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Modular Dedekind symbols associated to Fuchsian groups and higher-order Eisenstein series

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Let E(z, s) be the non-holomorphic Eisenstein series for the modular group \({\mathrm{SL}}(2,{{\mathbb {Z}}})\). The classical Kronecker limit formula shows that the second term in the Laurent expansion at \(s=1\) of E(z, s) is essentially the logarithm of the Dedekind eta function. This eta function is a weight 1/2 modular form and Dedekind expressed its multiplier system in terms of Dedekind sums. Building on work of Goldstein, we extend these results from the modular group to more general Fuchsian groups \({\Gamma }\). The analogue of the eta function has a multiplier system that may be expressed in terms of a map \(S:{\Gamma }\rightarrow {{\mathbb {R}}}\) which we call a modular Dedekind symbol. We obtain detailed properties of these symbols by means of the limit formula. Twisting the usual Eisenstein series with powers of additive homomorphisms from \({\Gamma }\) to \({{\mathbb {C}}}\) produces higher-order Eisenstein series. These series share many of the properties of E(z, s) though they have a more complicated automorphy condition. They satisfy a Kronecker limit formula and produce higher-order Dedekind symbols \(S^*:{\Gamma }\rightarrow {{\mathbb {R}}}\). As an application of our general results, we prove that higher-order Dedekind symbols associated to genus one congruence groups \({\Gamma }_0(N)\) are rational.

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Notes

  1. It should have been stated in this theorem that \(\xi _S^{m,m}(s) \not \equiv 0\) which is clear from its construction.

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Acknowledgements

The authors thank Holger Then for his assistance in preparing the example in Sect. 7.2. We are very grateful for his computational results and, in general, for the generosity by which he shares his mathematical insight.

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Authors and Affiliations

  1. Department of Mathematics, City College of New York, Convent Avenue at 138th Street, New York, NY, 10031, USA

    Jay Jorgenson

  2. Department of Mathematics, The CUNY Graduate Center, 365 Fifth Avenue, New York, NY, 10016, USA

    Cormac O’Sullivan

  3. Department of Mathematics, University of Sarajevo, Zmaja od Bosne 35, 71 000, Sarajevo, Bosnia and Herzegovina

    Lejla Smajlović

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  1. Jay Jorgenson
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  2. Cormac O’Sullivan
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  3. Lejla Smajlović
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Correspondence to Cormac O’Sullivan.

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J. Jorgenson and C. O’Sullivan are partially supported by PSC-CUNY grants.

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Jorgenson, J., O’Sullivan, C. & Smajlović, L. Modular Dedekind symbols associated to Fuchsian groups and higher-order Eisenstein series. Res. number theory 6, 22 (2020). https://doi.org/10.1007/s40993-020-00198-x

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