Overconvergent module of distributions

Initialize self. See help(type(self)) for accurate signature.

class darmonpoints.ocmodule.AddMeromorphicFunctions(K, twisting_matrix=None)

Bases: Parent, CachedRepresentation

TESTS:

sage: from darmonpoints.ocmodule import AddMeromorphicFunctions sage: from darmonpoints.homology import Divisors sage: from darmonpoints.sarithgroup import BigArithGroup sage: padic_printing.mode(‘val-unit’) sage: K.<a> = Qq(7^2,5) sage: G = BigArithGroup(7,1,1,use_shapiro=False, outfile=’/dev/null’) sage: M = AddMeromorphicFunctions(K) sage: Div = Divisors(K) sage: D = Div(a/7) - Div((a+1)/7) sage: f = M(D) sage: f.power_series() 7 * (1205 + 401*a) + O(7^5) + (7^2 * (59 + 229*a) + O(7^5))*t + (7^3 * (23 + 13*a) + O(7^5))*t^2 + (7^4 * (6 + 2*a) + O(7^5))*t^3 + O(7^5)*t^4 sage: E = Div((a+3)) - Div((a+2)) sage: f(E).log(0) == D.pair_with(E) True sage: g = G.Gpn.gen(1).quaternion_rep sage: M(g * D)(E) == (g * f)(E) True sage: M = AddMeromorphicFunctions(K, twisting_matrix = G.wp()) sage: Div = Divisors(K) sage: D = Div(a) - Div((a+1)) sage: f = M(D) sage: f.power_series() 7 * 2400 + O(7^5) + (7^2 * (1 + 2*a) + O(7^5))*t + (7^3 * (8 + 15*a) + O(7^5))*t^2 + (7^4 * (6 + 2*a) + O(7^5))*t^3 + O(7^5)*t^4 sage: E = Div((a+3)/7) - Div((a+2)/7) sage: f(E).log(0) == D.pair_with(E) True sage: g = G.Gpn.gen(1).quaternion_rep sage: A = M(g * D)(E) sage: B = (g * f)(E) sage: A == B True

Element

alias of AddMeromorphicFunctionsElement

acting_matrix(g, dim=None)

base_ring()

get_action_data(g, K=None)

is_additive()

power_series_ring()

twisting_matrix()

class darmonpoints.ocmodule.AddMeromorphicFunctionsElement(parent, data, check=True)

Bases: ModuleElement

evaluate_additive(D)

evaluate_multiplicative(D)

left_act_by_matrix(g)

pair_with(D)

power_series()

scale_by(k)

valuation(p=None)

class darmonpoints.ocmodule.OCVn(p, depth)

Bases: Module, UniqueRepresentation

Element

This class represents objects in the overconvergent approximation modules used to describe overconvergent p-adic automorphic forms.

INPUT:

• n - integer

• R - ring

• depth - integer (Default: None)

• basis - (Default: None)

AUTHORS:

• Cameron Franc (2012-02-20)

• Marc Masdeu (2012-02-20)

alias of OCVnElement

Sigma0()

acting_matrix(g, M)

approx_module(M=None)

base_ring()

This function returns the base ring of the overconvergent element.

basis()

A basis of the module.

INPUT:

• x - integer (default: 1) the description of the argument x goes here. If it contains multiple lines, all the lines after the first need to be indented.

• y - integer (default: 2) the …

clear_cache()

depth()

Returns the depth of the module.

dimension()

Returns the dimension (rank) of the module.

is_exact()

is_overconvergent()

precision_cap()

Returns the dimension (rank) of the module.

prime()

class darmonpoints.ocmodule.OCVnElement(parent, val=0, check=True, normalize=False)

Bases: ModuleElement

This class represents elements in an overconvergent coefficient module.

INPUT:

• parent - An overconvergent coefficient module.

• val - The value that it needs to store (default: 0). It can be another OCVnElement, in which case the values are copied. It can also be a column vector (or something coercible to a column vector) which represents the values of the element applied to the polynomials 1, x, x^2, … ,`x^n`.

• check - boolean (default: True). If set to False, no checks are done and val is assumed to be the a column vector.

AUTHORS:

• Cameron Franc (2012-02-20)

• Marc Masdeu (2012-02-20)

element()

The element self represents.

evaluate_at_poly(P, R=None, depth=None)

Evaluate self at a polynomial

lift(p=None, M=None)

list()

The element self represents.

matrix_rep()

Returns a matrix representation of self.

moment(i)

moments()

r_act_by(x)

Act on the right by a matrix.

reduce_mod(N=None)

valuation(l=None)

The l-adic valuation of self.

INPUT: a prime l. The default (None) uses the prime of the parent.

valuation_list(l=None)

The l-adic valuation of self, as a list.

