Jacobi forms of weight one on ${\mathrm{\Gamma }}_0(N)$

Let $J_{1,m}(N)$ be the vector space of Jacobi forms of weight one and index $m$ on ${\mathrm{\Gamma }}_0(N)$. 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 $\mathrm{g}\mathrm{c}\mathrm{d}(m,N)=1$. 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 $\mathrm{g}\mathrm{c}\mathrm{d}(m,N)=1$. 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.