The space of \(p\)-adic weights¶

A \(p\)-adic weight is a continuous character \(\ZZ_p^\times \to \CC_p^\times\). These are the \(\CC_p\)-points of a rigid space over \(\QQ_p\), which is isomorphic to a disjoint union of copies (indexed by \((\ZZ/p\ZZ)^\times\)) of the open unit \(p\)-adic disc.

Sage supports both “classical points”, which are determined by the data of a Dirichlet character modulo \(p^m\) for some \(m\) and an integer \(k\) (corresponding to the character \(z \mapsto z^k \chi(z)\)) and “non-classical points” which are determined by the data of an element of \((\ZZ/p\ZZ)^\times\) and an element \(w \in \CC_p\) with \(|w - 1| < 1\).

EXAMPLES:

Sage

sage: W = pAdicWeightSpace(17)
sage: W
Space of 17-adic weight-characters
 defined over 17-adic Field with capped relative precision 20
sage: R.<x> = QQ[]
sage: L = Qp(17).extension(x^2 - 17, names='a'); L.rename('L')
sage: W.base_extend(L)
Space of 17-adic weight-characters defined over L

Python

>>> from sage.all import *
>>> W = pAdicWeightSpace(Integer(17))
>>> W
Space of 17-adic weight-characters
 defined over 17-adic Field with capped relative precision 20
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> L = Qp(Integer(17)).extension(x**Integer(2) - Integer(17), names='a'); L.rename('L')
>>> W.base_extend(L)
Space of 17-adic weight-characters defined over L

Sage Live

W = pAdicWeightSpace(17)
W
R.<x> = QQ[]
L = Qp(17).extension(x^2 - 17, names='a'); L.rename('L')
W.base_extend(L)

We create a simple element of \(\mathcal{W}\): the algebraic character, \(x \mapsto x^6\):

Sage

sage: kappa = W(6)
sage: kappa(5)
15625
sage: kappa(5) == 5^6
True

Python

>>> from sage.all import *
>>> kappa = W(Integer(6))
>>> kappa(Integer(5))
15625
>>> kappa(Integer(5)) == Integer(5)**Integer(6)
True

Sage Live

kappa = W(6)
kappa(5)
kappa(5) == 5^6

A locally algebraic character, \(x \mapsto x^6 \chi(x)\) for \(\chi\) a Dirichlet character mod \(p\):

Sage

sage: kappa2 = W(6, DirichletGroup(17, Qp(17)).0^8)
sage: kappa2(5) == -5^6
True
sage: kappa2(13) == 13^6
True

Python

>>> from sage.all import *
>>> kappa2 = W(Integer(6), DirichletGroup(Integer(17), Qp(Integer(17))).gen(0)**Integer(8))
>>> kappa2(Integer(5)) == -Integer(5)**Integer(6)
True
>>> kappa2(Integer(13)) == Integer(13)**Integer(6)
True

Sage Live

kappa2 = W(6, DirichletGroup(17, Qp(17)).0^8)
kappa2(5) == -5^6
kappa2(13) == 13^6

A non-locally-algebraic character, sending the generator 18 of \(1 + 17 \ZZ_{17}\) to 35 and acting as \(\mu \mapsto \mu^4\) on the group of 16th roots of unity:

Sage

sage: kappa3 = W(35 + O(17^20), 4, algebraic=False)
sage: kappa3(2)
16 + 8*17 + ... + O(17^20)

Python

>>> from sage.all import *
>>> kappa3 = W(Integer(35) + O(Integer(17)**Integer(20)), Integer(4), algebraic=False)
>>> kappa3(Integer(2))
16 + 8*17 + ... + O(17^20)

Sage Live

kappa3 = W(35 + O(17^20), 4, algebraic=False)
kappa3(2)

AUTHORS:

David Loeffler (2008-9)

class sage.modular.overconvergent.weightspace.AlgebraicWeight(parent, k, chi=None)[source]¶

Bases: WeightCharacter

A point in weight space corresponding to a locally algebraic character, of the form \(x \mapsto \chi(x) x^k\) where \(k\) is an integer and \(\chi\) is a Dirichlet character modulo \(p^n\) for some \(n\).

