Hecke algebras

In Sage a “Hecke algebra” always refers to an algebra of endomorphisms of some explicit module, rather than the abstract Hecke algebra of double cosets attached to a subgroup of the modular group.

We distinguish between “anemic Hecke algebras”, which are algebras of Hecke operators whose indices do not divide some integer N (the level), and “full Hecke algebras”, which include Hecke operators coprime to the level. Morphisms in the category of Hecke modules are not required to commute with the action of the full Hecke algebra, only with the anemic algebra.

sage.modular.hecke.algebra.AnemicHeckeAlgebra(M)

Return the anemic Hecke algebra associated to the Hecke module M. This checks whether or not the object already exists in memory, and if so, returns the existing object rather than a new one.

EXAMPLES:

sage: CuspForms(1, 12).anemic_hecke_algebra() # indirect doctest
Anemic Hecke algebra acting on Cuspidal subspace of dimension 1 of Modular Forms space of dimension 2 for Modular Group SL(2,Z) of weight 12 over Rational Field

We test uniqueness:

sage: CuspForms(1, 12).anemic_hecke_algebra() is CuspForms(1, 12).anemic_hecke_algebra()
True

We can’t ensure uniqueness when loading and saving objects from files, but we can ensure equality:

sage: CuspForms(1, 12).anemic_hecke_algebra() is loads(dumps(CuspForms(1, 12).anemic_hecke_algebra()))
False
sage: CuspForms(1, 12).anemic_hecke_algebra() == loads(dumps(CuspForms(1, 12).anemic_hecke_algebra()))
True
sage.modular.hecke.algebra.HeckeAlgebra(M)

Return the full Hecke algebra associated to the Hecke module M. This checks whether or not the object already exists in memory, and if so, returns the existing object rather than a new one.

EXAMPLES:

sage: CuspForms(1, 12).hecke_algebra() # indirect doctest
Full Hecke algebra acting on Cuspidal subspace of dimension 1 of Modular Forms space of dimension 2 for Modular Group SL(2,Z) of weight 12 over Rational Field

We test uniqueness:

sage: CuspForms(1, 12).hecke_algebra() is CuspForms(1, 12).hecke_algebra()
True

We can’t ensure uniqueness when loading and saving objects from files, but we can ensure equality:

sage: CuspForms(1, 12).hecke_algebra() is loads(dumps(CuspForms(1, 12).hecke_algebra()))
False
sage: CuspForms(1, 12).hecke_algebra() == loads(dumps(CuspForms(1, 12).hecke_algebra()))
True
class sage.modular.hecke.algebra.HeckeAlgebra_anemic(M)

An anemic Hecke algebra, generated by Hecke operators with index coprime to the level.

__cmp__(other)

Compare self to other.

EXAMPLES:

sage: A = ModularForms(23).anemic_hecke_algebra()
sage: A == QQ
False
sage: A == ModularForms(23).hecke_algebra()
False
sage: A == A
True
_repr_()

EXAMPLE:

sage: H = CuspForms(3, 12).anemic_hecke_algebra()._repr_()
gens()

Return a generator over all Hecke operator T_n for n = 1, 2, 3, \ldots, with n coprime to the level. This is an infinite sequence.

EXAMPLES:

sage: T = ModularSymbols(12,2).anemic_hecke_algebra()
sage: g = T.gens()
sage: g.next()
Hecke operator T_1 on Modular Symbols space of dimension 5 for Gamma_0(12) of weight 2 with sign 0 over Rational Field
sage: g.next()
Hecke operator T_5 on Modular Symbols space of dimension 5 for Gamma_0(12) of weight 2 with sign 0 over Rational Field
hecke_operator(n)

Return the n-th Hecke operator, for n any positive integer coprime to the level.

EXAMPLES:

sage: T = ModularSymbols(Gamma1(5),3).anemic_hecke_algebra()
sage: T.hecke_operator(2)
Hecke operator T_2 on Modular Symbols space of dimension 4 for Gamma_1(5) of weight 3 with sign 0 and over Rational Field
sage: T.hecke_operator(5)
...
IndexError: Hecke operator T_5 not defined in the anemic Hecke algebra
is_anemic()

Return True, since this the anemic Hecke algebra.

EXAMPLES:

sage: H = CuspForms(3, 12).anemic_hecke_algebra()
sage: H.is_anemic()
True
class sage.modular.hecke.algebra.HeckeAlgebra_base(M)

Base class for algebras of Hecke operators on a fixed Hecke module.

__call__(x, check=True)

Convert x into an element of this Hecke algebra. Here x is either:

  • an element of a Hecke algebra equal to this one
  • an element of the corresponding anemic Hecke algebra, if x is a full Hecke algebra
  • an element of the corresponding full Hecke algebra of the form T_i where i is coprime to self.level(), if self is an anemic Hecke algebra
  • something that can be converted into an element of the underlying matrix space.

