Spaces of \(p\)-adic automorphic forms

Compute with harmonic cocycles and \(p\)-adic automorphic forms, including overconvergent \(p\)-adic automorphic forms.

For a discussion of nearly rigid analytic modular forms and the rigid analytic Shimura-Maass operator, see [Fra2011]. It is worth also looking at [FM2014] for information on how these are implemented in this code.

EXAMPLES:

Create a quotient of the Bruhat-Tits tree:

sage: X = BruhatTitsQuotient(13, 11)

>>> from sage.all import *
>>> X = BruhatTitsQuotient(Integer(13), Integer(11))

Declare the corresponding space of harmonic cocycles:

sage: H = X.harmonic_cocycles(2, prec=5)

>>> from sage.all import *
>>> H = X.harmonic_cocycles(Integer(2), prec=Integer(5))

And the space of \(p\)-adic automorphic forms:

sage: A = X.padic_automorphic_forms(2, prec=5, overconvergent=True)                 # needs sage.rings.padics

>>> from sage.all import *
>>> A = X.padic_automorphic_forms(Integer(2), prec=Integer(5), overconvergent=True)                 # needs sage.rings.padics

Harmonic cocycles, unlike \(p\)-adic automorphic forms, can be used to compute a basis:

sage: a = H.gen(0)                                                                  # needs sage.rings.padics

>>> from sage.all import *
>>> a = H.gen(Integer(0))                                                                  # needs sage.rings.padics

This can then be lifted to an overconvergent \(p\)-adic modular form:

sage: A.lift(a)  # long time                                                        # needs sage.rings.padics
p-adic automorphic form of cohomological weight 0

>>> from sage.all import *
>>> A.lift(a)  # long time                                                        # needs sage.rings.padics
p-adic automorphic form of cohomological weight 0
class sage.modular.btquotients.pautomorphicform.BruhatTitsHarmonicCocycleElement(_parent, vec)[source]

Bases: HeckeModuleElement

\(\Gamma\)-invariant harmonic cocycles on the Bruhat-Tits tree. \(\Gamma\)-invariance is necessary so that the cocycle can be stored in terms of a finite amount of data.

More precisely, given a BruhatTitsQuotient \(T\), harmonic cocycles are stored as a list of values in some coefficient module (e.g. for weight 2 forms can take \(\CC_p\)) indexed by edges of a fundamental domain for \(T\) in the Bruhat-Tits tree. Evaluate the cocycle at other edges using Gamma invariance (although the values may not be equal over an orbit of edges as the coefficient module action may be nontrivial).

EXAMPLES:

Harmonic cocycles form a vector space, so they can be added and/or subtracted from each other:

sage: X = BruhatTitsQuotient(5,23)
sage: H = X.harmonic_cocycles(2,prec=10)
sage: v1 = H.basis()[0]; v2 = H.basis()[1] # indirect doctest
sage: v3 = v1+v2
sage: v1 == v3-v2
True

>>> from sage.all import *
>>> X = BruhatTitsQuotient(Integer(5),Integer(23))
>>> H = X.harmonic_cocycles(Integer(2),prec=Integer(10))
>>> v1 = H.basis()[Integer(0)]; v2 = H.basis()[Integer(1)] # indirect doctest
>>> v3 = v1+v2
>>> v1 == v3-v2
True

and rescaled:

sage: v4 = 2*v1
sage: v1 == v4 - v1
True

>>> from sage.all import *
>>> v4 = Integer(2)*v1
>>> v1 == v4 - v1
True

AUTHORS:

  • Cameron Franc (2012-02-20)

  • Marc Masdeu

derivative(z=None, level=0, order=1)[source]

Integrate Teitelbaum’s \(p\)-adic Poisson kernel against the measure corresponding to self to evaluate the rigid analytic Shimura-Maass derivatives of the associated modular form at \(z\).

If z = None, a function is returned that encodes the derivative of the modular form.

Note

This function uses the integration method of Riemann summation and is incredibly slow! It should only be used for testing and bug-finding. Overconvergent methods are quicker.

INPUT:

  • z – an element in the quadratic unramified extension of \(\QQ_p\) that is not contained in \(\QQ_p\) (default: None); if z = None then a function encoding the derivative is returned.

  • level – integer; how fine of a mesh should the Riemann sum use

  • order – integer; how many derivatives to take

OUTPUT:

An element of the quadratic unramified extension of \(\QQ_p\), or a function encoding the derivative.

EXAMPLES:

sage: X = BruhatTitsQuotient(3,23)
sage: H = X.harmonic_cocycles(2,prec=5)
sage: b = H.basis()[0]
sage: R.<a> = Qq(9,prec=10)
sage: b.modular_form(a,level=0) == b.derivative(a,level=0,order=0)
True
sage: b.derivative(a,level=1,order=1)
(2*a + 2)*3 + (a + 2)*3^2 + 2*a*3^3 + 2*3^4 + O(3^5)
sage: b.derivative(a,level=2,order=1)
(2*a + 2)*3 + 2*a*3^2 + 3^3 + a*3^4 + O(3^5)

>>> from sage.all import *
>>> X = BruhatTitsQuotient(Integer(3),Integer(23))
>>> H = X.harmonic_cocycles(Integer(2),prec=Integer(5))
>>> b = H.basis()[Integer(0)]
>>> R = Qq(Integer(9),prec=Integer(10), names=('a',)); (a,) = R._first_ngens(1)
>>> b.modular_form(a,level=Integer(0)) == b.derivative(a,level=Integer(0),order=Integer(0))
True
>>> b.derivative(a,level=Integer(1),order=Integer(1))
(2*a + 2)*3 + (a + 2)*3^2 + 2*a*3^3 + 2*3^4 + O(3^5)
>>> b.derivative(a,level=Integer(2),order=Integer(1))
(2*a + 2)*3 + 2*a*3^2 + 3^3 + a*3^4 + O(3^5)
evaluate(e1)[source]

Evaluate a harmonic cocycle on an edge of the Bruhat-Tits tree.