Lvalue()[source]¶

Return the value of the \(p\)-adic \(L\)-function of \(\QQ\) evaluated at this weight-character.

If the character is \(x \mapsto x^k \chi(x)\) where \(k > 0\) and \(\chi\) has conductor a power of \(p\), this is an element of the number field generated by the values of \(\chi\), equal to the value of the complex \(L\)-function \(L(1-k, \chi)\). If \(\chi\) is trivial, it is equal to \((1 - p^{k-1})\zeta(1-k)\).

At present this is not implemented in any other cases, except the trivial character (for which the value is \(\infty\)).

Todo

Implement this more generally using the Amice transform machinery in sage/schemes/elliptic_curves/padic_lseries.py, which should clearly be factored out into a separate class.

EXAMPLES:

Sage

sage: pAdicWeightSpace(7)(4).Lvalue() == (1 - 7^3)*zeta__exact(-3)
True
sage: pAdicWeightSpace(7)(5, DirichletGroup(7, Qp(7)).0^4).Lvalue()
0
sage: pAdicWeightSpace(7)(6, DirichletGroup(7, Qp(7)).0^4).Lvalue()
1 + 2*7 + 7^2 + 3*7^3 + 3*7^5 + 4*7^6 + 2*7^7 + 5*7^8 + 2*7^9 + 3*7^10 + 6*7^11
 + 2*7^12 + 3*7^13 + 5*7^14 + 6*7^15 + 5*7^16 + 3*7^17 + 6*7^18 + O(7^19)

Python

>>> from sage.all import *
>>> pAdicWeightSpace(Integer(7))(Integer(4)).Lvalue() == (Integer(1) - Integer(7)**Integer(3))*zeta__exact(-Integer(3))
True
>>> pAdicWeightSpace(Integer(7))(Integer(5), DirichletGroup(Integer(7), Qp(Integer(7))).gen(0)**Integer(4)).Lvalue()
0
>>> pAdicWeightSpace(Integer(7))(Integer(6), DirichletGroup(Integer(7), Qp(Integer(7))).gen(0)**Integer(4)).Lvalue()
1 + 2*7 + 7^2 + 3*7^3 + 3*7^5 + 4*7^6 + 2*7^7 + 5*7^8 + 2*7^9 + 3*7^10 + 6*7^11
 + 2*7^12 + 3*7^13 + 5*7^14 + 6*7^15 + 5*7^16 + 3*7^17 + 6*7^18 + O(7^19)

Sage Live

pAdicWeightSpace(7)(4).Lvalue() == (1 - 7^3)*zeta__exact(-3)
pAdicWeightSpace(7)(5, DirichletGroup(7, Qp(7)).0^4).Lvalue()
pAdicWeightSpace(7)(6, DirichletGroup(7, Qp(7)).0^4).Lvalue()

chi()[source]¶

If this character is \(x \mapsto x^k \chi(x)\) for an integer \(k\) and a Dirichlet character \(\chi\), return \(\chi\).

EXAMPLES:

Sage

sage: kappa = pAdicWeightSpace(29)(13, DirichletGroup(29, Qp(29)).0^14)
sage: kappa.chi()
Dirichlet character modulo 29 of conductor 29
 mapping 2 |--> 28 + 28*29 + 28*29^2 + ... + O(29^20)

Python

>>> from sage.all import *
>>> kappa = pAdicWeightSpace(Integer(29))(Integer(13), DirichletGroup(Integer(29), Qp(Integer(29))).gen(0)**Integer(14))
>>> kappa.chi()
Dirichlet character modulo 29 of conductor 29
 mapping 2 |--> 28 + 28*29 + 28*29^2 + ... + O(29^20)

Sage Live

kappa = pAdicWeightSpace(29)(13, DirichletGroup(29, Qp(29)).0^14)
kappa.chi()

k()[source]¶

If this character is \(x \mapsto x^k \chi(x)\) for an integer \(k\) and a Dirichlet character \(\chi\), return \(k\).