In the last case, the parameter ``check’’ controls whether or not to check that this element really does lie in the appropriate algebra. At present, setting ``check=True’’ raises a NotImplementedError unless x is a scalar (or a diagonal matrix).

EXAMPLES:

sage: T = ModularSymbols(11).hecke_algebra()
sage: T.gen(2) in T
True
sage: 5 in T
True
sage: T.gen(2).matrix() in T
...
NotImplementedError: Membership testing for '...' not implemented
sage: T(T.gen(2).matrix(), check=False)
Hecke operator on Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field defined by:
[ 3  0 -1]
[ 0 -2  0]
[ 0  0 -2]
sage: A = ModularSymbols(11).anemic_hecke_algebra()
sage: A(T.gen(3))
Hecke operator T_3 on Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field
sage: A(T.gen(11))
...
TypeError: Don't know how to construct an element of Anemic Hecke algebra acting on Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field from Hecke operator T_11 on Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field

TESTS:

We test that coercion is OK between the Hecke algebras associated to two submodules which are equal but have different bases:

sage: M = CuspForms(Gamma0(57))                                                
sage: f1,f2,f3 = M.newforms()                                                  
sage: N1 = M.submodule(M.free_module().submodule_with_basis([f1.element().element(), f2.element().element()]))
sage: N2 = M.submodule(M.free_module().submodule_with_basis([f1.element().element(), (f1.element() + f2.element()).element()]))
sage: N1.hecke_operator(5).matrix_form()
Hecke operator on Modular Forms subspace of dimension 2 of ... defined by:
[-3  0]                                                                          
[ 0  1]                                                                          
sage: N2.hecke_operator(5).matrix_form()
Hecke operator on Modular Forms subspace of dimension 2 of ... defined by: 
[-3  0]                                                                          
[-4  1]
sage: N1.hecke_algebra()(N2.hecke_operator(5)).matrix_form()
Hecke operator on Modular Forms subspace of dimension 2 of ... defined by:
[-3  0]
[ 0  1]
sage: N1.hecke_algebra()(N2.hecke_operator(5).matrix_form())
Hecke operator on Modular Forms subspace of dimension 2 of ... defined by:
[-3  0]
[ 0  1]
__init__(M)

INPUT:

  • M - a Hecke module

EXAMPLE:

sage: CuspForms(1, 12).hecke_algebra() # indirect doctest
Full Hecke algebra acting on Cuspidal subspace of dimension 1 of Modular Forms space of dimension 2 for Modular Group SL(2,Z) of weight 12 over Rational Field
_an_element_impl()

Return an element of this algebra. Used by the coercion machinery.

EXAMPLE:

sage: CuspForms(1, 12).hecke_algebra().an_element() # indirect doctest
Hecke operator T_2 on Cuspidal subspace of dimension 1 of Modular Forms space of dimension 2 for Modular Group SL(2,Z) of weight 12 over Rational Field
_coerce_impl(x)

Implicit coercion of x into this Hecke algebra. The only things that coerce implicitly into self are: elements of Hecke algebras which are equal to self, or to the anemic subalgebra of self if self is not anemic; and elements that coerce into the base ring of self. Bare matrices do not coerce implicitly into self.

EXAMPLE:

sage: C = CuspForms(3, 12)                                            
sage: A = C.anemic_hecke_algebra()                                    
sage: F = C.hecke_algebra()
sage: F.coerce(A.2) # indirect doctest
Hecke operator T_2 on Cuspidal subspace of dimension 3 of Modular Forms space of dimension 5 for Congruence Subgroup Gamma0(3) of weight 12 over Rational Field
_latex_()

LaTeX representation of self.

EXAMPLES:

sage: latex(CuspForms(3, 24).hecke_algebra()) # indirect doctest
\mathbf{T}_{\text{Cuspidal subspace of dimension 7 of Modular Forms space of dimension 9 for Congruence Subgroup Gamma0(3) of weight 24 over Rational Field}}
basis()

Return a basis for this Hecke algebra as a free module over its base ring. Not implemented at present.

EXAMPLE:

sage: ModularSymbols(Gamma1(3), 3).hecke_algebra().basis()
...
NotImplementedError
discriminant()

Return the discriminant of this Hecke algebra, i.e. the determinant of the matrix {\rm Tr}(x_i x_j) where x_1,
\dots,x_d is a basis for self, and {\rm Tr}(x) signifies the trace (in the sense of linear algebra) of left multiplication by x on the algebra (not the trace of the operator x acting on the underlying Hecke module!). For further discussion and conjectures see Calegari + Stein, Conjectures about discriminants of Hecke algebras of prime level, Springer LNCS 3076.

