Jacobi forms of weight 1 on Γ0(N)Γ ₀(N)

Let be the vector space of Jacobi forms of weight one and index on . In 1985, Skoruppa proved that for all . In 2007, Ibukiyama and Skoruppa proved that for all and all squarefree with . This paper aims to extend their results. We determine all levels separately, such that for all ; or for all with . We also establish explicit dimension formulae of when and are relatively prime or 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.