EXAMPLES:

Sage

sage: kappa = pAdicWeightSpace(29)(13, DirichletGroup(29, Qp(29)).0^14)
sage: kappa.k()
13

Python

>>> from sage.all import *
>>> kappa = pAdicWeightSpace(Integer(29))(Integer(13), DirichletGroup(Integer(29), Qp(Integer(29))).gen(0)**Integer(14))
>>> kappa.k()
13

Sage Live

kappa = pAdicWeightSpace(29)(13, DirichletGroup(29, Qp(29)).0^14)
kappa.k()

teichmuller_type()[source]¶

Return the Teichmuller type of this weight-character \(\kappa\).

This is the unique \(t \in \ZZ/(p-1)\ZZ\) such that \(\kappa(\mu) = \mu^t\) for \(\mu\) a \((p-1)\)-st root of 1.

For \(p = 2\) this does not make sense, but we still want the Teichmuller type to correspond to the index of the component of weight space in which \(\kappa\) lies, so we return 1 if \(\kappa\) is odd and 0 otherwise.

EXAMPLES:

Sage

sage: pAdicWeightSpace(11)(2, DirichletGroup(11,QQ).0).teichmuller_type()
7
sage: pAdicWeightSpace(29)(13, DirichletGroup(29, Qp(29)).0).teichmuller_type()
14
sage: pAdicWeightSpace(2)(3, DirichletGroup(4,QQ).0).teichmuller_type()
0

Python

>>> from sage.all import *
>>> pAdicWeightSpace(Integer(11))(Integer(2), DirichletGroup(Integer(11),QQ).gen(0)).teichmuller_type()
7
>>> pAdicWeightSpace(Integer(29))(Integer(13), DirichletGroup(Integer(29), Qp(Integer(29))).gen(0)).teichmuller_type()
14
>>> pAdicWeightSpace(Integer(2))(Integer(3), DirichletGroup(Integer(4),QQ).gen(0)).teichmuller_type()
0

Sage Live

pAdicWeightSpace(11)(2, DirichletGroup(11,QQ).0).teichmuller_type()
pAdicWeightSpace(29)(13, DirichletGroup(29, Qp(29)).0).teichmuller_type()
pAdicWeightSpace(2)(3, DirichletGroup(4,QQ).0).teichmuller_type()

class sage.modular.overconvergent.weightspace.ArbitraryWeight(parent, w, t)[source]¶

Bases: WeightCharacter

Create the element of \(p\)-adic weight space in the given component mapping 1 + p to w.

Here w must be an element of a \(p\)-adic field, with finite precision.

EXAMPLES:

Sage

sage: pAdicWeightSpace(17)(1 + 17^2 + O(17^3), 11, False)
[1 + 17^2 + O(17^3), 11]

Python

>>> from sage.all import *
>>> pAdicWeightSpace(Integer(17))(Integer(1) + Integer(17)**Integer(2) + O(Integer(17)**Integer(3)), Integer(11), False)
[1 + 17^2 + O(17^3), 11]

Sage Live

pAdicWeightSpace(17)(1 + 17^2 + O(17^3), 11, False)

teichmuller_type()[source]¶

Return the Teichmuller type of this weight-character \(\kappa\).

This is the unique \(t \in \ZZ/(p-1)\ZZ\) such that \(\kappa(\mu) = \mu^t\) for mu a \((p-1)\)-st root of 1.

EXAMPLES:

Sage

sage: pAdicWeightSpace(17)(1 + 3*17 + 2*17^2 + O(17^3), 8, False).teichmuller_type()
8
sage: pAdicWeightSpace(2)(1 + 2 + O(2^2), 1, False).teichmuller_type()
1

Python

>>> from sage.all import *
>>> pAdicWeightSpace(Integer(17))(Integer(1) + Integer(3)*Integer(17) + Integer(2)*Integer(17)**Integer(2) + O(Integer(17)**Integer(3)), Integer(8), False).teichmuller_type()
8
>>> pAdicWeightSpace(Integer(2))(Integer(1) + Integer(2) + O(Integer(2)**Integer(2)), Integer(1), False).teichmuller_type()
1

Sage Live

pAdicWeightSpace(17)(1 + 3*17 + 2*17^2 + O(17^3), 8, False).teichmuller_type()
pAdicWeightSpace(2)(1 + 2 + O(2^2), 1, False).teichmuller_type()

class sage.modular.overconvergent.weightspace.WeightCharacter(parent)[source]¶

Bases: Element

Abstract base class representing an element of the \(p\)-adic weight space \(Hom(\ZZ_p^\times, \CC_p^\times)\).

