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

摘要

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