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Dedekind sums and power residue symbols. (English) Zbl 0604.10010
Let K be an imaginary quadratic field of discriminant \(D\equiv 1\) (mod 8) and \({\mathcal O}_ k\) the ring of algebraic integers in K. In a previous paper [Invent. Math. 76, 523-551 (1984; Zbl 0521.10022)] the author replacing the cotangent function in the definition of the classical Dedekind sums by an appropriate elliptic function introduced Dedekind sums with respect to the group \(SL_ 2 {\mathcal O}_ k\). In this work he discusses the relation between the power residue symbol in K and his Dedekind sums.
It is conjectured that there exists a canonical homomorphism \(\psi\) of \(SL_ 2 {\mathcal O}_ k\) with values in \({\mathcal O}_ H\), the ring of integers in the Hilbert class field H of K, such that \(\psi (SL_ 2 ({\mathcal O}_ k)\) reduced modulo 8 is a cyclic subgroup of \({\mathcal O}_ h/8{\mathcal O}_ H\). This conjecture generalizes the well known relation between classical Dedekind sums and the Legendre symbol. The paper concludes with the proof of this conjecture in some special cases such as \(D=-7\), -15, -23, -31, -39 and -55.
Reviewer: J.Antoniadis

MSC:
11F03 Modular and automorphic functions
11A15 Power residues, reciprocity
11R37 Class field theory
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11R11 Quadratic extensions
11R16 Cubic and quartic extensions
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