The elliptic Apostol-Dedekind sums generate odd Dedekind symbols with Laurent polynomial reciprocity laws

TL;DR

This paper introduces elliptic Apostol-Dedekind sums that generate all odd Dedekind symbols with Laurent polynomial reciprocity laws, linking them to modular forms and deriving new Eisenstein series identities.

Contribution

It defines a new elliptic analogue of Apostol-Dedekind sums and proves they generate all odd Dedekind symbols with Laurent polynomial reciprocity laws.

Findings

Generated all odd Dedekind symbols with Laurent polynomial reciprocity laws.

Derived Eisenstein series identities extending classical formulas.

Connected elliptic sums to modular forms and reciprocity laws.

Abstract

Dedekind symbols are generalizations of the classical Dedekind sums (symbols). There is a natural isomorphism between the space of Dedekind symbols with Laurent polynomial reciprocity laws and the space of modular forms. We will define a new elliptic analogue of the Apostol-Dedekind sums. Then we will show that the newly defined sums generate all odd Dedekind symbols with Laurent polynomial reciprocity laws. Our construction is based on Machide's result on his elliptic Dedekind-Rademacher sums. As an application of our results, we discover Eisenstein series identities which generalize certain formulas by Ramanujan, van der Pol, Rankin and Skoruppa.