Lvalue()[source]¶

Return the value of the \(p\)-adic \(L\)-function of \(\QQ\), which can be regarded as a rigid-analytic function on weight space, evaluated at this character.

EXAMPLES:

Sage

sage: W = pAdicWeightSpace(11)
sage: sage.modular.overconvergent.weightspace.WeightCharacter(W).Lvalue()
Traceback (most recent call last):
...
NotImplementedError

Python

>>> from sage.all import *
>>> W = pAdicWeightSpace(Integer(11))
>>> sage.modular.overconvergent.weightspace.WeightCharacter(W).Lvalue()
Traceback (most recent call last):
...
NotImplementedError

Sage Live

W = pAdicWeightSpace(11)
sage.modular.overconvergent.weightspace.WeightCharacter(W).Lvalue()

base_extend(R)[source]¶

Extend scalars to the base ring R.

The ring R must have a canonical map from the current base ring.

EXAMPLES:

Sage

sage: w = pAdicWeightSpace(17, QQ)(3)
sage: w.base_extend(Qp(17))
3

Python

>>> from sage.all import *
>>> w = pAdicWeightSpace(Integer(17), QQ)(Integer(3))
>>> w.base_extend(Qp(Integer(17)))
3

Sage Live

w = pAdicWeightSpace(17, QQ)(3)
w.base_extend(Qp(17))

is_even()[source]¶

Return True if this weight-character sends -1 to +1.

EXAMPLES:

Sage

sage: pAdicWeightSpace(17)(0).is_even()
True
sage: pAdicWeightSpace(17)(11).is_even()
False
sage: pAdicWeightSpace(17)(1 + 17 + O(17^20), 3, False).is_even()
False
sage: pAdicWeightSpace(17)(1 + 17 + O(17^20), 4, False).is_even()
True

Python

>>> from sage.all import *
>>> pAdicWeightSpace(Integer(17))(Integer(0)).is_even()
True
>>> pAdicWeightSpace(Integer(17))(Integer(11)).is_even()
False
>>> pAdicWeightSpace(Integer(17))(Integer(1) + Integer(17) + O(Integer(17)**Integer(20)), Integer(3), False).is_even()
False
>>> pAdicWeightSpace(Integer(17))(Integer(1) + Integer(17) + O(Integer(17)**Integer(20)), Integer(4), False).is_even()
True

Sage Live

pAdicWeightSpace(17)(0).is_even()
pAdicWeightSpace(17)(11).is_even()
pAdicWeightSpace(17)(1 + 17 + O(17^20), 3, False).is_even()
pAdicWeightSpace(17)(1 + 17 + O(17^20), 4, False).is_even()

is_trivial()[source]¶

Return True if and only if this is the trivial character.

EXAMPLES:

Sage

sage: pAdicWeightSpace(11)(2).is_trivial()
False
sage: pAdicWeightSpace(11)(2, DirichletGroup(11, QQ).0).is_trivial()
False
sage: pAdicWeightSpace(11)(0).is_trivial()
True

Python

>>> from sage.all import *
>>> pAdicWeightSpace(Integer(11))(Integer(2)).is_trivial()
False
>>> pAdicWeightSpace(Integer(11))(Integer(2), DirichletGroup(Integer(11), QQ).gen(0)).is_trivial()
False
>>> pAdicWeightSpace(Integer(11))(Integer(0)).is_trivial()
True

Sage Live

pAdicWeightSpace(11)(2).is_trivial()
pAdicWeightSpace(11)(2, DirichletGroup(11, QQ).0).is_trivial()
pAdicWeightSpace(11)(0).is_trivial()

one_over_Lvalue()[source]¶

Return the reciprocal of the \(p\)-adic \(L\)-function evaluated at this weight-character.

If the weight-character is odd, then the \(L\)-function is zero, so an error will be raised.