INPUT:

  • e1 – a matrix corresponding to an edge of the Bruhat-Tits tree

OUTPUT:

  • An element of the coefficient module of the cocycle which describes the value of the cocycle on e1

EXAMPLES:

sage: X = BruhatTitsQuotient(5,17)
sage: e0 = X.get_edge_list()[0]
sage: e1 = X.get_edge_list()[1]
sage: H = X.harmonic_cocycles(2,prec=10)
sage: b = H.basis()[0]
sage: b.evaluate(e0.rep)
1 + O(5^10)
sage: b.evaluate(e1.rep)
4 + 4*5 + 4*5^2 + 4*5^3 + 4*5^4 + 4*5^5 + 4*5^6 + 4*5^7 + 4*5^8 + 4*5^9 + O(5^10)

>>> from sage.all import *
>>> X = BruhatTitsQuotient(Integer(5),Integer(17))
>>> e0 = X.get_edge_list()[Integer(0)]
>>> e1 = X.get_edge_list()[Integer(1)]
>>> H = X.harmonic_cocycles(Integer(2),prec=Integer(10))
>>> b = H.basis()[Integer(0)]
>>> b.evaluate(e0.rep)
1 + O(5^10)
>>> b.evaluate(e1.rep)
4 + 4*5 + 4*5^2 + 4*5^3 + 4*5^4 + 4*5^5 + 4*5^6 + 4*5^7 + 4*5^8 + 4*5^9 + O(5^10)
modular_form(z=None, level=0)[source]

Integrate Teitelbaum’s \(p\)-adic Poisson kernel against the measure corresponding to self to evaluate the associated modular form at z.

If z = None, a function is returned that encodes the modular form.

Note

This function uses the integration method of Riemann summation and is incredibly slow! It should only be used for testing and bug-finding. Overconvergent methods are quicker.

INPUT:

  • z – an element in the quadratic unramified extension of \(\QQ_p\) that is not contained in \(\QQ_p\) (default: None)

  • level – integer; how fine of a mesh should the Riemann sum use

OUTPUT: an element of the quadratic unramified extension of \(\QQ_p\)

EXAMPLES:

sage: X = BruhatTitsQuotient(3,23)
sage: H = X.harmonic_cocycles(2,prec = 8)
sage: b = H.basis()[0]
sage: R.<a> = Qq(9,prec=10)
sage: x1 = b.modular_form(a,level = 0); x1
a + (2*a + 1)*3 + (a + 1)*3^2 + (a + 1)*3^3 + 3^4 + (a + 2)*3^5 + a*3^7 + O(3^8)
sage: x2 = b.modular_form(a,level = 1); x2
a + (a + 2)*3 + (2*a + 1)*3^3 + (2*a + 1)*3^4 + 3^5 + (a + 2)*3^6 + a*3^7 + O(3^8)
sage: x3 = b.modular_form(a,level = 2); x3
a + (a + 2)*3 + (2*a + 2)*3^2 + 2*a*3^4 + (a + 1)*3^5 + 3^6 + O(3^8)
sage: x4 = b.modular_form(a,level = 3);x4
a + (a + 2)*3 + (2*a + 2)*3^2 + (2*a + 2)*3^3 + 2*a*3^5 + a*3^6 + (a + 2)*3^7 + O(3^8)
sage: (x4-x3).valuation()
3

>>> from sage.all import *
>>> X = BruhatTitsQuotient(Integer(3),Integer(23))
>>> H = X.harmonic_cocycles(Integer(2),prec = Integer(8))
>>> b = H.basis()[Integer(0)]
>>> R = Qq(Integer(9),prec=Integer(10), names=('a',)); (a,) = R._first_ngens(1)
>>> x1 = b.modular_form(a,level = Integer(0)); x1
a + (2*a + 1)*3 + (a + 1)*3^2 + (a + 1)*3^3 + 3^4 + (a + 2)*3^5 + a*3^7 + O(3^8)
>>> x2 = b.modular_form(a,level = Integer(1)); x2
a + (a + 2)*3 + (2*a + 1)*3^3 + (2*a + 1)*3^4 + 3^5 + (a + 2)*3^6 + a*3^7 + O(3^8)
>>> x3 = b.modular_form(a,level = Integer(2)); x3
a + (a + 2)*3 + (2*a + 2)*3^2 + 2*a*3^4 + (a + 1)*3^5 + 3^6 + O(3^8)
>>> x4 = b.modular_form(a,level = Integer(3));x4
a + (a + 2)*3 + (2*a + 2)*3^2 + (2*a + 2)*3^3 + 2*a*3^5 + a*3^6 + (a + 2)*3^7 + O(3^8)
>>> (x4-x3).valuation()
3
monomial_coefficients(copy=True)[source]

Return a dictionary whose keys are indices of basis elements in the support of self and whose values are the corresponding coefficients.

EXAMPLES:

sage: M = BruhatTitsQuotient(3,5).harmonic_cocycles(2, prec=10)
sage: M.monomial_coefficients()
{}

>>> from sage.all import *
>>> M = BruhatTitsQuotient(Integer(3),Integer(5)).harmonic_cocycles(Integer(2), prec=Integer(10))
>>> M.monomial_coefficients()
{}
print_values()[source]

Print the values of the cocycle on all of the edges.

