Jacobi forms of weight 1 on Γ 0 ( N ) $\mathbf {\Gamma _0(N)}$

Let $J_{1,m}\left(N\right)$ be the vector space of Jacobi forms of weight one and index $m$ on ${\Gamma }_0\left(N\right)$. In 1985, Skoruppa proved that $J_{1,m}\left(1\right)=0$ for all $m$. In 2007, Ibukiyama and Skoruppa proved that $J_{1,m}\left(N\right)=0$ for all $m$ and all squarefree $N$ with $\gcd (m,N)=1$. This paper aims to extend their results. We determine all levels $N$ separately, such that $J_{1,m}\left(N\right)=0$ for all $m$; or $J_{1,m}\left(N\right)=0$ for all $m$ with $\gcd (m,N)=1$. We also establish explicit dimension formulae of $J_{1,m}\left(N\right)$ 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 2 and weight 1 in some cases.