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In this question I'll try to avoid using the words "Borcherd's Lift" only because I'm not sure in what setting it applies properly. What I will be asking about is sometimes called "second quantized elliptic genus" in the physics circles, or perhaps an "exponential lift of a Jacobi form".

One important result is that the elliptic genus $\chi(M_{d} ; q, y)$ of a $d$-dimensional, compact Calabi-Yau manifold $M_{d}$ is a weak Jacobi form of weight zero and index $d/2$. This means the elliptic genus admits a Fourier expansion

$$\chi(M_{d}; q,y) = \sum_{n \geq 0, k \in \mathbb{Z}} c(n, k) q^{n} y^{k},$$

where for a fixed power of $q$, finitely many powers of $y$ contribute. There is also symmetry under $y \to 1/y$. One can use the coefficients of the elliptic genus to produce a infinite product of the form

$$\prod_{m,n,k} \big(1-p^{m}q^{n}y^{k}\big)^{-c(nm, k)}.$$

It is known (https://arxiv.org/abs/math/9906190) that if $M_{d}$ Is Calabi-Yau, then this infinite product will be a Seigel modular form, at least up to some factor. Therefore, weight zero (arbitrary index) Jacobi forms seem to play a special role, at least in this sense.

However, index one (arbitrary weight) Jacobi forms also appear to have a special role in relation to Siegel modular forms, as indicated in the following figure:

enter image description here

(Figure taken from Eichler and Zagier, Theory of Jacobi Forms, page 4)

I'm specifically hoping someone can survey some results for me about the "lifting" of index one Jacobi forms. I have certain weight $2k$, index one Jacobi forms $\varphi_{2k,1}$. Do we know precisely which Siegel modular forms these will lift to? Moreover, assuming we know where each $\varphi_{2k, 1}$ goes to, do we know where a formal linear combination

$$\sum_{k=-1}^{\infty} \lambda^{2k} \varphi_{2k, 1}$$

will lift to? Here, $\lambda$ is a formal variable.

I would also be keenly interested if anyone has insight into why weight zero Jacobi forms, as well as index one Jacobi forms play seemingly such a special role.

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    $\begingroup$ I don't have an answer to your specific question, but one reason why index one Jacobi forms are special comes from the natural isomorphism (given in Example 2.3 of Borcherds's "arxiv.org/abs/alg-geom/9609022 and proved in Eichler-Zagier) between the space of holomorphic Jacobi forms of weight $k$ and index $m$ and the space of holomorphic vector-valued modular forms for $\bar{\rho}_M$, where $M$ is a one dimensional lattice generated by a vector of norm $2m$. The isomorphism is given in one direction by taking a suitable product with a vector-valued theta function for the lattice. $\endgroup$ – S. Carnahan Sep 2 '17 at 20:50
  • $\begingroup$ Thanks for the comment. You seem to state something for general index $m$, so why exactly is $m=1$ special in this example? What you're talking about is perhaps that "middle" isomorphism in the Figure of Eichler and Zagier? $\endgroup$ – Benighted Sep 2 '17 at 21:59
  • $\begingroup$ The only way $m=1$ is special that I had in mind is the small level - the lattice in question has determinant 2. You can get analogues of the middle isomorphism for larger $m$, but it is more cumbersome. Anyway, the Borcherds-Harvey-Moore lift gives you a way to write cusp expansions as infinite products, where the exponents are coefficients of the vector-valued form. I suppose you can uniquely identify the lifted form using finitely many such terms, but I don't have computational experience beyond some rather degenerate $O(2,2)$ forms in moonshine. $\endgroup$ – S. Carnahan Sep 3 '17 at 11:43

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