# 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, $$ T_p(x) = \left[ \begin{pmatrix}p&0\\ 0&1\end{pmatrix} + \sum_{r {\rm\, mod\,} p} \begin{pmatrix}1&r\\ 0&p\end{pmatrix} \right] x, $$ 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 ```magma > 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} ``` ## `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)` ```magma > 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 ${\mathbb{Q}}$-vector space. The entries will be integers if we compute $T_2$ with respect to an integral basis. ```magma > 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)`. ```magma > 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 $+1$. ```magma > 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 $-1$: ```magma > 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] ``` ## `Example: Theta Operator (ex-2cde37)` We compute an example of our analogue of the $\theta$-operator on modular symbols. ```magma > 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 $S_2(\Gamma_0(11))$. ```magma > 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 $n$th Hecke eigenvalue of $y=\theta(x)$ equals $n\cdot a_n(f)$, where $f$ is as above. ```magma > [y*HeckeOperator(M2,n) - n*Coefficient(f,n)*y : n in [1..9]]; [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ```