Module: sage.modular.abvar.morphism
Morphisms between modular abelian varieties, including Hecke operators acting on modular abelian varieties.
Sage can compute with Hecke operators on modular abelian varieties. A Hecke operator is defined by given a modular abelian variety and an index. Given a Hecke operator, Sage can compute the characteristic polynomial, and the action of the Hecke operator on various homology groups.
Author Log:
sage: A = J0(54) sage: t5 = A.hecke_operator(5); t5 Hecke operator T_5 on Abelian variety J0(54) of dimension 4 sage: t5.charpoly().factor() (x - 3) * (x + 3) * x^2 sage: B = A.new_subvariety(); B Abelian subvariety of dimension 2 of J0(54) sage: t5 = B.hecke_operator(5); t5 Hecke operator T_5 on Abelian subvariety of dimension 2 of J0(54) sage: t5.charpoly().factor() (x - 3) * (x + 3) sage: t5.action_on_homology().matrix() [ 0 3 3 -3] [-3 3 3 0] [ 3 3 0 -3] [-3 6 3 -3]
Class: DegeneracyMap
self, parent, A, t) |
Create the degeneracy map of index t in parent defined by the matrix A.
Input:
sage: J0(44).degeneracy_map(11,2) Degeneracy map from Abelian variety J0(44) of dimension 4 to Abelian variety J0(11) of dimension 1 defined by [2] sage: J0(44)[0].degeneracy_map(88,2) Degeneracy map from Simple abelian subvariety 11a(1,44) of dimension 1 of J0(44) to Abelian variety J0(88) of dimension 9 defined by [2]
Functions: t
self) |
Return the list of indices defining self.
sage: J0(22).degeneracy_map(44).t() [1] sage: J = J0(22) * J0(11) sage: J.degeneracy_map([44,44], [2,1]) Degeneracy map from Abelian variety J0(22) x J0(11) of dimension 3 to Abelian variety J0(44) x J0(44) of dimension 8 defined by [2, 1] sage: J.degeneracy_map([44,44], [2,1]).t() [2, 1]
Special Functions: __init__,
_repr_
self) |
Return the string representation of self.
sage: J0(22).degeneracy_map(44)._repr_() 'Degeneracy map from Abelian variety J0(22) of dimension 2 to Abelian variety J0(44) of dimension 4 defined by [1]'
Class: HeckeOperator
self, abvar, n) |
Create the Hecke operator of index
acting on the abelian
variety abvar.
Input:
sage: J = J0(37) sage: T2 = J.hecke_operator(2); T2 Hecke operator T_2 on Abelian variety J0(37) of dimension 2
Functions: action_on_homology,
characteristic_polynomial,
charpoly,
index,
matrix,
n
self, [R=Integer Ring]) |
Return the action of this Hecke operator on the homology
of this abelian variety with coefficients in
.
sage: A = J0(43) sage: t2 = A.hecke_operator(2); t2 Hecke operator T_2 on Abelian variety J0(43) of dimension 3 sage: h2 = t2.action_on_homology(); h2 Hecke operator T_2 on Integral Homology of Abelian variety J0(43) of dimension 3 sage: h2.matrix() [-2 1 0 0 0 0] [-1 1 1 0 -1 0] [-1 0 -1 2 -1 1] [-1 0 1 1 -1 1] [ 0 -2 0 2 -2 1] [ 0 -1 0 1 0 -1] sage: h2 = t2.action_on_homology(GF(2)); h2 Hecke operator T_2 on Homology with coefficients in Finite Field of size 2 of Abelian variety J0(43) of dimension 3 sage: h2.matrix() [0 1 0 0 0 0] [1 1 1 0 1 0] [1 0 1 0 1 1] [1 0 1 1 1 1] [0 0 0 0 0 1] [0 1 0 1 0 1]
self, [var=x]) |
Return the characteristic polynomial of this Hecke operator in the given variable.