INPUT: a prime l. The default (None) uses the prime of the parent.

class darmonpoints.ocmodule.Sigma0Action(G, M)

Bases: Action

class darmonpoints.ocmodule.our_adjuster

Bases: Sigma0ActionAdjuster

Callable object that turns matrices into 4-tuples.

EXAMPLES:

sage: from sage.modular.btquotients.pautomorphicform import _btquot_adjuster
sage: adj = _btquot_adjuster()
sage: adj(matrix(ZZ,2,2,[1..4]))
(4, 2, 3, 1)

class darmonpoints.ocmodule.ps_adjuster

Bases: Sigma0ActionAdjuster

Callable object that turns matrices into 4-tuples.

Representations

Initialize self. See help(type(self)) for accurate signature.

class darmonpoints.representations.CoIndAction(algebra, V, G)

Bases: Action

class darmonpoints.representations.CoIndElement(parent, data, check=True)

Bases: ModuleElement

Elements in a co-induced module are represented by lists [v_1,ldots v_r] indexed by the cosets of G(p) G(1).

act_and_evaluate_at_identity(g)

evaluate(x)

evaluate_at_identity()

is_zero()

values()

class darmonpoints.representations.CoIndModule(G, V)

Bases: Parent

A co-induced module.

TESTS:

sage: from darmonpoints.homology import *
sage: from darmonpoints.cohomology_abstract import *
sage: from darmonpoints.sarithgroup import *
sage: G = BigArithGroup(5,6,1,outfile='/tmp/darmonpoints.tmp') #  optional - magma

Element

alias of CoIndElement

act_by_genpow(i, a, v)

acting_matrix(g, prec=None)

EXAMPLES:

sage: class Foo:
....:     def __init__(self, x):
....:         self._x = x
....:     @cached_method
....:     def f(self,*args):
....:         return self._x^2
sage: a = Foo(2)
sage: a.f.cache
{}
sage: a.f()
4
sage: a.f.cache
{((), ()): 4}

base_field()

base_ring()

basis()

coefficient_module()

dimension()

gen(i)

EXAMPLES:

sage: class Foo:
....:     def __init__(self, x):
....:         self._x = x
....:     @cached_method
....:     def f(self,*args):
....:         return self._x^2
sage: a = Foo(2)
sage: a.f.cache
{}
sage: a.f()
4
sage: a.f.cache
{((), ()): 4}

gens()

EXAMPLES:

sage: class Foo:
....:     def __init__(self, x):
....:         self._x = x
....:     @cached_method
....:     def f(self):
....:         return self._x^2
sage: a = Foo(2)
sage: print(a.f.cache)
None
sage: a.f()
4
sage: a.f.cache
4

get_action_data(g, idx=None)

EXAMPLES:

sage: class Foo:
....:     def __init__(self, x):
....:         self._x = x
....:     @cached_method
....:     def f(self,*args):
....:         return self._x^2
sage: a = Foo(2)
sage: a.f.cache
{}
sage: a.f()
4
sage: a.f.cache
{((), ()): 4}

group()

class darmonpoints.representations.IndAction(algebra, V, G)

Bases: CoIndAction

class darmonpoints.representations.IndElement(parent, data, check=True)

Bases: CoIndElement

evaluate_at_identity()

class darmonpoints.representations.IndModule(G, V)

Bases: CoIndModule

Element

alias of IndElement

class darmonpoints.representations.MatrixAction(G, M)

Bases: Action

class darmonpoints.representations.Scaling(G, M)

Bases: Action

class darmonpoints.representations.TrivialAction(G, M)

Bases: Action

Meromorphic functions

Initialize self. See help(type(self)) for accurate signature.

class darmonpoints.meromorphic.MeromorphicFunctions(K, p, prec)

Bases: Parent, UniqueRepresentation

TESTS:

Element

alias of MeromorphicFunctionsElement

base_ring()

get_action_data(g, oldparam, param, prec=None)

EXAMPLES:

sage: class Foo:
....:     def __init__(self, x):
....:         self._x = x
....:     @cached_method
....:     def f(self,*args):
....:         return self._x^2
sage: a = Foo(2)
sage: a.f.cache
{}
sage: a.f()
4
sage: a.f.cache
{((), ()): 4}

power_series_ring()

prime()

class darmonpoints.meromorphic.MeromorphicFunctionsElement(parent, data, parameter, check=True)

Bases: ModuleElement

eval_derivative(D)

evaluate(D)

fast_act(key)

left_act_by_matrix(g, param=None)

power_series()

scale_by(k)

valuation(p=None)

darmonpoints.meromorphic.divisor_to_pseries(parameter, Ps, data, prec)

darmonpoints.meromorphic.evalpoly(poly, x)