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Pell surfaces

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In 1826 Abel started the study of the polynomial Pell equationx2 − g(u)y2 = 1. Its solvability in polynomials x(u), y(u) depends on a certain torsion point on the Jacobian of the hyperelliptic curve v2 = g(u). In this paper we study the affine surfaces defined by the Pell equations in 3-space with coordinatesx, y, u, and aim to describe all affine lines on it. These are polynomial solutions of the equation x(t)2 − g(u(t))y(t)2 = 1. Our results are rather complete when the degree of g is even but the odd degree cases are left completely open. For even degrees we also describe all curves on these Pell surfaces that have only 1 place at infinity.

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Acknowledgements

I thank A.A. Chilikov and A.J. Kanel-Belov for posing the original question, D. Gabai, S. Mullane and Z. Scherr for help with the literature, L. Chen, S. Kovács, A. Libgober, M. Lieblich, J.-P. Serre, B. Totaro and J. Waldron for helpful conversations and comments. U. Zannier gave numerous comments, corrections and examples.

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  1. Princeton University, Princeton, NJ, 08544-1000, USA

    J. Kollár

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Correspondence to J. Kollár.

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Partial financial support was provided by the NSF under grant numbers DMS-1362960 and DMS-1440140 while the author was in residence at MSRI during the Spring 2019 semester.

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Kollár, J. Pell surfaces. Acta Math. Hungar. 160, 478–518 (2020). https://doi.org/10.1007/s10474-019-01008-2

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