Jacobi forms of weight one on $\Gamma_0(N)$

TL;DR

This paper extends previous results on the vanishing of Jacobi forms of weight one by classifying levels where the space is zero and providing explicit dimension formulas, with applications to Siegel modular forms.

Contribution

It determines all levels N for which the space of Jacobi forms of weight one vanishes and derives explicit dimension formulas, refining Skoruppa's method and analyzing Weil representations.

Findings

Identifies all levels N with zero Jacobi form spaces for all m.

Provides explicit dimension formulas for Jacobi forms when gcd(m,N)=1 or m is squarefree.

Proves vanishing of certain Siegel modular forms of degree two and weight one.

Abstract

Let $J_{1,m}(N)$ be the vector space of Jacobi forms of weight one and index $m$ on $\Gamma_0(N)$. In 1985, Skoruppa proved that $J_{1,m}(1)=0$ for all $m$. In 2007, Ibukiyama and Skoruppa proved that $J_{1,m}(N)=0$ for all $m$ and all squarefree $N$ with $\mathrm{gcd}(m,N)=1$. This paper aims to extend their results. We determine all levels $N$ separately, such that $J_{1,m}(N)=0$ for all $m$; or $J_{1,m}(N)=0$ for all $m$ with $\mathrm{gcd}(m,N)=1$. We also establish explicit dimension formulas of $J_{1,m}(N)$ when $m$ and $N$ are relatively prime or $m$ is squarefree. These results are obtained by refining Skoruppa's method and analyzing local invariants of Weil representations. As applications, we prove the vanishing of Siegel modular forms of degree two and weight one in some cases.