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

Abstract We find group cochains valued in currents giving explicit representatives for the $$\textrm{GL}_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>GL</mml:mtext> <mml:mn>2</mml:mn> </mml:msub> </mml:math> -equivariant polylogarithm class of a torus. Based on the construction of weight-2 Eisenstein series for $$\textrm{GL}_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>GL</mml:mtext> <mml:mn>2</mml:mn> </mml:msub> </mml:math> 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 $$E^2_{\alpha ,\beta }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>E</mml:mi> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mi>β</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> for various nonzero torsion sections $$(\alpha ,\beta )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mi>β</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> .