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

We find group cochains valued in currents giving explicit representatives for the ${\text{GL}}_2$-equivariant polylogarithm class of a torus. Based on the construction of weight-$2$ Eisenstein series for ${\text{GL}}_2$ 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 $E_{\alpha ,\beta }^2$ for various nonzero torsion sections $(\alpha ,\beta )$.