EXAMPLES:

sage: X = BruhatTitsQuotient(5,23)
sage: H = X.harmonic_cocycles(2,prec=10)
sage: H.basis()[0].print_values()
0   |1 + O(5^10)
1   |0
2   |0
3   |4 + 4*5 + 4*5^2 + 4*5^3 + 4*5^4 + 4*5^5 + 4*5^6 + 4*5^7 + 4*5^8 + 4*5^9 + O(5^10)
4   |0
5   |0
6   |0
7   |0
8   |0
9   |0
10  |0
11  |0

>>> from sage.all import *
>>> X = BruhatTitsQuotient(Integer(5),Integer(23))
>>> H = X.harmonic_cocycles(Integer(2),prec=Integer(10))
>>> H.basis()[Integer(0)].print_values()
0   |1 + O(5^10)
1   |0
2   |0
3   |4 + 4*5 + 4*5^2 + 4*5^3 + 4*5^4 + 4*5^5 + 4*5^6 + 4*5^7 + 4*5^8 + 4*5^9 + O(5^10)
4   |0
5   |0
6   |0
7   |0
8   |0
9   |0
10  |0
11  |0
riemann_sum(f, center=1, level=0, E=None)[source]

Evaluate the integral of the function f with respect to the measure determined by self over \(\mathbf{P}^1(\QQ_p)\).

INPUT:

  • f – a function on \(\mathbf{P}^1(\QQ_p)\)

  • center – integer (default: 1); Center of integration

  • level – integer (default: 0); Determines the size of the covering when computing the Riemann sum. Runtime is exponential in the level.

  • E – list of edges (default: None); They should describe a covering of \(\mathbf{P}^1(\QQ_p)\)

OUTPUT: a \(p\)-adic number

EXAMPLES:

sage: X = BruhatTitsQuotient(5,7)
sage: H = X.harmonic_cocycles(2,prec=10)
sage: b = H.basis()[0]
sage: R.<z> = PolynomialRing(QQ,1)
sage: f = z^2

>>> from sage.all import *
>>> X = BruhatTitsQuotient(Integer(5),Integer(7))
>>> H = X.harmonic_cocycles(Integer(2),prec=Integer(10))
>>> b = H.basis()[Integer(0)]
>>> R = PolynomialRing(QQ,Integer(1), names=('z',)); (z,) = R._first_ngens(1)
>>> f = z**Integer(2)

Note that \(f\) has a pole at infinity, so that the result will be meaningless:

sage: b.riemann_sum(f,level=0)
1 + 5 + 2*5^3 + 4*5^4 + 2*5^5 + 3*5^6 + 3*5^7 + 2*5^8 + 4*5^9 + O(5^10)

>>> from sage.all import *
>>> b.riemann_sum(f,level=Integer(0))
1 + 5 + 2*5^3 + 4*5^4 + 2*5^5 + 3*5^6 + 3*5^7 + 2*5^8 + 4*5^9 + O(5^10)
valuation()[source]

Return the valuation of the cocycle, defined as the minimum of the values it takes on a set of representatives.

OUTPUT: integer

EXAMPLES:

sage: X = BruhatTitsQuotient(3,17)
sage: H = X.harmonic_cocycles(2,prec=10)
sage: b1 = H.basis()[0]
sage: b2 = 3*b1
sage: b1.valuation()
0
sage: b2.valuation()
1
sage: H(0).valuation()
+Infinity

>>> from sage.all import *
>>> X = BruhatTitsQuotient(Integer(3),Integer(17))
>>> H = X.harmonic_cocycles(Integer(2),prec=Integer(10))
>>> b1 = H.basis()[Integer(0)]
>>> b2 = Integer(3)*b1
>>> b1.valuation()
0
>>> b2.valuation()
1
>>> H(Integer(0)).valuation()
+Infinity
class sage.modular.btquotients.pautomorphicform.BruhatTitsHarmonicCocycles(X, k, prec=None, basis_matrix=None, base_field=None)[source]

Bases: AmbientHeckeModule, UniqueRepresentation

Ensure unique representation.

EXAMPLES:

sage: X = BruhatTitsQuotient(3,5)
sage: M1 = X.harmonic_cocycles( 2, prec = 10)
sage: M2 = X.harmonic_cocycles( 2, 10)
sage: M1 is M2
True

>>> from sage.all import *
>>> X = BruhatTitsQuotient(Integer(3),Integer(5))
>>> M1 = X.harmonic_cocycles( Integer(2), prec = Integer(10))
>>> M2 = X.harmonic_cocycles( Integer(2), Integer(10))
>>> M1 is M2
True
Element[source]

alias of BruhatTitsHarmonicCocycleElement

base_extend(base_ring)[source]

Extend the base ring of the coefficient module.

INPUT:

  • base_ring – a ring that has a coerce map from the current base ring

OUTPUT: a new space of HarmonicCocycles with the base extended

EXAMPLES:

sage: X = BruhatTitsQuotient(3,19)
sage: H = X.harmonic_cocycles(2,10)
sage: H.base_ring()
3-adic Field with capped relative precision 10
sage: H1 = H.base_extend(Qp(3,prec=15))
sage: H1.base_ring()
3-adic Field with capped relative precision 15

>>> from sage.all import *
>>> X = BruhatTitsQuotient(Integer(3),Integer(19))
>>> H = X.harmonic_cocycles(Integer(2),Integer(10))
>>> H.base_ring()
3-adic Field with capped relative precision 10
>>> H1 = H.base_extend(Qp(Integer(3),prec=Integer(15)))
>>> H1.base_ring()
3-adic Field with capped relative precision 15
basis_matrix()[source]

Return a basis of self in matrix form.

If the coefficient module \(M\) is of finite rank then the space of Gamma invariant \(M\) valued harmonic cocycles can be represented as a subspace of the finite rank space of all functions from the finitely many edges in the corresponding BruhatTitsQuotient into \(M\). This function computes this representation of the space of cocycles.

OUTPUT:

  • A basis matrix describing the cocycles in the spaced of all \(M\) valued Gamma invariant functions on the tree.

