# 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.

AUTHORS:

• William Stein (2007-03)
• Craig Citro (2008-03)

EXAMPLES:

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 sage.modular.abvar.morphism.DegeneracyMap(parent, A, t)

Create the degeneracy map of index t in parent defined by the matrix A.

INPUT:

• parent - a space of homomorphisms of abelian varieties
• A - a matrix defining self
• t - a list of indices defining the degeneracy map

EXAMPLES:

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]

t()

Return the list of indices defining self.

EXAMPLES:

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]

class sage.modular.abvar.morphism.HeckeOperator(abvar, n)

A Hecke operator acting on a modular abelian variety.

action_on_homology(R=Integer Ring)

Return the action of this Hecke operator on the homology $$H_1(A; R)$$ of this abelian variety with coefficients in $$R$$.

EXAMPLES:

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]

characteristic_polynomial(var='x')

Return the characteristic polynomial of this Hecke operator in the given variable.

INPUT:

• var - a string (default: ‘x’)

OUTPUT: a polynomial in var over the rational numbers.

EXAMPLES:

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

charpoly(var='x')

Synonym for self.characteristic_polynomial(var).

INPUT:

• var - string (default: ‘x’)

EXAMPLES:

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

index()

Return the index of this Hecke operator. (For example, if this is the operator $$T_n$$, then the index is the integer $$n$$.)

OUTPUT:

• n - a (Sage) Integer

EXAMPLES:

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'>

matrix()

Return the matrix of self acting on the homology $$H_1(A, ZZ)$$ of this abelian variety with coefficients in $$\ZZ$$.

EXAMPLES:

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]

n()

Alias for self.index().

EXAMPLES:

sage: J = J0(17)
sage: J.hecke_operator(5).n()
5

class sage.modular.abvar.morphism.Morphism(parent, A, copy_matrix=True)

Initialize self.

EXAMPLES:

sage: from sage.modules.matrix_morphism import MatrixMorphism
sage: T = End(ZZ^3)
sage: M = MatrixSpace(ZZ,3)
sage: I = M.identity_matrix()
sage: A = MatrixMorphism(T, I)
True

restrict_domain(sub)

Restrict self to the subvariety sub of self.domain().

EXAMPLES:

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 sage.modular.abvar.morphism.Morphism_abstract(parent)

A morphism between modular abelian varieties. EXAMPLES:

sage: t = J0(11).hecke_operator(2)
sage: from sage.modular.abvar.morphism import Morphism
sage: isinstance(t, Morphism)
True

cokernel()

Return the cokernel of self.

OUTPUT:

• A - an abelian variety (the cokernel)
• phi - a quotient map from self.codomain() to the cokernel of self

EXAMPLES:

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)

complementary_isogeny()

Returns the complementary isogeny of self.

EXAMPLES:

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]

factor_out_component_group()

View self as a morphism $$f:A \to B$$. Then $$\ker(f)$$ is an extension of an abelian variety $$C$$ by a finite component group $$G$$. This function constructs a morphism $$g$$ with domain $$A$$ and codomain Q isogenous to $$C$$ such that $$\ker(g)$$ is equal to $$C$$.

OUTPUT: a morphism

EXAMPLES:

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 $$G$$ of $$B$$ so that the composition $$h: A\to B \to B/G$$ has kernel that contains $$A[n]$$ and component group isomorphic to $$(\ZZ/n\ZZ)^{2d}$$, where $$d$$ is the dimension of $$A$$. Then $$h$$ factors through multiplication by $$n$$, so there is a morphism $$g: A\to B/G$$ such that $$g \circ [n] = h$$. Then $$g$$ is the desired morphism. We give more details below about how to transform this into linear algebra.

image()

Return the image of this morphism.

OUTPUT: an abelian variety

EXAMPLES: We compute the image of projection onto a factor of $$J_0(33)$$:

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

is_isogeny()

Return True if this morphism is an isogeny of abelian varieties.

EXAMPLES:

sage: J = J0(39)
sage: Id = J.hecke_operator(1)
sage: Id.is_isogeny()
True
sage: J.hecke_operator(19).is_isogeny()
False

kernel()

Return the kernel of this morphism.

OUTPUT:

• G - a finite group
• A - an abelian variety (identity component of the kernel)

EXAMPLES: 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))


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