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On a property of elliptic Dedekind sums. (English) Zbl 0624.10018
Let L be a lattice in \({\mathbb{C}}\) with multiplier ring \({\mathcal O}=\{m\in {\mathbb{C}}|\) mL\(\subset L\}\). Let \(E_ k(x)\) be the value of the continuation of \(\sum_{w\in L}(w+x)^{-k} | w+x|^{-s}\) at \(s=0\). R. Sczech [Invent. Math. 76, 523-551 (1984; Zbl 0521.10021)] introduced the sums \[ D(a,c)=(1/c)\sum_{k\in L/cL}E_ 1(ak/c)E_ 1(k/c) \] for a, c in \({\mathcal O}\) with \(c\neq 0\), an analog of the classical Dedekind sums, and conjectured that \(D(a,c)=-D(\bar a,\bar c).\) This formula can be derived from previous work of the author [J. Reine Angew. Math. 373, 148-165 (1987; Zbl 0601.10021)]. Now the author gives a short proof of this formula which rests upon a well-known functional equation of an Eisenstein series. He also discusses some consequences.
Reviewer: G.Köhler

MSC:
11F11 Holomorphic modular forms of integral weight
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
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[1] Ito, H., A function on the upper half space which is analogous to the imaginary part of log ν(z), J. reine angew. math., 373, 148-165, (1987) · Zbl 0601.10021
[2] Rademacher, H.; Grosswald, E., (), Carus Mathematical Monographs, No. 16 · JFM 49.0022.03
[3] Sczech, R., Dedekindsummen mit elliptischen funktionen, Invent. math., 76, 523-551, (1984) · Zbl 0521.10021
[4] Weil, A., ()
[5] Weselmann, U., Eisenstein-kohomologie und dedekindsummen für GL2über imaginarquadratischen zahlkörpern, Diplomarbeit, (1985), Bonn
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