EXAMPLES:

sage: X = BruhatTitsQuotient(5,3)
sage: M = X.harmonic_cocycles(4,prec = 20)
sage: B = M.basis() # indirect doctest
sage: len(B) == X.dimension_harmonic_cocycles(4)
True

>>> from sage.all import *
>>> X = BruhatTitsQuotient(Integer(5),Integer(3))
>>> M = X.harmonic_cocycles(Integer(4),prec = Integer(20))
>>> B = M.basis() # indirect doctest
>>> len(B) == X.dimension_harmonic_cocycles(Integer(4))
True

AUTHORS:

  • Cameron Franc (2012-02-20)

  • Marc Masdeu (2012-02-20)

change_ring(new_base_ring)[source]

Change the base ring of the coefficient module.

INPUT:

  • new_base_ring – a ring that has a coerce map from the current base ring

OUTPUT: new space of HarmonicCocycles with different base ring

EXAMPLES:

sage: X = BruhatTitsQuotient(5,17)
sage: H = X.harmonic_cocycles(2,10)
sage: H.base_ring()
5-adic Field with capped relative precision 10
sage: H1 = H.base_extend(Qp(5,prec=15)) # indirect doctest
sage: H1.base_ring()
5-adic Field with capped relative precision 15

>>> from sage.all import *
>>> X = BruhatTitsQuotient(Integer(5),Integer(17))
>>> H = X.harmonic_cocycles(Integer(2),Integer(10))
>>> H.base_ring()
5-adic Field with capped relative precision 10
>>> H1 = H.base_extend(Qp(Integer(5),prec=Integer(15))) # indirect doctest
>>> H1.base_ring()
5-adic Field with capped relative precision 15
character()[source]

The trivial character.

OUTPUT: the identity map

EXAMPLES:

sage: X = BruhatTitsQuotient(3,7)
sage: H = X.harmonic_cocycles(2,prec = 10)
sage: f = H.character()
sage: f(1)
1
sage: f(2)
2

>>> from sage.all import *
>>> X = BruhatTitsQuotient(Integer(3),Integer(7))
>>> H = X.harmonic_cocycles(Integer(2),prec = Integer(10))
>>> f = H.character()
>>> f(Integer(1))
1
>>> f(Integer(2))
2
embed_quaternion(g, scale=1, exact=None)[source]

Embed the quaternion element g into the matrix algebra.

INPUT:

  • g – a quaternion, expressed as a 4x1 matrix

OUTPUT: a 2x2 matrix with \(p\)-adic entries

EXAMPLES:

sage: X = BruhatTitsQuotient(7,2)
sage: q = X.get_stabilizers()[0][1][0]
sage: H = X.harmonic_cocycles(2,prec = 5)
sage: Hmat = H.embed_quaternion(q)
sage: Hmat.matrix().trace() == X._conv(q).reduced_trace() and Hmat.matrix().determinant() == 1
True

>>> from sage.all import *
>>> X = BruhatTitsQuotient(Integer(7),Integer(2))
>>> q = X.get_stabilizers()[Integer(0)][Integer(1)][Integer(0)]
>>> H = X.harmonic_cocycles(Integer(2),prec = Integer(5))
>>> Hmat = H.embed_quaternion(q)
>>> Hmat.matrix().trace() == X._conv(q).reduced_trace() and Hmat.matrix().determinant() == Integer(1)
True
free_module()[source]

Return the underlying free module.

OUTPUT: a free module

EXAMPLES:

sage: X = BruhatTitsQuotient(3,7)
sage: H = X.harmonic_cocycles(2,prec=10)
sage: H.free_module()
Vector space of dimension 1 over 3-adic Field with
capped relative precision 10

>>> from sage.all import *
>>> X = BruhatTitsQuotient(Integer(3),Integer(7))
>>> H = X.harmonic_cocycles(Integer(2),prec=Integer(10))
>>> H.free_module()
Vector space of dimension 1 over 3-adic Field with
capped relative precision 10
is_simple()[source]

Whether self is irreducible.

OUTPUT: boolean; True if and only if self is irreducible

EXAMPLES:

sage: X = BruhatTitsQuotient(3, 29)
sage: H = X.harmonic_cocycles(4, prec=10)
sage: H.rank()
14
sage: H.is_simple()
False
sage: X = BruhatTitsQuotient(7, 2)
sage: H = X.harmonic_cocycles(2, prec=10)
sage: H.rank()
1
sage: H.is_simple()
True

>>> from sage.all import *
>>> X = BruhatTitsQuotient(Integer(3), Integer(29))
>>> H = X.harmonic_cocycles(Integer(4), prec=Integer(10))
>>> H.rank()
14
>>> H.is_simple()
False
>>> X = BruhatTitsQuotient(Integer(7), Integer(2))
>>> H = X.harmonic_cocycles(Integer(2), prec=Integer(10))
>>> H.rank()
1
>>> H.is_simple()
True
monomial_coefficients()[source]

Void method to comply with pickling.

EXAMPLES:

sage: M = BruhatTitsQuotient(3,5).harmonic_cocycles(2,prec=10)
sage: M.monomial_coefficients()
{}

>>> from sage.all import *
>>> M = BruhatTitsQuotient(Integer(3),Integer(5)).harmonic_cocycles(Integer(2),prec=Integer(10))
>>> M.monomial_coefficients()
{}
rank()[source]

Return the rank (dimension) of self.

OUTPUT: integer

EXAMPLES:

sage: X = BruhatTitsQuotient(7,11)
sage: H = X.harmonic_cocycles(2,prec = 10)
sage: X.genus() == H.rank()
True
sage: H1 = X.harmonic_cocycles(4,prec = 10)
sage: H1.rank()
16

>>> from sage.all import *
>>> X = BruhatTitsQuotient(Integer(7),Integer(11))
>>> H = X.harmonic_cocycles(Integer(2),prec = Integer(10))
>>> X.genus() == H.rank()
True
>>> H1 = X.harmonic_cocycles(Integer(4),prec = Integer(10))
>>> H1.rank()
16
submodule(v, check=False)[source]

Return the submodule of self spanned by v.

