Timeline for "Lifting" of Jacobi forms of weight zero vs. index one?
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 3, 2017 at 11:43 | comment | added | S. Carnahan♦ | 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. | |
| Sep 2, 2017 at 21:59 | comment | added | Benighted | 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? | |
| Sep 2, 2017 at 20:50 | comment | added | S. Carnahan♦ | 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. | |
| Sep 2, 2017 at 7:50 | history | edited | Benighted | CC BY-SA 3.0 |
added 1 character in body
|
| Sep 2, 2017 at 2:10 | history | asked | Benighted | CC BY-SA 3.0 |