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 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 (J Number Theory 128:1060–1073, 2008) on his elliptic Dedekind–Rademacher sums. As an application of our results, we discover Eisenstein series identities which generalize certain formulas by Ramanujan (Collected Papers of Srinivasa Ramanujan, pp. 136–162. AMS Chelsea Publishing, Providence, 2000), van der Pol (Indag Math 13:261–271, 272–284, 1951), Rankin (Proc R Soc Edinburgh Sect A 76:107–117, 1976) and Skoruppa (J Number Theory 43:68–73, 1993). Mathematics Subject Classification (2000) Primary 11F20 - Secondary 11F11 - 33E05 This work was supported by Grant-in-Aid for Scientific Research (No.19540101), Japan Society for the Promotion of Science.