INPUT:

  • v – submodule of self.free_module()

  • check – boolean (default: False)

OUTPUT: subspace of harmonic cocycles

EXAMPLES:

sage: X = BruhatTitsQuotient(3,17)
sage: H = X.harmonic_cocycles(2,prec=10)
sage: H.rank()
3
sage: v = H.gen(0)
sage: N = H.free_module().span([v.element()])
sage: H1 = H.submodule(N)
Traceback (most recent call last):
...
NotImplementedError

>>> from sage.all import *
>>> X = BruhatTitsQuotient(Integer(3),Integer(17))
>>> H = X.harmonic_cocycles(Integer(2),prec=Integer(10))
>>> H.rank()
3
>>> v = H.gen(Integer(0))
>>> N = H.free_module().span([v.element()])
>>> H1 = H.submodule(N)
Traceback (most recent call last):
...
NotImplementedError
sage.modular.btquotients.pautomorphicform.eval_dist_at_powseries(phi, f)[source]

Evaluate a distribution on a powerseries.

A distribution is an element in the dual of the Tate ring. The elements of coefficient modules of overconvergent modular symbols and overconvergent \(p\)-adic automorphic forms give examples of distributions in Sage.

INPUT:

  • phi – a distribution

  • f – a power series over a ring coercible into a \(p\)-adic field

OUTPUT:

The value of phi evaluated at f, which will be an element in the ring of definition of f

EXAMPLES:

sage: from sage.modular.btquotients.pautomorphicform import eval_dist_at_powseries
sage: R.<X> = PowerSeriesRing(ZZ,10)
sage: f = (1 - 7*X)^(-1)

sage: D = OverconvergentDistributions(0,7,10)                                   # needs sage.rings.padics
sage: phi = D(list(range(1,11)))                                                # needs sage.rings.padics
sage: eval_dist_at_powseries(phi,f)                                             # needs sage.rings.padics
1 + 2*7 + 3*7^2 + 4*7^3 + 5*7^4 + 6*7^5 + 2*7^7 + 3*7^8 + 4*7^9 + O(7^10)

>>> from sage.all import *
>>> from sage.modular.btquotients.pautomorphicform import eval_dist_at_powseries
>>> R = PowerSeriesRing(ZZ,Integer(10), names=('X',)); (X,) = R._first_ngens(1)
>>> f = (Integer(1) - Integer(7)*X)**(-Integer(1))

>>> D = OverconvergentDistributions(Integer(0),Integer(7),Integer(10))                                   # needs sage.rings.padics
>>> phi = D(list(range(Integer(1),Integer(11))))                                                # needs sage.rings.padics
>>> eval_dist_at_powseries(phi,f)                                             # needs sage.rings.padics
1 + 2*7 + 3*7^2 + 4*7^3 + 5*7^4 + 6*7^5 + 2*7^7 + 3*7^8 + 4*7^9 + O(7^10)
class sage.modular.btquotients.pautomorphicform.pAdicAutomorphicFormElement(parent, vec)[source]

Bases: ModuleElement

Rudimentary implementation of a class for a \(p\)-adic automorphic form on a definite quaternion algebra over \(\QQ\). These are required in order to compute moments of measures associated to harmonic cocycles on the Bruhat-Tits tree using the overconvergent modules of Darmon-Pollack and Matt Greenberg. See Greenberg’s thesis [Gr2006] for more details.

INPUT:

  • vec – a preformatted list of data

EXAMPLES:

sage: X = BruhatTitsQuotient(17,3)
sage: H = X.harmonic_cocycles(2,prec=10)
sage: h = H.an_element()
sage: HH = X.padic_automorphic_forms(2,10)
sage: a = HH(h)
sage: a
p-adic automorphic form of cohomological weight 0

>>> from sage.all import *
>>> X = BruhatTitsQuotient(Integer(17),Integer(3))
>>> H = X.harmonic_cocycles(Integer(2),prec=Integer(10))
>>> h = H.an_element()
>>> HH = X.padic_automorphic_forms(Integer(2),Integer(10))
>>> a = HH(h)
>>> a
p-adic automorphic form of cohomological weight 0

AUTHORS:

  • Cameron Franc (2012-02-20)

  • Marc Masdeu

coleman(t1, t2, E=None, method='moments', mult=False)[source]

If self is a \(p\)-adic automorphic form that corresponds to a rigid modular form, then this computes the Coleman integral of this form between two points on the boundary \(P^1(\QQ_p)\) of the \(p\)-adic upper half plane.

INPUT:

  • t1, t2 – elements of \(P^1(\QQ_p)\) (the endpoints of integration)

  • E – (default: None) if specified, will not compute the covering adapted to t1 and t2 and instead use the given one. In that case, E should be a list of matrices corresponding to edges describing the open balls to be considered.

