Duke Mathematical Journal

Modular cocycles and linking numbers

W. Duke, Ö. Imamoḡlu, and Á. Tóth

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It is known that the 3-manifold SL(2,Z)\SL(2,R) is diffeomorphic to the complement of the trefoil knot in S3. E. Ghys showed that the linking number of this trefoil knot with a modular knot is given by the Rademacher symbol, which is a homogenization of the classical Dedekind symbol. The Dedekind symbol arose historically in the transformation formula of the logarithm of Dedekind’s eta function under SL(2,Z). In this paper we give a generalization of the Dedekind symbol associated to a fixed modular knot. This symbol also arises in the transformation formula of a certain modular function. It can be computed in terms of a special value of a certain Dirichlet series and satisfies a reciprocity law. The homogenization of this symbol, which generalizes the Rademacher symbol, gives the linking number between two distinct symmetric links formed from modular knots.

Article information

Duke Math. J., Volume 166, Number 6 (2017), 1179-1210.

Received: 3 June 2015
Revised: 8 July 2016
First available in Project Euclid: 7 January 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 11F12: Automorphic forms, one variable 11F20: Dedekind eta function, Dedekind sums

Modular forms modular integrals Dedekind symbol linking number


Duke, W.; Imamoḡlu, Ö.; Tóth, Á. Modular cocycles and linking numbers. Duke Math. J. 166 (2017), no. 6, 1179--1210. doi:10.1215/00127094-3793032. https://projecteuclid.org/euclid.dmj/1483758030

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