Jacobi forms of weight one on Γ₀(N)

2024/10/17 by Jialin Li, Li, Jialin, Haowu Wang +1
Mathematics · #11F27 #11F46 #11F50 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2410.13208

openalex publication_date 2024/10/17 · openalex created_date 2024/10/21 · openalex updated_date 2026/07/28

Abstract

Let J_1,m(N) be the vector space of Jacobi forms of weight one and index m on Γ₀(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 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 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.

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