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

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

2009-07-23

arxiv.org/abs/0907.4058v1

Mathematics

Number Theory

AMS-LaTeX, 21 pages

Scientific paper

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.

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