Operators

Each space \(\bf M\) of modular symbols comes equipped with a commuting family \(T_1, T_2, T_3, \ldots\) of linear operators acting on it called the Hecke operators.

The Hecke operators are defined recursively, as follows. First, \(T_1=1\). When \(n=p\) is prime,

\[\begin{split}T_p(x) = \left[ \begin{pmatrix}p&0\\ 0&1\end{pmatrix} + \sum_{r {\rm\, mod\,} p} \begin{pmatrix}1&r\\ 0&p\end{pmatrix} \right] x,\end{split}\]

where the first matrix is omitted if \(p\) divides the level \(N\) of \(M\). If \(m\) and \(n\) are coprime, then \(T_{mn} = T_mT_n\). If \(p\) is a prime, \(r\geq 2\) is an integer, \(\varepsilon\) is the Dirichlet character associated to \(M\), and \(k\) is the weight of \(M\), then

\[T_{p^r} = T_p T_{p^{r-1}} - \varepsilon(p) p^{k-1} T_{p^{r-2}}.\]

Example: Hecke Operators (ex-223884)

In Magma, Hecke operators are represented as \(n\times n\)-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}

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HeckeOperator(M, n): ModSym, RngIntElt -> AlgMatElt

Compute a matrix representing the \(n\)th Hecke operator \(T_n\) with respect to Basis(M) where \(M\) is a space of modular symbols.

HeckePolynomial(M, n): ModSym, RngIntElt -> RngUPolResElt

Compute the characteristic polynomial of the Hecke operator \(T_n\) 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)).

IntegralHeckeOperator(M, n): ModSym, RngIntElt -> AlgMatElt

A matrix representing the \(n\)th Hecke operator with respect to Basis(Lattice(M)) where \(M\) is a space of modular symbols.

DualHeckeOperator(M, n): ModSym, RngIntElt -> AlgMatElt

Compute a matrix representing the Hecke operator \(T_n\) 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)).

AtkinLehner(M, q): ModSym, RngIntElt -> AlgMatElt

A matrix representing the \(q\)th Atkin-Lehner involution \(W_q\) on the space of modular symbols \(M\), when it is defined. The involution \(W_q\) 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 \({\rm gcd}(q,N/q)=1\) there is an Atkin-Lehner involution \(W_q\) 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 \(g=\begin{pmatrix}dx & y\\ Nz& qw\end{pmatrix}\) and define

\[W_q(x) = g(x) / q^{k-2 \over 2}.\]

For example, when \(q=N\) we have \(g=\begin{pmatrix}0&-1\\ N& 0\end{pmatrix}\).

DualAtkinLehner(M, q): ModSym, RngIntElt -> AlgMatElt

The action of the Atkin-Lehner involution on the dual representation of the space of modular symbols \(M\), when it is defined.

StarInvolution(M): ModSym -> AlgMatElt

The conjugation involution \(*\) on the space of modular symbols \(M\) that sends the modular symbol \(X^iY^j\{u,v\}\) to \((-1)^jX^iY^j \{-u,-v\}\).

DualStarInvolution(M): ModSym -> AlgMatElt

The conjugation involution \(*\) on the dual representation of the space of modular symbols \(M\) (see the documentation for StarInvolution.)

ThetaOperator(M1, M2): ModSym, ModSym -> Map

Multiplication by \(X^{p}Y - XY^{p}\), which is a possible analogue of the \(\theta\)-operator. (On mod \(p\) modular forms, the \(\theta\)-operator is the map given by \(f \mapsto q {df\over dq}\).) Both \(M_1\) and \(M_2\) 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 \(M_2\) must equal the weight of \(M_1\) plus \(p+1\).

Example: Operators (ex-e83c93)

> 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]

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The entries of \(T_2\) are not guaranteed to be integers because Basis(M) is just a basis of a \({\mathbb{Q}}\)-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]

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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!

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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]

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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]

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Example: Theta Operator (ex-2cde37)

We compute an example of our analogue of the \(\theta\)-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]

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Now that we have computed theta, we can apply it to one of the modular symbols corresponding to the newform in \(S_2(\Gamma_0(11))\).

> 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}

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Finally, we verify for \(n<10\) that the \(n\)th Hecke eigenvalue of \(y=\theta(x)\) equals \(n\cdot a_n(f)\), 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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