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<p>We prove that vector-valued Siegel cusp forms for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\Gamma _0^n(N)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi mathvariant="normal">Γ</mi> <mn>0</mn> <mi>n</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with certain nebentypus are determined by their fundamental Fourier coefficients with discriminants coprime to <i>N</i>, assuming <i>N</i> is odd and square-free. In the case of genus 3, we strengthen this to Fourier coefficients corresponding to maximal orders in quaternion algebras. We also prove that Jacobi forms with odd, square-free level <i>N</i> and odd, square-free index with discriminant coprime to <i>N</i> are determined by their primitive theta components.</p>

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Certain Siegel Cusp Forms with Level are Determined by their Fundamental Fourier Coefficients

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  • 发表日期：

    2026年4月28日

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  • Sidney Washburn

摘要

We prove that vector-valued Siegel cusp forms for \(\Gamma _0^n(N)\) Γ 0 n ( N ) with certain nebentypus are determined by their fundamental Fourier coefficients with discriminants coprime to N, assuming N is odd and square-free. In the case of genus 3, we strengthen this to Fourier coefficients corresponding to maximal orders in quaternion algebras. We also prove that Jacobi forms with odd, square-free level N and odd, square-free index with discriminant coprime to N are determined by their primitive theta components.