In something I've been thinking about recently, the following object appears:

$$\mathcal{F}_{g} = \sum_{n=0}^{\infty} Q^{n} T_{n}\big( \phi_{2g-2}(\tau, z) \big)$$

where $T_{n}$ is the $n$-th Hecke operator and $\phi_{2g-2}(\tau, z)$ is a weak Jacobi form of weight $2g-2$ and index one. For context, this arises from a purely geometric standpoint in enumerative geometry. Therefore, I'm wondering what sort of geometric interpretation exists for these Hecke operators, specifically of this sort of infinite generating function of them which I show above?

My interest was peaked by the following. On page 68 of (https://books.google.ca/books?id=Oy2n7wVuREwC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false) Cheng and Duncan's article in String-Math 2011, they consider a Hecke operator acting on a weak Jacobi form of weight *zero* and index one:

$$T_{n} \phi_{0}(\tau, z) = \frac{1}{n}\sum_{ad=n, \,\,b\text{mod} d} \phi \big(\frac{a \tau + b}{d}, az\big).$$

They provide the following geometrical interpretation: "*the right-hand side looks like a sum over degree $n$ maps between elliptic curves $E' \to E$. Summing over $n$ gives the instanton contribution to the free energy of the free string theory.*"

**Is there a similar geometric interpretation for all weak Jacobi forms of even weight (larger than -2) and index 1?**

and more simply, about the above example specifically...

**How exactly does that right-hand side correspond to degree $n$ covers of an elliptic curve? I certainly understand that the range of the sum looks exactly like the degree $n$ covers of a torus, but what about the summand? What role does the Jacobi form play with this interpretation?**

To be honest, my formula arises in a similar context to Cheng-Duncan (Gromov-Witten theory, string theory) so I'm hoping for a similar geometrical analogy.