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: 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

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

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

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

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

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

AUTHORS:

• David Loeffler (2008-9)

class sage.modular.overconvergent.weightspace.AlgebraicWeight(parent, k, chi=None)

Bases: sage.modular.overconvergent.weightspace.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()

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: 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)

chi()

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

EXAMPLES:

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)

k()

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

EXAMPLES:

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

teichmuller_type()

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: 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

class sage.modular.overconvergent.weightspace.ArbitraryWeight(parent, w, t)

Bases: sage.modular.overconvergent.weightspace.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: pAdicWeightSpace(17)(1 + 17^2 + O(17^3), 11, False)
[1 + 17^2 + O(17^3), 11]

teichmuller_type()

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: 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

class sage.modular.overconvergent.weightspace.WeightCharacter(parent)

Bases: sage.structure.element.Element

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

Lvalue()

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

base_extend(R)

Extend scalars to the base ring R.

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

EXAMPLES:

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

is_even()

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

EXAMPLES:

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

is_trivial()

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

EXAMPLES:

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

one_over_Lvalue()

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: 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(_)
<type 'sage.rings.integer.Integer'>

pAdicEisensteinSeries(ring, prec=20)

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

EXAMPLES:

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)

values_on_gens()

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: 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)

class sage.modular.overconvergent.weightspace.WeightSpace_class(p, base_ring)

Bases: sage.structure.parent.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)

Extend scalars to the ring R.

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

EXAMPLES:

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

prime()

Return the prime $p$ such that this is a $p$-adic weight space.

EXAMPLES:

sage: pAdicWeightSpace(17).prime()
17

zero()

Return the zero of this weight space.

EXAMPLES:

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

sage.modular.overconvergent.weightspace.WeightSpace_constructor(p, base_ring=None)

Construct the p-adic weight space for the given prime p.

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.

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: 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