On the non-existence of singular Borcherds products
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2023
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866916458196369408 |
|---|---|
| author | Wang, Haowu Williams, Brandon |
| author_facet | Wang, Haowu Williams, Brandon |
| contents | Let $l\geq 3$ and $F$ be a modular form of weight $l/2-1$ on $\mathrm{O}(l,2)$ which vanishes only on rational quadratic divisors. We prove that $F$ has only simple zeros and that $F$ is anti-invariant under every reflection fixing a quadratic divisor in the zeros of $F$. In particular, $F$ is a reflective modular form. As a corollary, the existence of $F$ leads to $l\leq 20$ or $l=26$, in which case $F$ equals the Borcherds form on $\mathrm{II}_{26,2}$. This answers a question posed by Borcherds in 1995. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_13367 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the non-existence of singular Borcherds products Wang, Haowu Williams, Brandon Number Theory Quantum Algebra 11F22, 11F27, 11F55 Let $l\geq 3$ and $F$ be a modular form of weight $l/2-1$ on $\mathrm{O}(l,2)$ which vanishes only on rational quadratic divisors. We prove that $F$ has only simple zeros and that $F$ is anti-invariant under every reflection fixing a quadratic divisor in the zeros of $F$. In particular, $F$ is a reflective modular form. As a corollary, the existence of $F$ leads to $l\leq 20$ or $l=26$, in which case $F$ equals the Borcherds form on $\mathrm{II}_{26,2}$. This answers a question posed by Borcherds in 1995. |
| title | On the non-existence of singular Borcherds products |
| topic | Number Theory Quantum Algebra 11F22, 11F27, 11F55 |
| url | https://arxiv.org/abs/2301.13367 |