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Each space Mm of modular symbols comes equipped with a commuting
family T1, T2, T3, ... of
linear operators acting on it called the Hecke operators.
The Hecke operators are defined recursively, as follows.
First, T1=1. When n=p is prime,
Tp(x) = [ pmatrix(p&0
0&1) + ∑_(r (mod ) p)
pmatrix(1&r
0&p) ] x,
where the first matrix is omitted if p divides the
level N of M.
If m and n are coprime, then Tmn = TmTn.
If p is a prime, r≥2 is an integer, ε is
the Dirichlet character associated to M, and k is the weight
of M, then
Tpr = Tp T_(pr - 1) - ε(p) pk - 1 T_(pr - 2).
In Magma, Hecke operators are represented as n x n-matrices,
acting from the right, with respect to the basis Basis(M).
For example
> M := ModularSymbols(12);
> T2 := HeckeOperator(M,2);
> M.1;
{oo, 0}
> T2;
[ 2 0 -1 0 0]
[ 2 0 -1 0 0]
[ 0 0 1 -2 -2]
[ 0 -1 1 -1 -2]
[ 0 1 -1 1 2]
> M.1*T2;
2*{oo, 0} + -1*{-1/10, 0}
Compute a matrix representing the nth Hecke operator Tn
with respect to Basis(M) where M is a space of modular symbols.
Compute the characteristic polynomial of the Hecke operator Tn with
respect to the space of modular symbols M.
When n is prime, the Deligne bound on the sizes of Hecke eigenvalues
is used, so HeckePolynomial is frequently much faster than
CharacteristicPolynomial(HeckeOperator(M,n)).
A matrix representing the nth Hecke operator with
respect to Basis(Lattice(M)) where M is a space of modular symbols.
Compute a matrix representing the Hecke operator Tn on
the dual vector space representation of the space of modular symbols M. This function
is much more efficient than HeckeOperator(M,n)
when the dimension of M is small relative to the dimension of
the AmbientSpace(M). Note that DualHeckeOperator(M,n)
is not guaranteed to be the transpose of HeckeOperator(M,n)
because DualHeckeOperator(M,n) is computed with respect
to Basis(DualVectorSpace(M)).
A matrix representing the qth Atkin-Lehner involution Wq on the space of modular symbols M,
when it is defined. The involution Wq is defined when M has
trivial character and even weight. When possible, the Atkin-Lehner
map is normalized so that it is an involution; such normalization may
not be possible when k>2 and the characteristic of the base field
of M divides q.
To each divisor q of N such that (gcd)(q, N/q)=1
there is an Atkin-Lehner involution Wq on M,
which is defined as follows.
Using the Euclidean algorithm,
choose integers x, y, z, w such that qxw - (N/q)yz = 1;
let g=pmatrix(dx & y
Nz& qw) and define
Wq(x) = g(x) / qk - 2 /2.
For example, when q=N we have
g=pmatrix(0& - 1
N& 0).
The action of the Atkin-Lehner involution on the dual
representation of the space of modular symbols M, when it is defined.
The conjugation involution * on the space of modular symbols M that
sends the modular symbol XiYj{u, v}
to ( - 1)jXiYj {-u, - v}.
The conjugation involution * on the dual representation of the space of modular symbols M
(see the documentation for StarInvolution.)
Multiplication by XpY - XYp, which is
a possible analogue of the θ-operator.
(On mod p modular forms, the θ-operator
is the map given by f |-> q (df/dq).)
Both M1 and M2 must be spaces of modular symbols over a
field of positive characteristic p; they must have the same
level and character, and the weight of M2 must equal
the weight of M1 plus p + 1.
> M := ModularSymbols(11,4,+1); M;
Full modular symbols space for Gamma_0(11) of weight 4 and dimension 4
over Rational Field
> HeckeOperator(M,2);
[ 9 0 2/5 -2/5]
[ 0 5 9/5 11/5]
[ 0 5 7/5 13/5]
[ 0 0 22/5 23/5]
The entries of T 2 are not guaranteed to be
integers because Basis(M) is just a basis
of a Q-vector space. The entries will be integers
if we compute T 2 with respect to an integral basis.
> IntegralHeckeOperator(M,2);
[ 0 2 0 0]
[ 1 2 0 0]
[-5 6 9 0]
[ 2 0 0 9]
The matrix for the Hecke operator on the dual of M is the transpose of
T 2.
However, the chosen basis for the cuspidal subspace of the dual
of M need not satisfy any compatibility with
CuspidalSubspace(M).
> DualHeckeOperator(M,2);
[ 9 0 0 0]
[ 0 5 5 0]
[ 2/5 9/5 7/5 22/5]
[-2/5 11/5 13/5 23/5]
> S := CuspidalSubspace(M);
> HeckeOperator(S, 2);
[ 5 -13/5]
[ 5 -3]
> DualHeckeOperator(S, 2);
[-3/4 1/8]
[-1/2 11/4]
> // NOT the transpose!
We can also compute the Atkin-Lehner and the *-involution.
The *-involution is the identity because we are working
in the +1-quotient, which is the largest quotient
of ModularSymbols(11,4) where * acts as +1.
> AtkinLehner(S, 11);
[1 0]
[0 1]
> StarInvolution(S);
[1 0]
[0 1]
On the -1 quotient the Atkin-Lehner involution is the same,
but * acts as -1:
> M := ModularSymbols(11,4,-1); M;
Full modular symbols space for Gamma_0(11) of weight 4 and dimension 2
over Rational Field
> S := CuspidalSubspace(M);
> AtkinLehner(S, 11);
[1 0]
[0 1]
> StarInvolution(S);
[-1 0]
[ 0 -1]
We compute an example of our analogue of the
θ-operator on modular symbols.
> N := 11; p := 3;
> k1 := 2; k2 := k1 + (p+1);
> M1 := ModularSymbols(11,k1,GF(p));
> M2 := ModularSymbols(11,k2,GF(p));
> theta := ThetaOperator(M1,M2); theta;
Mapping from: ModSym: M1 to ModSym: M2 given by a rule [no inverse]
Now that we have computed theta, we can apply it to one of the
modular symbols corresponding to the newform in S2(Γ0(11)).
> D := Decomposition(M1,2);
> f := qEigenform(D[2],10); f;
q + q^2 + 2*q^3 + 2*q^4 + q^5 + 2*q^6 + q^7 + q^9 + O(q^10)
> x := D[2].1;
> y := theta(x); y;
(X^4 + X*Y^3)*{-1/7, 0} + (X^4 + X^3*Y + X*Y^3 + Y^4)*{-1/7, 0} + (X^4
+ 2*X^3*Y + 2*X*Y^3 + Y^4)*{-1/5, 0} + Y^4*{oo, 0}
Finally, we verify for n<10 that the nth Hecke eigenvalue of
y=θ(x) equals n.an(f), where f is as above.
> [y*HeckeOperator(M2,n) - n*Coefficient(f,n)*y : n in [1..9]];
[
0,
0,
0,
0,
0,
0,
0,
0,
0
]
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