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<p>We define a theta lift between the homology in degree <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(N-1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> of a locally symmetric space associated to <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\textrm{SL}_N(\mathbb {R})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>SL</mtext> <mi>N</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and the space of modular forms of weight <i>N</i>, similar to the Kudla–Millson lift in the orthogonal setting. We show that the Fourier coefficients of this lift are Poincaré duals to modular symbols associated to maximal parabolic subgroups. The constant term is a canonical cohomology classes obtained by transgressing the Euler class of a torus bundle. When <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(N=2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, we show that the lift surjects on the space of weight 2 modular forms spanned by an Eisenstein series and the eigenforms with non-vanishing <i>L</i>-function.</p>

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Eisenstein classes and generating series of modular symbols in \(\textrm{SL}_N\)

  • 期刊论文

  • 发表日期：

    2026年4月3日

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  • Romain Branchereau

摘要

We define a theta lift between the homology in degree \(N-1\) N - 1 of a locally symmetric space associated to \(\textrm{SL}_N(\mathbb {R})\) SL N ( R ) and the space of modular forms of weight N, similar to the Kudla–Millson lift in the orthogonal setting. We show that the Fourier coefficients of this lift are Poincaré duals to modular symbols associated to maximal parabolic subgroups. The constant term is a canonical cohomology classes obtained by transgressing the Euler class of a torus bundle. When \(N=2\) N = 2 , we show that the lift surjects on the space of weight 2 modular forms spanned by an Eisenstein series and the eigenforms with non-vanishing L-function.