Input:
sage: A = J0(43)[1]; A Simple abelian subvariety 43b(1,43) of dimension 2 of J0(43) sage: t2 = A.hecke_operator(2); t2 Hecke operator T_2 on Simple abelian subvariety 43b(1,43) of dimension 2 of J0(43) sage: f = t2.characteristic_polynomial(); f x^2 - 2 sage: f.parent() Univariate Polynomial Ring in x over Integer Ring sage: f.factor() x^2 - 2 sage: t2.characteristic_polynomial('y') y^2 - 2
self, [var=x]) |
Synonym for self.characteristic_polynomial(var)
.
Input:
sage: A = J1(13) sage: t2 = A.hecke_operator(2); t2 Hecke operator T_2 on Abelian variety J1(13) of dimension 2 sage: f = t2.charpoly(); f x^2 + 3*x + 3 sage: f.factor() x^2 + 3*x + 3 sage: t2.charpoly('y') y^2 + 3*y + 3
self) |
Return the index of this Hecke operator. (For example, if this
is the operator
, then the index is the integer
.)
Output:
sage: J = J0(15) sage: t = J.hecke_operator(53) sage: t Hecke operator T_53 on Abelian variety J0(15) of dimension 1 sage: t.index() 53 sage: t = J.hecke_operator(54) sage: t Hecke operator T_54 on Abelian variety J0(15) of dimension 1 sage: t.index() 54
sage: J = J1(12345) sage: t = J.hecke_operator(997) ; t Hecke operator T_997 on Abelian variety J1(12345) of dimension 5405473 sage: t.index() 997 sage: type(t.index()) <type 'sage.rings.integer.Integer'>
self) |
Return the matrix of self acting on the homology
of this abelian variety with coefficients in
.
sage: J0(47).hecke_operator(3).matrix() [ 0 0 1 -2 1 0 -1 0] [ 0 0 1 0 -1 0 0 0] [-1 2 0 0 2 -2 1 -1] [-2 1 1 -1 3 -1 -1 0] [-1 -1 1 0 1 0 -1 1] [-1 0 0 -1 2 0 -1 0] [-1 -1 2 -2 2 0 -1 0] [ 0 -1 0 0 1 0 -1 1]
sage: J0(11).hecke_operator(7).matrix() [-2 0] [ 0 -2] sage: (J0(11) * J0(33)).hecke_operator(7).matrix() [-2 0 0 0 0 0 0 0] [ 0 -2 0 0 0 0 0 0] [ 0 0 0 0 2 -2 2 -2] [ 0 0 0 -2 2 0 2 -2] [ 0 0 0 0 2 0 4 -4] [ 0 0 -4 0 2 2 2 -2] [ 0 0 -2 0 2 2 0 -2] [ 0 0 -2 0 0 2 0 -2]
sage: J0(23).hecke_operator(2).matrix() [ 0 1 -1 0] [ 0 1 -1 1] [-1 2 -2 1] [-1 1 0 -1]
self) |
Alias for self.index()
.
sage: J = J0(17) sage: J.hecke_operator(5).n() 5
Special Functions: __init__,
_repr_
self) |
String representation of this Hecke operator.
sage: J = J0(37) sage: J.hecke_operator(2)._repr_() 'Hecke operator T_2 on Abelian variety J0(37) of dimension 2'
Class: Morphism
Functions: restrict_domain
self, sub) |
Restrict self to the subvariety sub of self.domain().
sage: J = J0(37) ; A, B = J.decomposition() sage: A.lattice().matrix() [ 1 -1 1 0] [ 0 0 2 -1] sage: B.lattice().matrix() [1 1 1 0] [0 0 0 1] sage: T = J.hecke_operator(2) ; T.matrix() [-1 1 1 -1] [ 1 -1 1 0] [ 0 0 -2 1] [ 0 0 0 0] sage: T.restrict_domain(A) Abelian variety morphism: From: Simple abelian subvariety 37a(1,37) of dimension 1 of J0(37) To: Abelian variety J0(37) of dimension 2 sage: T.restrict_domain(A).matrix() [-2 2 -2 0] [ 0 0 -4 2] sage: T.restrict_domain(B) Abelian variety morphism: From: Simple abelian subvariety 37b(1,37) of dimension 1 of J0(37) To: Abelian variety J0(37) of dimension 2 sage: T.restrict_domain(B).matrix() [0 0 0 0] [0 0 0 0]
Class: Morphism_abstract
sage: t = J0(11).hecke_operator(2) sage: from sage.modular.abvar.morphism import Morphism sage: isinstance(t, Morphism) True
Functions: cokernel,
complementary_isogeny,
factor_out_component_group,
image,
is_isogeny,
kernel
self) |
Return the cokernel of self.