  • method – string (default: 'moments'); tells which algorithm to use (alternative is 'riemann_sum', which is unsuitable for computations requiring high precision)

  • mult – boolean (default: False); whether to compute the multiplicative version

OUTPUT: the result of the Coleman integral

EXAMPLES:

sage: p = 7
sage: lev = 2
sage: prec = 10
sage: X = BruhatTitsQuotient(p, lev)
sage: k = 2
sage: M = X.harmonic_cocycles(k, prec)
sage: B = M.basis()
sage: f = 3*B[0]
sage: MM = X.padic_automorphic_forms(k, prec, overconvergent=True)
sage: D = -11
sage: X.is_admissible(D)
True
sage: K.<a> = QuadraticField(D)
sage: Kp.<g> = Qq(p**2, prec)
sage: P = Kp.gen()
sage: Q = 2 + Kp.gen() + p*(Kp.gen()+1)
sage: F = MM.lift(f)  # long time
sage: J0 = F.coleman(P, Q, mult=True)  # long time

>>> from sage.all import *
>>> p = Integer(7)
>>> lev = Integer(2)
>>> prec = Integer(10)
>>> X = BruhatTitsQuotient(p, lev)
>>> k = Integer(2)
>>> M = X.harmonic_cocycles(k, prec)
>>> B = M.basis()
>>> f = Integer(3)*B[Integer(0)]
>>> MM = X.padic_automorphic_forms(k, prec, overconvergent=True)
>>> D = -Integer(11)
>>> X.is_admissible(D)
True
>>> K = QuadraticField(D, names=('a',)); (a,) = K._first_ngens(1)
>>> Kp = Qq(p**Integer(2), prec, names=('g',)); (g,) = Kp._first_ngens(1)
>>> P = Kp.gen()
>>> Q = Integer(2) + Kp.gen() + p*(Kp.gen()+Integer(1))
>>> F = MM.lift(f)  # long time
>>> J0 = F.coleman(P, Q, mult=True)  # long time

AUTHORS:

  • Cameron Franc (2012-02-20)

  • Marc Masdeu (2012-02-20)

derivative(z=None, level=0, method='moments', order=1)[source]

Return the derivative of the modular form corresponding to self.

INPUT:

  • z – (default: None) if specified, evaluates the derivative at the point z in the \(p\)-adic upper half plane

  • level – integer (default: 0); if method is ‘riemann_sum’, will use a covering of \(P^1(\QQ_p)\) with balls of size \(p^-\mbox{level}\).

  • method – string (default: 'moments'); it must be either 'moments' or 'riemann_sum'

  • order – integer (default: 1); the order of the derivative to be computed

OUTPUT:

A function from the \(p\)-adic upper half plane to \(\CC_p\). If an argument z was passed, returns instead the value of the derivative at that point.

EXAMPLES:

Integrating the Poisson kernel against a measure yields a value of the associated modular form. Such values can be computed efficiently using the overconvergent method, as long as one starts with an ordinary form:

sage: X = BruhatTitsQuotient(7, 2)
sage: X.genus()
1

>>> from sage.all import *
>>> X = BruhatTitsQuotient(Integer(7), Integer(2))
>>> X.genus()
1

Since the genus is 1, the space of weight 2 forms is 1 dimensional. Hence any nonzero form will be a \(U_7\) eigenvector. By Jacquet-Langlands and Cerednik-Drinfeld, in this case the Hecke eigenvalues correspond to that of any nonzero form on \(\Gamma_0(14)\) of weight \(2\). Such a form is ordinary at \(7\), and so we can apply the overconvergent method directly to this form without \(p\)-stabilizing:

sage: H = X.harmonic_cocycles(2,prec=5)
sage: h = H.gen(0)
sage: A = X.padic_automorphic_forms(2,prec=5,overconvergent=True)
sage: f0 = A.lift(h)

>>> from sage.all import *
>>> H = X.harmonic_cocycles(Integer(2),prec=Integer(5))
>>> h = H.gen(Integer(0))
>>> A = X.padic_automorphic_forms(Integer(2),prec=Integer(5),overconvergent=True)
>>> f0 = A.lift(h)

Now that we’ve lifted our harmonic cocycle to an overconvergent automorphic form, we extract the associated modular form as a function and test the modular property:

sage: T.<x> = Qq(49,prec=10)
sage: f = f0.modular_form()
sage: g = X.get_embedding_matrix()*X.get_units_of_order()[1]
sage: a,b,c,d = g.change_ring(T).list()
sage: (c*x +d)^2*f(x)-f((a*x + b)/(c*x + d))
O(7^5)

>>> from sage.all import *
>>> T = Qq(Integer(49),prec=Integer(10), names=('x',)); (x,) = T._first_ngens(1)
>>> f = f0.modular_form()
>>> g = X.get_embedding_matrix()*X.get_units_of_order()[Integer(1)]
>>> a,b,c,d = g.change_ring(T).list()
>>> (c*x +d)**Integer(2)*f(x)-f((a*x + b)/(c*x + d))
O(7^5)

We can also compute the Shimura-Maass derivative, which is a nearly rigid analytic modular forms of weight 4:

sage: f = f0.derivative()
sage: (c*x + d)^4*f(x)-f((a*x + b)/(c*x + d))
O(7^5)

>>> from sage.all import *
>>> f = f0.derivative()
>>> (c*x + d)**Integer(4)*f(x)-f((a*x + b)/(c*x + d))
O(7^5)
evaluate(e1)[source]

Evaluate a \(p\)-adic automorphic form on a matrix in \(GL_2(\QQ_p)\).

INPUT:

  • e1 – a matrix in \(GL_2(\QQ_p)\)

OUTPUT: the value of self evaluated on e1

EXAMPLES:

sage: X = BruhatTitsQuotient(7,5)
sage: M = X.harmonic_cocycles(2,prec=5)
sage: A = X.padic_automorphic_forms(2,prec=5)
sage: a = A(M.basis()[0])
sage: a.evaluate(Matrix(ZZ,2,2,[1,2,3,1]))
4 + 6*7 + 6*7^2 + 6*7^3 + 6*7^4 + O(7^5)
sage: a.evaluate(Matrix(ZZ,2,2,[17,0,0,1]))
1 + O(7^5)

>>> from sage.all import *
>>> X = BruhatTitsQuotient(Integer(7),Integer(5))
>>> M = X.harmonic_cocycles(Integer(2),prec=Integer(5))
>>> A = X.padic_automorphic_forms(Integer(2),prec=Integer(5))
>>> a = A(M.basis()[Integer(0)])
>>> a.evaluate(Matrix(ZZ,Integer(2),Integer(2),[Integer(1),Integer(2),Integer(3),Integer(1)]))
4 + 6*7 + 6*7^2 + 6*7^3 + 6*7^4 + O(7^5)
>>> a.evaluate(Matrix(ZZ,Integer(2),Integer(2),[Integer(17),Integer(0),Integer(0),Integer(1)]))
1 + O(7^5)
integrate(f, center=1, level=0, method='moments')[source]

Calculate

\[\int_{\mathbf{P}^1(\QQ_p)} f(x)d\mu(x)\]

were \(\mu\) is the measure associated to self.