EXAMPLES:

Sage

sage: pAdicWeightSpace(11)(4).one_over_Lvalue()
-12/133
sage: pAdicWeightSpace(11)(3, DirichletGroup(11, QQ).0).one_over_Lvalue()
-1/6
sage: pAdicWeightSpace(11)(3).one_over_Lvalue()
Traceback (most recent call last):
...
ZeroDivisionError: rational division by zero
sage: pAdicWeightSpace(11)(0).one_over_Lvalue()
0
sage: type(_)
<class 'sage.rings.integer.Integer'>

Python

>>> from sage.all import *
>>> pAdicWeightSpace(Integer(11))(Integer(4)).one_over_Lvalue()
-12/133
>>> pAdicWeightSpace(Integer(11))(Integer(3), DirichletGroup(Integer(11), QQ).gen(0)).one_over_Lvalue()
-1/6
>>> pAdicWeightSpace(Integer(11))(Integer(3)).one_over_Lvalue()
Traceback (most recent call last):
...
ZeroDivisionError: rational division by zero
>>> pAdicWeightSpace(Integer(11))(Integer(0)).one_over_Lvalue()
0
>>> type(_)
<class 'sage.rings.integer.Integer'>

Sage Live

pAdicWeightSpace(11)(4).one_over_Lvalue()
pAdicWeightSpace(11)(3, DirichletGroup(11, QQ).0).one_over_Lvalue()
pAdicWeightSpace(11)(3).one_over_Lvalue()
pAdicWeightSpace(11)(0).one_over_Lvalue()
type(_)

pAdicEisensteinSeries(ring, prec=20)[source]¶

Calculate the \(q\)-expansion of the \(p\)-adic Eisenstein series of given weight-character, normalised so the constant term is 1.

EXAMPLES:

Sage

sage: kappa = pAdicWeightSpace(3)(3, DirichletGroup(3,QQ).0)
sage: kappa.pAdicEisensteinSeries(QQ[['q']], 20)
1 - 9*q + 27*q^2 - 9*q^3 - 117*q^4 + 216*q^5 + 27*q^6 - 450*q^7 + 459*q^8
 - 9*q^9 - 648*q^10 + 1080*q^11 - 117*q^12 - 1530*q^13 + 1350*q^14 + 216*q^15
 - 1845*q^16 + 2592*q^17 + 27*q^18 - 3258*q^19 + O(q^20)

Python

>>> from sage.all import *
>>> kappa = pAdicWeightSpace(Integer(3))(Integer(3), DirichletGroup(Integer(3),QQ).gen(0))
>>> kappa.pAdicEisensteinSeries(QQ[['q']], Integer(20))
1 - 9*q + 27*q^2 - 9*q^3 - 117*q^4 + 216*q^5 + 27*q^6 - 450*q^7 + 459*q^8
 - 9*q^9 - 648*q^10 + 1080*q^11 - 117*q^12 - 1530*q^13 + 1350*q^14 + 216*q^15
 - 1845*q^16 + 2592*q^17 + 27*q^18 - 3258*q^19 + O(q^20)

Sage Live

kappa = pAdicWeightSpace(3)(3, DirichletGroup(3,QQ).0)
kappa.pAdicEisensteinSeries(QQ[['q']], 20)

values_on_gens()[source]¶

If \(\kappa\) is this character, calculate the values \((\kappa(r), t)\) where \(r\) is \(1 + p\) (or 5 if \(p = 2\)) and \(t\) is the unique element of \(\ZZ/(p-1)\ZZ\) such that \(\kappa(\mu) = \mu^t\) for \(\mu\) a (p-1)st root of unity. (If \(p = 2\), we take \(t\) to be 0 or 1 according to whether \(\kappa\) is odd or even.) These two values uniquely determine the character \(\kappa\).

EXAMPLES:

Sage

sage: W = pAdicWeightSpace(11); W(2).values_on_gens()
(1 + 2*11 + 11^2 + O(11^20), 2)
sage: W(2, DirichletGroup(11, QQ).0).values_on_gens()
(1 + 2*11 + 11^2 + O(11^20), 7)
sage: W(1 + 2*11 + O(11^5), 4, algebraic = False).values_on_gens()
(1 + 2*11 + O(11^5), 4)