Output:
sage: t = J0(33).hecke_operator(2) sage: (t-1).cokernel() (Abelian subvariety of dimension 1 of J0(33), Abelian variety morphism: From: Abelian variety J0(33) of dimension 3 To: Abelian subvariety of dimension 1 of J0(33))
Projection will always have cokernel zero.
sage: J0(37).projection(J0(37)[0]).cokernel() (Simple abelian subvariety of dimension 0 of J0(37), Abelian variety morphism: From: Simple abelian subvariety 37a(1,37) of dimension 1 of J0(37) To: Simple abelian subvariety of dimension 0 of J0(37))
Here we have a nontrivial cokernel of a Hecke operator, as the T_2-eigenvalue for the newform 37b is 0.
sage: J0(37).hecke_operator(2).cokernel() (Abelian subvariety of dimension 1 of J0(37), Abelian variety morphism: From: Abelian variety J0(37) of dimension 2 To: Abelian subvariety of dimension 1 of J0(37)) sage: AbelianVariety('37b').newform().q_expansion(5) q + q^3 - 2*q^4 + O(q^5)
self) |
Returns the complementary isogeny of self.
sage: J = J0(43) sage: A = J[1] sage: T5 = A.hecke_operator(5) sage: T5.is_isogeny() True sage: T5.complementary_isogeny() Abelian variety endomorphism of Simple abelian subvariety 43b(1,43) of dimension 2 of J0(43) sage: (T5.complementary_isogeny() * T5).matrix() [2 0 0 0] [0 2 0 0] [0 0 2 0] [0 0 0 2]
self) |
View self as a morphism
. Then
is an
extension of an abelian variety
by a finite component
group
. This function constructs a morphism
with
domain
and codomain Q isogenous to
such that
is equal to
.
Output: a morphism
sage: A,B,C = J0(33) sage: pi = J0(33).projection(A) sage: pi.kernel() (Finite subgroup with invariants [5] over QQbar of Abelian variety J0(33) of dimension 3, Abelian subvariety of dimension 2 of J0(33)) sage: psi = pi.factor_out_component_group() sage: psi.kernel() (Finite subgroup with invariants [] over QQbar of Abelian variety J0(33) of dimension 3, Abelian subvariety of dimension 2 of J0(33))
ALGORITHM: We compute a subgroup
of
so that the
composition
has kernel that contains
and component group isomorphic to
,
where
is the dimension of
. Then
factors through
multiplication by
, so there is a morphism
such that
. Then
is the desired
morphism. We give more details below about how to transform
this into linear algebra.
self) |
Return the image of this morphism.
Output: an abelian variety
We compute the image of projection onto a factor of
:
sage: A,B,C = J0(33) sage: A Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33) sage: f = J0(33).projection(A) sage: f.image() Abelian subvariety of dimension 1 of J0(33) sage: f.image() == A True
We compute the image of a Hecke operator:
sage: t2 = J0(33).hecke_operator(2); t2.fcp() (x - 1) * (x + 2)^2 sage: phi = t2 + 2 sage: phi.image() Abelian subvariety of dimension 1 of J0(33)
The sum of the image and the kernel is the whole space:
sage: phi.kernel()[1] + phi.image() == J0(33) True
self) |
Return True if this morphism is an isogeny of abelian varieties.
sage: J = J0(39) sage: Id = J.hecke_operator(1) sage: Id.is_isogeny() True sage: J.hecke_operator(19).is_isogeny() False
self) |
Return the kernel of this morphism.
