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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.