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

TL;DR

This paper constructs explicit representatives for the equivariant polylogarithm class of a torus using group cochains, linking Eisenstein series and Bernoulli polynomials to derive classical formulas for Dedekind-Rademacher homomorphisms.

Contribution

It provides a geometric construction of the equivariant polylogarithm class and derives classical period formulas for Eisenstein series in a new, explicit manner.

Findings

Explicit group cochain representatives for the polylogarithm class.

Derivation of classical Dedekind-Rademacher formulas from geometric constructions.

Connection between Eisenstein series, Bernoulli polynomials, and equivariant cohomology.

Abstract

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 $E^2_{\alpha,\beta}$ for various nonzero torsion sections $(\alpha, \beta)$.