Fundamental Fourier coefficients of Siegel modular forms of higher degrees and levels

2025/09/15 by Pramath Anamby, Anamby, Pramath, Soumya Das +1
Mathematics · #11F37 #11F46 #Advanced Algebra and Geometry #Analytic Number Theory Research #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #Primary 11F30 #Secondary 11F50

paper · pdf · doi:10.48550/arxiv.2509.12148

openalex publication_date 2025/09/15 · openalex created_date 2025/10/12 · openalex updated_date 2026/07/28

Abstract

We prove the following statement about any Siegel modular form F of degree n and arbitrary odd level N on the group Γ₀⁽ⁿ⁾(N). Let A(F,T) denote the Fourier coefficients of F and write T=(T(i,j)). Suppose that F has a non-zero Fourier coefficient A(F,T₀) such that (T₀(n,n),N)=1. Then there exist infinitely many odd and square-free (and thus fundamental) integers m such that m=discriminant(T) and A(F,T)≠ 0. In the case of odd degrees, we prove a stronger result by replacing odd and square-free with odd and prime. We also prove quantitative results in this direction. As a consequence, we can show in particular that the statement of the main result in arXiv:2408.03442 about the algebraicity of certain critical values (and the expected functional equation) of the spinor L-functions of holomorphic newforms (in the ambit of Deligne's conjectures) on congruence subgroups of GSp(3) is unconditional.

Citations

• Jacobi Forms of Affine Weight in Higher Cogenus and Nearly Holomorphic Functions
• Certain Siegel Cusp Forms with Level are Determined by their Fundamental Fourier Coefficients
• Non-vanishing of theta components of Jacobi forms with level and an application
• On sign changes of primitive Fourier coefficients of Siegel cusp forms
• Determination of modular forms by fundamental Fourier coefficients
• Siegel cusp forms of degree 2 are determined by their fundamental Fourier coefficients
• Siegel cusp forms of degree 2 are determined by their fundamental Fourier coefficients

Related

• Siegel cusp forms of degree 2 are determined by their fundamental Fourier coefficients
• Some relations on Fourier coefficients of degree 2 Siegel forms of arbitrary level
• Certain Siegel Cusp Forms with Level are Determined by their Fundamental Fourier Coefficients
• New bounds for fundamental Fourier coefficients of Siegel modular forms
• Improved bounds for Fourier coefficients of Siegel modular forms