Roughly speaking, my question is the following: Let $\wp(\tau, z)$ be the Weierstrass $\wp$-function, where $\tau \in \mathbf{H}$ and $z \in \mathbf{C}$. If $p$ is a prime number, can we define $T_p\,\wp(\tau, z)$ where $T_p$ is the $p$-th Hecke operator?
More details: If we evaluate the function $\wp(\tau, z)$ 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 have thought of so far: I came across the book "The Theory of Jacobi Forms" 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.
