Let $\wp(\tau, z)$ be the Weierstrass $\wp$-function attached to the lattice $\langle1, \tau \rangle$ where $\tau \in \mathbf{H}$ belongs to the complex upper half-plane. If we evaluate $\wp(\tau, z)$ at $N$-torsion points of the lattice $\mathbf{C}/(\mathbf{Z} + \mathbf{Z}\tau)$, we obtain a modular form of level $\Gamma(N)$. Precisely, if we define the function

$$f_{\frac{a}{N}, \frac{b}{N}}(\tau) = \wp \left(\tau, \dfrac{a}{N}\tau + \dfrac{b}{N} \right)$$

then it turns out that $f_{\frac{a}{N}, \frac{b}{N}}$ is a weight $2$ modular form of level $\Gamma(N)$:
$$f_{\frac{a}{N}, \frac{b}{N}}(\tau) \in M_2(\Gamma(N)).$$

**My question is:** Can we define a "Hecke operator" $T_p$ acting on the Weierstrass $\wp$-function such that 
 
$$T_p\,\wp\left(\tau, \dfrac{a}{N}\tau + \dfrac{b}{N} \right) = T_p \,f_{\frac{a}{N}, \frac{b}{N}}?$$

The $T_p$ on the right is the usual $p$-th Hecke operator acting on $M_2(\Gamma(N))$. That is, can we define a function $T_p\,\wp\,(\tau, z)$ such that if we evaluate $T_p\,\wp\,(\tau, z)$ at the $N$-torsion point $z = \frac{a}{N}\tau + \frac{b}{N}$, we recover the usual Hecke operator $T_p$ acting on the modular form $f_{\frac{a}{N}, \frac{b}{N}}$? If so, how would $T_p \, \wp$ be defined?

**What I've thought of so far:** I came across the book "[The Theory of Jacobi Forms](https://doi.org/10.1007/978-1-4684-9162-3)" by Eichler and Zagier where they define Hecke operators on Jacobi forms. But their definition of Hecke operators does not seem to commute with the usual definition of Hecke operators when I specialize the $\wp$ function to torsion points, so I think another definition would be needed. Any insights / references would be appreciated.