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Markov spectra for modular billiards

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Abstract

The connection between Markov’s theory of minima of indefinite binary quadratic forms and hyperbolic geodesics is well-known. We introduce some new analogues of the Markov spectrum defined in terms of modular billiards and consider the problem of characterizing that part of the spectrum below the lowest limit point.

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Acknowledgements

The second author thanks Alex Kontorovich for some enlightening discussions on the topics of this paper. The authors thank the referee for several constructive comments that have improved the exposition of this paper.

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Authors and Affiliations

  1. UCLA Mathematics Department, Box 951555, Los Angeles, CA, 90095-1555, USA

    Nickolas Andersen & William Duke

Authors
  1. Nickolas Andersen
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  2. William Duke
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Corresponding author

Correspondence to Nickolas Andersen.

Additional information

Communicated by Kannan Soundararajan.

In memory of Harvey Cohn (1923–2014)

Supported by NSF Grant DMS 1701638.

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Andersen, N., Duke, W. Markov spectra for modular billiards. Math. Ann. 373, 1151–1175 (2019). https://doi.org/10.1007/s00208-018-1781-x

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  • DOI: https://doi.org/10.1007/s00208-018-1781-x

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