times\) returned by unit_gens().

EXAMPLES:

sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupUnramifiedQuadratic
sage: SmoothCharacterGroupUnramifiedQuadratic(7, QQ).exponents(2)
[48, 7, 7, 0]
sage: SmoothCharacterGroupUnramifiedQuadratic(2, QQ).exponents(3)
[3, 4, 2, 2, 0]
sage: SmoothCharacterGroupUnramifiedQuadratic(2, QQ).exponents(2)
[3, 2, 2, 0]

ideal(c)

Return the ideal \(p^c\) of self.number_field(). The result is cached, since we use the methods idealstar() and ideallog() which cache a Pari bid structure.

EXAMPLES:

sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupUnramifiedQuadratic
sage: G = SmoothCharacterGroupUnramifiedQuadratic(7, QQ, 'a'); I = G.ideal(3); I
Fractional ideal (343)
sage: I is G.ideal(3)
True

number_field()

Return a number field of which this is the completion at \(p\), defined by a polynomial whose discriminant is not divisible by \(p\).

EXAMPLES:

sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupUnramifiedQuadratic
sage: SmoothCharacterGroupUnramifiedQuadratic(7, QQ, 'a').number_field()
Number Field in a with defining polynomial x^2 + 6*x + 3
sage: SmoothCharacterGroupUnramifiedQuadratic(5, QQ, 'b').number_field()
Number Field in b with defining polynomial x^2 + 4*x + 2
sage: SmoothCharacterGroupUnramifiedQuadratic(2, QQ, 'c').number_field()
Number Field in c with defining polynomial x^2 + x + 1

subgroup_gens(level)

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 SmoothCharacterGroupUnramifiedQuadratic
sage: SmoothCharacterGroupUnramifiedQuadratic(7, QQ).subgroup_gens(1)
[s]
sage: SmoothCharacterGroupUnramifiedQuadratic(7, QQ).subgroup_gens(2)
[8, 7*s + 1]
sage: SmoothCharacterGroupUnramifiedQuadratic(2, QQ).subgroup_gens(2)
[3, 2*s + 1]

unit_gens(c)

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 exponents()). We adopt the convention that the final generator \(x_d\) is a uniformiser (and \(n_d = 0\)).

ALGORITHM: Use Teichmueller lifts.

EXAMPLES:

sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupUnramifiedQuadratic
sage: SmoothCharacterGroupUnramifiedQuadratic(7, QQ).unit_gens(0)
[7]
sage: SmoothCharacterGroupUnramifiedQuadratic(7, QQ).unit_gens(1)
[s, 7]
sage: SmoothCharacterGroupUnramifiedQuadratic(7, QQ).unit_gens(2)
[22*s, 8, 7*s + 1, 7]
sage: SmoothCharacterGroupUnramifiedQuadratic(7, QQ).unit_gens(3)
[169*s + 49, 8, 7*s + 1, 7]

In the 2-adic case there can be more than 4 generators:

sage: SmoothCharacterGroupUnramifiedQuadratic(2, QQ).unit_gens(0)
[2]
sage: SmoothCharacterGroupUnramifiedQuadratic(2, QQ).unit_gens(1)
[s, 2]
sage: SmoothCharacterGroupUnramifiedQuadratic(2, QQ).unit_gens(2)
[s, 2*s + 1, -1, 2]
sage: SmoothCharacterGroupUnramifiedQuadratic(2, QQ).unit_gens(3)
[s, 2*s + 1, 4*s + 1, -1, 2]