Modular forms and Hecke operators

Let $N$ and $k$ be positive integers and let $M_k(N)=M_k(\Gamma_0(N))$ be the $\mathbb{C}$-vector space of weight $k$ modular forms on $X_0(N)$. This space can be viewed as the set of functions $f(z)$, holomorphic on the upper half-plane, such that

$$f(z)=f\vert[\gamma]_k(z):=(cz+d)^{-k}f\left(\frac{az+b}{cz+d}\right)$$

for all $\gamma\in\Gamma_0(N)$, and such that $f$ satisfies a certain holomorphic condition at the cusps.

Any $f\in M_k(N)$ has a Fourier expansion

$$f = a_0(f) + a_1(f) q + a_2(f)q^2 + \cdots =\sum a_n q^n \in \mathbb{C}[[q]]$$

where $q=e^{2\pi i z}$. The map sending $f$ to its $q$-expansion is an injective map $M_k(N)\hookrightarrow\mathbb{C}[[q]]$ called the $q$-expansion map. Define $M_k(N;\mathbb{Z})$ to be the inverse image of $\mathbb{Z}[[q]]$ under this map. It is known (see §12.3, [DI]) that

$$M_k(N)=M_k(N;\mathbb{Z})\otimes \mathbb{C}.$$

For any ring $R$, define $M_k(N;R):=M_k(N;\mathbb{Z})\otimes _{\mathbb{Z}} R.$

Let $p$ be a prime. Define two operators on $\mathbb{C}[[q]]$:

$$V_p(\sum a_n q^n) = \sum a_n q^{np}$$

and

$$U_p(\sum a_n q^n) = \sum a_{np} q^n.$$

The Hecke operator $T_p$ acts on $q$-expansions by

$$T_p = U_p + \varepsilon (p) p^{k-1} V_p$$

where $\varepsilon (p) = 1$, unless $p\vert N$ in which case $\varepsilon (p)=0$. If $m$ and $n$ are coprime, the Hecke operators satisfy $T_{nm}=T_n T_m = T_m T_n$. If $p$ is a prime and $r\geq 2$,

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

The $T_n$ are linear maps which preserves $M_k(N;\mathbb{Z})$. The Hecke algebra $\mathbb{T}=\mathbb{T}(N)=\mathbb{Z}[T_1,T_2,T_3,\ldots]$, which is viewed as a subring of the ring of linear endomorphisms of $M_k(N)$, is a finite commutative $\mathbb{Z}$-algebra.

Proposition 1.1   Let $\sum a_n q^n$ be the $q$-expansion of $f\in M_k(N)$ and let $\sum b_n q^n$ be the $q$-expansion of $T_m f$. Then the coefficients $b_n$ are given by

$$b_n = \sum_{d\vert(m,n)} \varepsilon (d) d^{k-1} a_{mn/d^2}.$$

Note in particular that $a_1(T_m f) = a_m(f)$.

Proof. Proposition 3.4.3, [DI]. $\qedsymbol$

Proposition 1.2   For any ring $R$, there is a perfect pairing

$$\mathbb{T}_R\otimes _RM_k(N;R) \rightarrow R,\qquad (T,f)\mapsto a_1(Tf),$$

where $\mathbb{T}_R = \mathbb{T}\otimes _{\mathbb{Z}} R$.

Proof. Proposition 12.4.13, [DI]. $\qedsymbol$