Output:
We compute the kernel of a projection map. Notice that the kernel has a nontrivial abelian variety part.
sage: A, B, C = J0(33) sage: pi = J0(33).projection(B) sage: pi.kernel() (Finite subgroup with invariants [20] over QQbar of Abelian variety J0(33) of dimension 3, Abelian subvariety of dimension 2 of J0(33))
We compute the kernels of some Hecke operators:
sage: t2 = J0(33).hecke_operator(2) sage: t2 Hecke operator T_2 on Abelian variety J0(33) of dimension 3 sage: t2.kernel() (Finite subgroup with invariants [2, 2, 2, 2] over QQ of Abelian variety J0(33) of dimension 3, Abelian subvariety of dimension 0 of J0(33)) sage: t3 = J0(33).hecke_operator(3) sage: t3.kernel() (Finite subgroup with invariants [3, 3] over QQ of Abelian variety J0(33) of dimension 3, Abelian subvariety of dimension 0 of J0(33))
Special Functions: __call__,
_image_of_abvar,
_image_of_element,
_image_of_finite_subgroup,
_repr_,
_repr_type
self, X) |
Input:
We apply morphisms to elements:
sage: t2 = J0(33).hecke_operator(2) sage: G = J0(33).torsion_subgroup(2); G Finite subgroup with invariants [2, 2, 2, 2, 2, 2] over QQ of Abelian variety J0(33) of dimension 3 sage: t2(G.0) [(-1/2, 0, 1/2, -1/2, 1/2, -1/2)] sage: t2(G.0) in G True sage: t2(G.1) [(0, -1, 1/2, 0, 1/2, -1/2)] sage: t2(G.2) [(0, 0, 0, 0, 0, 0)] sage: K = t2.kernel()[0]; K Finite subgroup with invariants [2, 2, 2, 2] over QQ of Abelian variety J0(33) of dimension 3 sage: t2(K.0) [(0, 0, 0, 0, 0, 0)]
We apply morphisms to subgroups:
sage: t2 = J0(33).hecke_operator(2) sage: G = J0(33).torsion_subgroup(2); G Finite subgroup with invariants [2, 2, 2, 2, 2, 2] over QQ of Abelian variety J0(33) of dimension 3 sage: t2(G) Finite subgroup with invariants [2, 2] over QQ of Abelian variety J0(33) of dimension 3 sage: t2.fcp() (x - 1) * (x + 2)^2
We apply morphisms to abelian subvarieties:
sage: E11a0, E11a1, B = J0(33) sage: t2 = J0(33).hecke_operator(2) sage: t3 = J0(33).hecke_operator(3) sage: E11a0 Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33) sage: t3(E11a0) Abelian subvariety of dimension 1 of J0(33) sage: t3(E11a0).decomposition() [ Simple abelian subvariety 11a(3,33) of dimension 1 of J0(33) ] sage: t3(E11a0) == E11a1 True sage: t2(E11a0) == E11a0 True sage: t3(E11a0) == E11a0 False sage: t3(E11a0 + E11a1) == E11a0 + E11a1 True
We apply some Hecke operators to the cuspidal subgroup and split it up:
sage: C = J0(33).cuspidal_subgroup(); C Finite subgroup with invariants [10, 10] over QQ of Abelian variety J0(33) of dimension 3 sage: t2 = J0(33).hecke_operator(2); t2.fcp() (x - 1) * (x + 2)^2 sage: (t2 - 1)(C) Finite subgroup with invariants [5, 5] over QQ of Abelian variety J0(33) of dimension 3 sage: (t2 + 2)(C) Finite subgroup with invariants [2, 2] over QQ of Abelian variety J0(33) of dimension 3
Same but on a simple new factor:
sage: C = J0(33)[2].cuspidal_subgroup(); C Finite subgroup with invariants [2, 2] over QQ of Simple abelian subvariety 33a(1,33) of dimension 1 of J0(33) sage: t2 = J0(33)[2].hecke_operator(2); t2.fcp() x - 1 sage: t2(C) Finite subgroup with invariants [2, 2] over QQ of Simple abelian subvariety 33a(1,33) of dimension 1 of J0(33)
self, A) |
Compute the image of the abelian variety
under this
morphism.
Input:
OUTPUT an abelian variety
sage: t = J0(33).hecke_operator(2) sage: t._image_of_abvar(J0(33).new_subvariety()) Abelian subvariety of dimension 1 of J0(33)
sage: t = J0(33).hecke_operator(3) sage: A = J0(33)[0] sage: B = t._image_of_abvar(A); B Abelian subvariety of dimension 1 of J0(33) sage: B == A False sage: A + B == J0(33).old_subvariety() True
sage: J = J0(37) ; A, B = J.decomposition() sage: J.projection(A)._image_of_abvar(A) Abelian subvariety of dimension 1 of J0(37) sage: J.projection(A)._image_of_abvar(B) Abelian subvariety of dimension 0 of J0(37) sage: J.projection(B)._image_of_abvar(A) Abelian subvariety of dimension 0 of J0(37) sage: J.projection(B)._image_of_abvar(B) Abelian subvariety of dimension 1 of J0(37) sage: J.projection(B)._image_of_abvar(J) Abelian subvariety of dimension 1 of J0(37)
self, x) |
Return the image of the torsion point
under this morphism.
The parent of the image element is always the group of all torsion elements of the abelian variety.
Input:
sage: A = J0(11); t = A.hecke_operator(2) sage: t.matrix() [-2 0] [ 0 -2] sage: P = A.cuspidal_subgroup().0; P [(0, 1/5)] sage: t._image_of_element(P) [(0, -2/5)] sage: -2*P [(0, -2/5)]
sage: J = J0(37) ; phi = J._isogeny_to_product_of_simples() sage: phi._image_of_element(J.torsion_subgroup(5).gens()[0]) [(1/5, -1/5, -1/5, 1/5, 1/5, 1/5, 1/5, -1/5)]
sage: K = J[0].intersection(J[1])[0] sage: K.list() [[(0, 0, 0, 0)], [(1/2, -1/2, 1/2, 0)], [(0, 0, 1, -1/2)], [(1/2, -1/2, 3/2, -1/2)]] sage: [ phi.restrict_domain(J[0])._image_of_element(k) for k in K ] [[(0, 0, 0, 0, 0, 0, 0, 0)], [(0, 0, 0, 0, 0, 0, 0, 0)], [(0, 0, 0, 0, 0, 0, 0, 0)], [(0, 0, 0, 0, 0, 0, 0, 0)]]
self, G) |
Return the image of the finite group
under this morphism.
Input:
sage: J = J0(33); A = J[0]; B = J[1] sage: C = A.intersection(B)[0] ; C Finite subgroup with invariants [5] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33) sage: t = J.hecke_operator(3) sage: D = t(C); D Finite subgroup with invariants [5] over QQ of Abelian variety J0(33) of dimension 3 sage: D == C True
Or we directly test this function:
sage: D = t._image_of_finite_subgroup(C); D Finite subgroup with invariants [5] over QQ of Abelian variety J0(33) of dimension 3 sage: phi = J0(11).degeneracy_map(22,2) sage: J0(11).rational_torsion_subgroup().order() 5 sage: phi._image_of_finite_subgroup(J0(11).rational_torsion_subgroup()) Finite subgroup with invariants [5] over QQ of Abelian variety J0(22) of dimension 2
self) |
Return string representation of this morphism.
sage: t = J0(11).hecke_operator(2) sage: sage.modular.abvar.morphism.Morphism_abstract._repr_(t) 'Abelian variety endomorphism of Abelian variety J0(11) of dimension 1' sage: J0(42).projection(J0(42)[0])._repr_() 'Abelian variety morphism: From: Abelian variety J0(42) of dimension 5 To: Simple abelian subvariety 14a(1,42) of dimension 1 of J0(42)'
self) |
Return type of morphism.
sage: t = J0(11).hecke_operator(2) sage: sage.modular.abvar.morphism.Morphism_abstract._repr_type(t) 'Abelian variety'
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