INPUT:

  • f – an analytic function

  • center – 2x2 matrix over \(\QQ_p\) (default: 1)

  • level – integer (default: 0)

  • method – string (default: 'moments'); which method of integration to use. Either 'moments' or 'riemann_sum'.

EXAMPLES:

Integrating the Poisson kernel against a measure yields a value of the associated modular form. Such values can be computed efficiently using the overconvergent method, as long as one starts with an ordinary form:

sage: X = BruhatTitsQuotient(7,2)
sage: X.genus()
1

>>> from sage.all import *
>>> X = BruhatTitsQuotient(Integer(7),Integer(2))
>>> X.genus()
1

Since the genus is 1, the space of weight 2 forms is 1 dimensional. Hence any nonzero form will be a \(U_7\) eigenvector. By Jacquet-Langlands and Cerednik-Drinfeld, in this case the Hecke eigenvalues correspond to that of any nonzero form on \(\Gamma_0(14)\) of weight \(2\). Such a form is ordinary at \(7\), and so we can apply the overconvergent method directly to this form without \(p\)-stabilizing:

sage: H = X.harmonic_cocycles(2,prec = 5)
sage: h = H.gen(0)
sage: A = X.padic_automorphic_forms(2,prec = 5,overconvergent=True)
sage: a = A.lift(h)
sage: a._value[0].moment(2)
2 + 6*7 + 4*7^2 + 4*7^3 + 6*7^4 + O(7^5)

>>> from sage.all import *
>>> H = X.harmonic_cocycles(Integer(2),prec = Integer(5))
>>> h = H.gen(Integer(0))
>>> A = X.padic_automorphic_forms(Integer(2),prec = Integer(5),overconvergent=True)
>>> a = A.lift(h)
>>> a._value[Integer(0)].moment(Integer(2))
2 + 6*7 + 4*7^2 + 4*7^3 + 6*7^4 + O(7^5)

Now that we’ve lifted our harmonic cocycle to an overconvergent automorphic form we simply need to define the Teitelbaum-Poisson Kernel, and then integrate:

sage: Kp.<x> = Qq(49,prec = 5)
sage: z = Kp['z'].gen()
sage: f = 1/(z-x)
sage: a.integrate(f)
(5*x + 5) + (4*x + 4)*7 + (5*x + 5)*7^2 + (5*x + 6)*7^3 + O(7^5)

>>> from sage.all import *
>>> Kp = Qq(Integer(49),prec = Integer(5), names=('x',)); (x,) = Kp._first_ngens(1)
>>> z = Kp['z'].gen()
>>> f = Integer(1)/(z-x)
>>> a.integrate(f)
(5*x + 5) + (4*x + 4)*7 + (5*x + 5)*7^2 + (5*x + 6)*7^3 + O(7^5)

AUTHORS:

  • Cameron Franc (2012-02-20)

  • Marc Masdeu (2012-02-20)

modular_form(z=None, level=0, method='moments')[source]

Return the modular form corresponding to self.

INPUT:

  • z – (default: None) if specified, returns the value of the form at the point z in the \(p\)-adic upper half plane

  • level – integer (default: 0); if method is ‘riemann_sum’, will use a covering of \(P^1(\QQ_p)\) with balls of size \(p^-\mbox{level}\)

  • method – string (default: 'moments'); it must be either 'moments' or 'riemann_sum'

OUTPUT:

A function from the \(p\)-adic upper half plane to \(\CC_p\). If an argument z was passed, returns instead the value at that point.

EXAMPLES:

Integrating the Poisson kernel against a measure yields a value of the associated modular form. Such values can be computed efficiently using the overconvergent method, as long as one starts with an ordinary form:

sage: X = BruhatTitsQuotient(7, 2)
sage: X.genus()
1

>>> from sage.all import *
>>> X = BruhatTitsQuotient(Integer(7), Integer(2))
>>> X.genus()
1

Since the genus is 1, the space of weight 2 forms is 1 dimensional. Hence any nonzero form will be a \(U_7\) eigenvector. By Jacquet-Langlands and Cerednik-Drinfeld, in this case the Hecke eigenvalues correspond to that of any nonzero form on \(\Gamma_0(14)\) of weight \(2\). Such a form is ordinary at \(7\), and so we can apply the overconvergent method directly to this form without \(p\)-stabilizing:

sage: H = X.harmonic_cocycles(2,prec = 5)
sage: A = X.padic_automorphic_forms(2,prec = 5,overconvergent=True)
sage: f0 = A.lift(H.basis()[0])

>>> from sage.all import *
>>> H = X.harmonic_cocycles(Integer(2),prec = Integer(5))
>>> A = X.padic_automorphic_forms(Integer(2),prec = Integer(5),overconvergent=True)
>>> f0 = A.lift(H.basis()[Integer(0)])

Now that we’ve lifted our harmonic cocycle to an overconvergent automorphic form, we extract the associated modular form as a function and test the modular property:

sage: T.<x> = Qq(7^2,prec = 5)
sage: f = f0.modular_form(method = 'moments')
sage: a,b,c,d = X.embed_quaternion(X.get_units_of_order()[1]).change_ring(T.base_ring()).list()
sage: ((c*x + d)^2*f(x)-f((a*x + b)/(c*x + d))).valuation()
5

>>> from sage.all import *
>>> T = Qq(Integer(7)**Integer(2),prec = Integer(5), names=('x',)); (x,) = T._first_ngens(1)
>>> f = f0.modular_form(method = 'moments')
>>> a,b,c,d = X.embed_quaternion(X.get_units_of_order()[Integer(1)]).change_ring(T.base_ring()).list()
>>> ((c*x + d)**Integer(2)*f(x)-f((a*x + b)/(c*x + d))).valuation()
5
valuation()[source]

The valuation of self, defined as the minimum of the valuations of the values that it takes on a set of edge representatives.

