Fourier coefficients and cuspidality of modular forms: a new approach

We provide a simple and new induction based treatment of the problem of distinguishing cusp forms from the growth of the Fourier coefficients of modular forms. Our approach gives the best possible ranges of the weights for this problem, and has wide adaptability. We propose a conjecture which asks the same converse question based on information on the Fourier-Jacobi coefficients, and answer it partially. We also discuss how to recover cuspidality from the poles of the allied Rankin-Selberg $L$-series.

💡 Research Summary

The paper addresses the classical problem of detecting cuspidality of Siegel modular forms solely from the growth of their Fourier coefficients. While the statement is relatively easy for elliptic modular forms, it becomes highly non‑trivial in higher rank because the space of Eisenstein series is large and their Fourier expansions are poorly understood. The author proposes a new, short, and conceptually simple induction‑based method that works for any degree (n), any congruence subgroup (\Gamma\subset Sp_n(\mathbb Z)), and even for half‑integral weights.

The main theorem (Theorem 1.2) states that if a Siegel modular form (F\in M_k^{(n)}(\Gamma)) satisfies \