Not implemented at present.

EXAMPLE:

sage: BrandtModule(3, 4).hecke_algebra().discriminant()
...
NotImplementedError
gen(n)

Return the n-th Hecke operator.

EXAMPLES:

sage: T = ModularSymbols(11).hecke_algebra()
sage: T.gen(2)
Hecke operator T_2 on Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field
gens()

Return a generator over all Hecke operator T_n for n = 1, 2, 3, \ldots. This is infinite.

EXAMPLES:

sage: T = ModularSymbols(1,12).hecke_algebra()
sage: g = T.gens()
sage: g.next()
Hecke operator T_1 on Modular Symbols space of dimension 3 for Gamma_0(1) of weight 12 with sign 0 over Rational Field
sage: g.next()
Hecke operator T_2 on Modular Symbols space of dimension 3 for Gamma_0(1) of weight 12 with sign 0 over Rational Field
hecke_matrix(n)

Return the matrix of the n-th Hecke operator T_n.

EXAMPLES:

sage: T = ModularSymbols(1,12).hecke_algebra()
sage: T.hecke_matrix(2)
[ -24    0    0]
[   0  -24    0]
[4860    0 2049]
hecke_operator(n)

Return the n-th Hecke operator T_n.

EXAMPLES:

sage: T = ModularSymbols(1,12).hecke_algebra()
sage: T.hecke_operator(2)
Hecke operator T_2 on Modular Symbols space of dimension 3 for Gamma_0(1) of weight 12 with sign 0 over Rational Field
is_noetherian()

Return True if this Hecke algebra is Noetherian as a ring. This is true if and only if the base ring is Noetherian.

EXAMPLES:

sage: CuspForms(1, 12).anemic_hecke_algebra().is_noetherian()
True
level()

Return the level of this Hecke algebra, which is (by definition) the level of the Hecke module on which it acts.

EXAMPLE:

sage: ModularSymbols(37).hecke_algebra().level()
37
matrix_space()

Return the underlying matrix space of this module.

EXAMPLES:

sage: CuspForms(3, 24, base_ring=Qp(5)).anemic_hecke_algebra().matrix_space()
Full MatrixSpace of 7 by 7 dense matrices over 5-adic Field with capped relative precision 20
module()

The Hecke module on which this algebra is acting.

EXAMPLES:

sage: T = ModularSymbols(1,12).hecke_algebra()
sage: T.module()
Modular Symbols space of dimension 3 for Gamma_0(1) of weight 12 with sign 0 over Rational Field
ngens()

The size of the set of generators returned by gens(), which is clearly infinity. (This is not necessarily a minimal set of generators.)

EXAMPLES:

sage: CuspForms(1, 12).anemic_hecke_algebra().ngens()
+Infinity
rank()

The rank of this Hecke algebra as a module over its base ring. Not implemented at present.

EXAMPLE:

sage: ModularSymbols(Gamma1(3), 3).hecke_algebra().rank()
...
NotImplementedError
class sage.modular.hecke.algebra.HeckeAlgebra_full(M)

A full Hecke algebra (including the operators T_n where n is not assumed to be coprime to the level).

__cmp__(other)

Compare self to other.

EXAMPLES:

sage: A = ModularForms(37).hecke_algebra()
sage: A == QQ
False
sage: A == ModularForms(37).anemic_hecke_algebra()
False
sage: A == A
True
_repr_()

String representation of self.

EXAMPLE:

sage: ModularForms(37).hecke_algebra()._repr_()
'Full Hecke algebra acting on Modular Forms space of dimension 3 for Congruence Subgroup Gamma0(37) of weight 2 over Rational Field'
anemic_subalgebra()

The subalgebra of self generated by the Hecke operators of index coprime to the level.

EXAMPLE:

sage: H = CuspForms(3, 12).hecke_algebra()
sage: H.anemic_subalgebra()
Anemic Hecke algebra acting on Cuspidal subspace of dimension 3 of Modular Forms space of dimension 5 for Congruence Subgroup Gamma0(3) of weight 12 over Rational Field
is_anemic()

Return False, since this the full Hecke algebra.

EXAMPLES:

sage: H = CuspForms(3, 12).hecke_algebra()
sage: H.is_anemic()
False
sage.modular.hecke.algebra.is_HeckeAlgebra(x)

Return True if x is of type HeckeAlgebra.

EXAMPLES:

sage: from sage.modular.hecke.algebra import is_HeckeAlgebra
sage: is_HeckeAlgebra(CuspForms(1, 12).anemic_hecke_algebra())
True
sage: is_HeckeAlgebra(ZZ)
False

Previous topic

Degeneracy maps

Next topic

Hecke operators

This Page