Jacobi Forms of Critical Weight and Weil Representations

2007/07/05 by Nils-Peter Skoruppa, Skoruppa, Nils-Peter
Mathematics · #11F03 #11F27 #11F50 #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT) #math.NT #msc:11F03 #msc:11F27 #msc:11F50

paper · pdf · doi:10.48550/arxiv.0707.0718

arxiv created 2007/07/05 · openalex publication_date 2007/07/05 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Jacobi forms can be considered as vector valued modular forms, and Jacobi forms of critical weight correspond to vector valued modular forms of weight \frac12. Since the only modular forms of weight \frac12 on congruence subgroups of \SL are theta series the theory of Jacobi forms of critical weight is intimately related to the theory of Weil representations of finite quadratic modules. This article explains this relation in detail, gives an account of various facts about Weil representations which are useful in this context, and it gives some applications of the theory developed herein by proving various vanishing theorems and by proving a conjecture on Jacobi forms of weight one on \SL with character.

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