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.

  • 1
    $\begingroup$ The Cheng-Duncan paper is available on the ArXiv (arxiv.org/abs/1201.4140), where the equation you consider is on page 16. $\endgroup$
    – S. Carnahan
    Jun 3 '17 at 20:35

The interpretation of Hecke operators on modular forms and Jacobi forms in terms of a sum over isogenies is classical - this is the worldsheet interpretation. Their "target space" interpretation in terms of genera of symmetric products and second quantization of strings was given by Dijkgraaf-Moore-Verlinde-Verlinde in 1996.

The generating function of Hecke operators actually appeared earlier, in the 1980s, in the Koike-Norton-Zagier formula: $$ J(\sigma) - J(\tau) = p^{-1}\prod_{m>0,n \in \mathbb{Z}}(1-p^mq^n)^{c(mn)} $$ where $c(n)$ is the $q^n$ coefficient of $J(\tau)$. To prove this identity, one multiplies by $p$ and takes the logarithm of the right side to get $\sum_{m>0} T_m J(\tau) p^m$. Borcherds used this formula as the denominator formula of the Monster Lie algebra on the way to proving the Monstrous Moonshine conjectures. There are more recent interpretations of this formula in terms of second quantized strings, following the lead of DMVV.

If you just want to know how the summand works in the weight zero case, replace the $\tau$ and $z$ variables with pairs given by an elliptic curve $\mathbb{C}/\langle 1, \tau \rangle$ and a point on it, i.e., consider the function as a function on isomorphism classes of marked elliptic curves. Degree $n$ isogenies to the curve are given by index $n$ sublattices of $\langle 1, \tau \rangle$, and if we set $d$ to be a positive generator of the intersection of the sublattice with $\mathbb{Z}$, these are given by $\langle d, a\tau + b \rangle$ as the variables range over non-negative integers satisfying $ad=n$ and $0 \leq b < d$. The point $z$ gets pulled back to $az$. If you consider nonzero weights, then you need to consider a contribution from a tensor power of the canonical bundle, and some powers of $n$ appear.

  • $\begingroup$ Thanks, that's extremely helpful :) So if I'm understanding your last paragraph correctly, if I want to consider the case of non-zero weight $2g-2$, I should study degree $n$ isogenies to a genus $g$ curve, and account for the term coming from the canonical bundle? $\endgroup$
    – Benighted
    Jun 3 '17 at 21:58
  • $\begingroup$ @StephenPietromonaco I think the $g$ in your GW expansion is distinct from the genus of the curve that takes isogenies, which should just be 1 (as long as you are looking at Jacobi forms). I suppose that means this isn't quite the same as a worldsheet interpretation. $\endgroup$
    – S. Carnahan
    Jun 4 '17 at 7:26
  • $\begingroup$ I see; but if we should only be using the isogenies to genus 1 curves, then what canonical bundle were you referring to? If you looks at the general formula for a Hecke operator acting on a weak Jacobi form, of possibly non-zero weight, there is this extra factor that looks like it might be related to the canonical bundle of a genus $g$ curve, as you mention. $\endgroup$
    – Benighted
    Jun 4 '17 at 19:01
  • $\begingroup$ @StephenPietromonaco According to section 1.1 of Katz's "p-adic properties" paper, a weight k modular form is a rule that assigns to any elliptic curve $(p: E \to S, e:S \to E)$ a section of $(p_*(\Omega_{E/S}))^{\otimes k}$, such that the assignment is invariant under base change, and depends only on the isomorphism class of the elliptic curve. By the Kodaira-Spencer isomorphism this is basically a section of the pluricanonical bundle $\Omega_S^{\otimes k/2}$ on the base. $\endgroup$
    – S. Carnahan
    Jun 5 '17 at 4:16

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