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<p>We find group cochains valued in currents giving explicit representatives for the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\textrm{GL}_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mtext>GL</mtext> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-equivariant polylogarithm class of a torus. Based on the construction of weight-2 Eisenstein series for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\textrm{GL}_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mtext>GL</mtext> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> from this polylogarithm class, we give a geometrically-flavored derivation of the classical formulas for the associated Dedekind-Rademacher homomorphisms, i.e. the periods of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(E^2_{\alpha ,\beta }\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>E</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mrow> <mn>2</mn> </msubsup> </math></EquationSource> </InlineEquation> for various nonzero torsion sections <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\((\alpha ,\beta )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Classical periods of Eisenstein series and Bernoulli polynomials in the equivariant cohomology of a torus

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  • 发表日期：

    2026年7月17日

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  • Peter Xu

摘要

We find group cochains valued in currents giving explicit representatives for the \(\textrm{GL}_2\) GL 2 -equivariant polylogarithm class of a torus. Based on the construction of weight-2 Eisenstein series for \(\textrm{GL}_2\) GL 2 from this polylogarithm class, we give a geometrically-flavored derivation of the classical formulas for the associated Dedekind-Rademacher homomorphisms, i.e. the periods of \(E^2_{\alpha ,\beta }\) E α , β 2 for various nonzero torsion sections \((\alpha ,\beta )\) ( α , β ) .