OUTPUT: integer

EXAMPLES:

sage: X = BruhatTitsQuotient(17,3)
sage: M = X.harmonic_cocycles(2,prec=10)
sage: A = X.padic_automorphic_forms(2,prec=10)
sage: a = A(M.gen(0))
sage: a.valuation()
0
sage: (17*a).valuation()
1

>>> from sage.all import *
>>> X = BruhatTitsQuotient(Integer(17),Integer(3))
>>> M = X.harmonic_cocycles(Integer(2),prec=Integer(10))
>>> A = X.padic_automorphic_forms(Integer(2),prec=Integer(10))
>>> a = A(M.gen(Integer(0)))
>>> a.valuation()
0
>>> (Integer(17)*a).valuation()
1
class sage.modular.btquotients.pautomorphicform.pAdicAutomorphicForms(domain, U, prec=None, t=None, R=None, overconvergent=False)[source]

Bases: Module, UniqueRepresentation

Create a space of \(p\)-automorphic forms.

EXAMPLES:

sage: X = BruhatTitsQuotient(11,5)
sage: H = X.harmonic_cocycles(2,prec=10)
sage: A = X.padic_automorphic_forms(2,prec=10)
sage: TestSuite(A).run()

>>> from sage.all import *
>>> X = BruhatTitsQuotient(Integer(11),Integer(5))
>>> H = X.harmonic_cocycles(Integer(2),prec=Integer(10))
>>> A = X.padic_automorphic_forms(Integer(2),prec=Integer(10))
>>> TestSuite(A).run()
Element[source]

alias of pAdicAutomorphicFormElement

lift(f)[source]

Lift the harmonic cocycle f to a p-automorphic form.

If one is using overconvergent coefficients, then this will compute all of the moments of the measure associated to f.

INPUT:

  • f – a harmonic cocycle

OUTPUT: a \(p\)-adic automorphic form

EXAMPLES:

If one does not work with an overconvergent form then lift does nothing:

sage: X = BruhatTitsQuotient(13,5)
sage: H = X.harmonic_cocycles(2,prec=10)
sage: h = H.gen(0)
sage: A = X.padic_automorphic_forms(2,prec=10)
sage: A.lift(h) # long time
p-adic automorphic form of cohomological weight 0

>>> from sage.all import *
>>> X = BruhatTitsQuotient(Integer(13),Integer(5))
>>> H = X.harmonic_cocycles(Integer(2),prec=Integer(10))
>>> h = H.gen(Integer(0))
>>> A = X.padic_automorphic_forms(Integer(2),prec=Integer(10))
>>> A.lift(h) # long time
p-adic automorphic form of cohomological weight 0

With overconvergent forms, the input is lifted naively and its moments are computed:

sage: X = BruhatTitsQuotient(13,11)
sage: H = X.harmonic_cocycles(2,prec=5)
sage: A2 = X.padic_automorphic_forms(2,prec=5,overconvergent=True)
sage: a = H.gen(0)
sage: A2.lift(a) # long time
p-adic automorphic form of cohomological weight 0

>>> from sage.all import *
>>> X = BruhatTitsQuotient(Integer(13),Integer(11))
>>> H = X.harmonic_cocycles(Integer(2),prec=Integer(5))
>>> A2 = X.padic_automorphic_forms(Integer(2),prec=Integer(5),overconvergent=True)
>>> a = H.gen(Integer(0))
>>> A2.lift(a) # long time
p-adic automorphic form of cohomological weight 0
precision_cap()[source]

Return the precision of self.

OUTPUT: integer

EXAMPLES:

sage: X = BruhatTitsQuotient(13,11)
sage: A = X.padic_automorphic_forms(2,prec=10)
sage: A.precision_cap()
10

>>> from sage.all import *
>>> X = BruhatTitsQuotient(Integer(13),Integer(11))
>>> A = X.padic_automorphic_forms(Integer(2),prec=Integer(10))
>>> A.precision_cap()
10
prime()[source]

Return the underlying prime.

OUTPUT: p – prime integer

EXAMPLES:

sage: X = BruhatTitsQuotient(11,5)
sage: H = X.harmonic_cocycles(2,prec = 10)
sage: A = X.padic_automorphic_forms(2,prec = 10)
sage: A.prime()
11

>>> from sage.all import *
>>> X = BruhatTitsQuotient(Integer(11),Integer(5))
>>> H = X.harmonic_cocycles(Integer(2),prec = Integer(10))
>>> A = X.padic_automorphic_forms(Integer(2),prec = Integer(10))
>>> A.prime()
11
zero()[source]

Return the zero element of self.

EXAMPLES:

sage: X = BruhatTitsQuotient(5, 7)
sage: H1 = X.padic_automorphic_forms( 2, prec=10)
sage: H1.zero() == 0
True

>>> from sage.all import *
>>> X = BruhatTitsQuotient(Integer(5), Integer(7))
>>> H1 = X.padic_automorphic_forms( Integer(2), prec=Integer(10))
>>> H1.zero() == Integer(0)
True