Python

>>> from sage.all import *
>>> W = pAdicWeightSpace(Integer(11)); W(Integer(2)).values_on_gens()
(1 + 2*11 + 11^2 + O(11^20), 2)
>>> W(Integer(2), DirichletGroup(Integer(11), QQ).gen(0)).values_on_gens()
(1 + 2*11 + 11^2 + O(11^20), 7)
>>> W(Integer(1) + Integer(2)*Integer(11) + O(Integer(11)**Integer(5)), Integer(4), algebraic = False).values_on_gens()
(1 + 2*11 + O(11^5), 4)

Sage Live

W = pAdicWeightSpace(11); W(2).values_on_gens()
W(2, DirichletGroup(11, QQ).0).values_on_gens()
W(1 + 2*11 + O(11^5), 4, algebraic = False).values_on_gens()

class sage.modular.overconvergent.weightspace.WeightSpace_class(p, base_ring)[source]¶

Bases: Parent

The space of \(p\)-adic weight-characters \(\mathcal{W} = {\rm Hom}(\ZZ_p^\times, \CC_p^\times)\).

This is isomorphic to a disjoint union of \((p-1)\) open discs of radius 1 (or 2 such discs if \(p = 2\)), with the parameter on the open disc corresponding to the image of \(1 + p\) (or 5 if \(p = 2\))

base_extend(R)[source]¶

Extend scalars to the ring R.

There must be a canonical coercion map from the present base ring to R.

EXAMPLES:

Sage

sage: W = pAdicWeightSpace(3, QQ)
sage: W.base_extend(Qp(3))
Space of 3-adic weight-characters
 defined over 3-adic Field with capped relative precision 20
sage: W.base_extend(IntegerModRing(12))
Traceback (most recent call last):
...
TypeError: No coercion map from 'Rational Field'
to 'Ring of integers modulo 12' is defined

Python

>>> from sage.all import *
>>> W = pAdicWeightSpace(Integer(3), QQ)
>>> W.base_extend(Qp(Integer(3)))
Space of 3-adic weight-characters
 defined over 3-adic Field with capped relative precision 20
>>> W.base_extend(IntegerModRing(Integer(12)))
Traceback (most recent call last):
...
TypeError: No coercion map from 'Rational Field'
to 'Ring of integers modulo 12' is defined

Sage Live

W = pAdicWeightSpace(3, QQ)
W.base_extend(Qp(3))
W.base_extend(IntegerModRing(12))

prime()[source]¶

Return the prime \(p\) such that this is a \(p\)-adic weight space.

EXAMPLES:

Sage

sage: pAdicWeightSpace(17).prime()
17

Python

>>> from sage.all import *
>>> pAdicWeightSpace(Integer(17)).prime()
17

Sage Live

pAdicWeightSpace(17).prime()

zero()[source]¶

Return the zero of this weight space.

EXAMPLES:

Sage

sage: W = pAdicWeightSpace(17)
sage: W.zero()
0

Python

>>> from sage.all import *
>>> W = pAdicWeightSpace(Integer(17))
>>> W.zero()
0

Sage Live

W = pAdicWeightSpace(17)
W.zero()

sage.modular.overconvergent.weightspace.WeightSpace_constructor(p, base_ring=None)[source]¶

Construct the \(p\)-adic weight space for the given prime p.

Note that the “base ring” of a \(p\)-adic weight is the smallest ring containing the image of \(\ZZ\); in particular, although the default base ring is \(\QQ_p\), base ring \(\QQ\) will also work.

EXAMPLES:

Sage

sage: pAdicWeightSpace(3) # indirect doctest
Space of 3-adic weight-characters
 defined over 3-adic Field with capped relative precision 20
sage: pAdicWeightSpace(3, QQ)
Space of 3-adic weight-characters defined over Rational Field
sage: pAdicWeightSpace(10)
Traceback (most recent call last):
...
ValueError: p must be prime

Python

>>> from sage.all import *
>>> pAdicWeightSpace(Integer(3)) # indirect doctest
Space of 3-adic weight-characters
 defined over 3-adic Field with capped relative precision 20
>>> pAdicWeightSpace(Integer(3), QQ)
Space of 3-adic weight-characters defined over Rational Field
>>> pAdicWeightSpace(Integer(10))
Traceback (most recent call last):
...
ValueError: p must be prime

Sage Live

pAdicWeightSpace(3) # indirect doctest
pAdicWeightSpace(3, QQ)
pAdicWeightSpace(10)