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# 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

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

  Bases: `sage.structure.element.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.

    EXAMPLE:

        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.

    EXAMPLE:

        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 \(F^\times\).

    EXAMPLE:

        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: `sage.structure.parent_base.ParentWithBase`

  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.

    EXAMPLE:

        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).

    EXAMPLE:

        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

    TESTS:

        sage: G.character(1, [2])
        Traceback (most recent call last):
        ...
        AssertionError: 2 images must be given

  `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 \(\QQ_p^\times\).

    EXAMPLE:

    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.

    EXAMPLE:

        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()`.

    EXAMPLE:

        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.

    EXAMPLE:

        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 ...>

  `prime`()

    The residue characteristic of the underlying field.

    EXAMPLE:

        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 \((\mathcal{O}_F / \mathfrak{p}^{c-1})^\times\).

    EXAMPLE:

        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 \(n_d = 0\)).

    EXAMPLE:

        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: `sage.modular.local_comp.smoothchar.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.

    EXAMPLE:

        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()`.

    EXAMPLE:

        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()`.

    EXAMPLE:

        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]

  `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 \(\ZZ\).

    EXAMPLE:

        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.

    EXAMPLE:

        sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupQp
        sage: SmoothCharacterGroupQp(7, Zmod(3)).number_field()
        Rational Field

  `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.

    EXAMPLE:

        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.``SmoothCharacterGroupRamifiedQuadratic`(prime, flag, base_ring, names='s')

  Bases: `sage.modular.local_comp.smoothchar.SmoothCharacterGroupGeneric`

  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.

    EXAMPLE:

        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 + 7 = 0) with values in Complex Field with 53 bits of precision

  `discrete_log`(level, x)

    Solve the discrete log problem in the unit group.

    EXAMPLE:

        sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupRamifiedQuadratic
        sage: G = SmoothCharacterGroupRamifiedQuadratic(3, 1, QQ)
        sage: s = G.number_field().gen()
        sage: G.discrete_log(4, 3 + 2*s)
        [5, 2, 1, 1]
        sage: gs = G.unit_gens(4); gs[0]^5 * gs[1]^2 * gs[2] * gs[3] - (3 + 2*s) in G.ideal(4)
        True

  `exponents`(c)

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

    EXAMPLE:

        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 + 7

  `subgroup_gens`(level)

    A set of elements of \((\mathcal{O}_F / \mathfrak{p}^c)^\times\) generating the kernel of the reduction map to \((\mathcal{O}_F / \mathfrak{p}^{c-1})^\times\).

    EXAMPLE:

        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.

    EXAMPLE:

        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: `sage.modular.local_comp.smoothchar.SmoothCharacterGroupGeneric`

  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.

  EXAMPLE:

      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.

    EXAMPLE:

        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

  `discrete_log`(level, x)

    Express the class of \(x\) in \(F^\times / (1 + \mathfrak{p}^c)^\times\) in terms of the generators returned by `self.unit_gens(level)`.

    EXAMPLE:

        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

  `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()`.

    EXAMPLE:

        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]

  `extend_character`(level, chi, x, check=True)

    Return the unique character of \(F^\times\) which coincides with \(\chi\) on \(\QQ_p^\times\) and maps the generator \(\alpha\) returned by `quotient_gen()` to \(x\).

    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`)
    - `x`: an element of the base ring of self (or some other ring coercible to it).

    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, 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^6, 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: Value at s must satisfy x^6 = chi(2) = -1, but it does not

        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

  `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

  `quotient_gen`(level)

    Find an element generating the quotient

    \[\mathcal{O}_F^\times / \ZZ_p^\times \cdot (1 + p^c \mathcal{O}_F),\]

    where \(c\) is the given level.

    EXAMPLE:

        sage: from sage.modular.local_comp.smoothchar import SmoothCharacterGroupUnramifiedQuadratic
        sage: G = SmoothCharacterGroupUnramifiedQuadratic(7,QQ)
        sage: G.quotient_gen(1)
        s
        sage: G.quotient_gen(2)
        -20*s - 21
        sage: G.quotient_gen(3)
        -69*s - 70

    For \(p = 2\) an error will be raised for level \(\ge 3\), as the quotient is not cyclic:

        sage: G = SmoothCharacterGroupUnramifiedQuadratic(2,QQ)
        sage: G.quotient_gen(1)
        s
        sage: G.quotient_gen(2)
        -s + 2
        sage: G.quotient_gen(3)
        Traceback (most recent call last):
        ...
        ValueError: Quotient group not cyclic

  `subgroup_gens`(level)

    A set of elements of \((\mathcal{O}_F / \mathfrak{p}^c)^\times\) generating the kernel of the reduction map to \((\mathcal{O}_F / \mathfrak{p}^{c-1})^\times\).

    EXAMPLE:

        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.

    EXAMPLE:

        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]

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