Abstract
We introduce interaction entropies, which can be represented as logarithmic couplings of certain cycles on a class of algebraic curves of arithmetic interest. In particular, via interaction entropies for Legendre–Ramanujan curves \( Y^n=(1-X)^{n-1}X(1-\alpha X)\) (\( n\in \{6,4,3,2\}\)), we reformulate the Kontsevich–Zagier integral representations of weight-4 automorphic Green’s functions \( G_2^{\mathfrak H/\overline{\varGamma }_0(N)}(z_1,z_2)\) (\(N=4\sin ^2(\pi /n )\in \{1,2,3,4\}\)), in a geometric context. These geometric entropies allow us to establish algebraic relations between certain weight-4 automorphic self-energies and special values of weight-6 automorphic Green’s functions.
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Notes
Hereafter, as in Part I [19], we maintain the distinction between lowercase backslash (“\(\smallsetminus \)” for set minus operations) and uppercase backslash (“\( \backslash \)” for orbit spaces).
We write “a.e.” for “almost every” point in question, so as to accommodate to possible exceptions that form a set of zero measure.
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Masanobu Kaneko)Acknowledgments
The manuscript was completed during the author’s visit to Prof. Weinan E at Princeton in 2014 and at BICMR in 2015. The author thanks Prof. E for discussions on renormalization group theory, and Prof. Shou-Wu Zhang for his comments on Theorem 1.1.3 at Princeton in 2014. The author is especially grateful to Prof. Don B. Zagier (MPIM, Bonn) for his encouragements on this series of works. The author appreciates the suggestions from Dr. Qingtao Chen (ETH Zürich) and an anonymous referee on improving the organization of this paper.
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This work was partly supported by the Applied Mathematics Program within the Department of Energy (DOE) Office of Advanced Scientific Computing Research (ASCR) as part of the Collaboratory on Mathematics for Mesoscopic Modeling of Materials (CM4).
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Zhou, Y. Kontsevich–Zagier integrals for automorphic Green’s functions. II. Ramanujan J 42, 623–688 (2017). https://doi.org/10.1007/s11139-016-9818-9
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DOI: https://doi.org/10.1007/s11139-016-9818-9
Keywords
- Kontsevich–Zagier periods
- Automorphic Green’s functions
- Interaction entropies
- Gross–Zagier renormalization