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<p>We give closed formulas for the first few expansion coefficients of the basic modular forms for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\({{\,\textrm{GL}\,}}(r, \mathbb {F}_{q}[T])\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mspace width="0.166667em" /> <mtext>GL</mtext> <mspace width="0.166667em" /> </mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">[</mo> <mi>T</mi> <mo stretchy="false">]</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Here the rank <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(r\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>r</mi> </math></EquationSource> </InlineEquation> is larger or equal to <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, and the forms in question include the coefficient forms <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(g_{1}, \dots , g_{r}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>g</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>g</mi> <mi>r</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> and the Eisenstein series <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(E_{q^{i}-1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>E</mi> <mrow> <msup> <mi>q</mi> <mi>i</mi> </msup> <mo>-</mo> <mn>1</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq12"> <EquationSource Format="TEX">\(i \in \mathbb {N}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>).</p>

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Modular forms for \({{\,\textrm{GL}\,}}(r, \mathbb {F}_{q}[T])\): \(t\)-expansions of the basic forms

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  • 发表日期：

    2026年3月31日

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  • Ernst-Ulrich Gekeler

摘要

We give closed formulas for the first few expansion coefficients of the basic modular forms for \({{\,\textrm{GL}\,}}(r, \mathbb {F}_{q}[T])\) GL ( r , F q [ T ] ) . Here the rank \(r\) r is larger or equal to \(3\) 3 , and the forms in question include the coefficient forms \(g_{1}, \dots , g_{r}\) g 1 , ⋯ , g r and the Eisenstein series \(E_{q^{i}-1}\) E q i - 1 ( \(i \in \mathbb {N}\) i ∈ N ).