Quadratic Chabauty for Atkin-Lehner Quotients of Modular Curves
Quadratic Chabauty for Atkin-Lehner Quotients of Modular Curves of Prime Level and Genus 4, 5, 6 Nikola Adˇzaga1, Vishal Arul2, Lea Beneish3, Mingjie Chen4, Shiva Chidambaram5, Timo Keller6, and Boya Wen7 1Department of Mathematics, Faculty of Civil Engineering, University of Zagreb∗ 2Department of Mathematics, University College London† 3Department of Mathematics and Statistics, McGill University‡ 4Department of Mathematics, University of California at San Diego§ 5Department of Mathematics, University of Chicago¶ 6Department of Mathematics, Universit¨at Bayreuth‖ 7Department of Mathematics, Princeton University∗∗ May 12, 2021 Abstract + We use the method of quadratic Chabauty on the quotients X0 (N) of modular curves X0(N) by their Fricke involutions to provably compute all the rational points of these curves for prime levels N of genus four, five, and six. We find that the only such curves with exceptional rational points are of levels 137 and 311. In particular there are no exceptional rational points on those curves of genus five and six. More precisely, we determine the rational + points on the curves X0 (N) for N = 137, 173, 199, 251, 311, 157, 181, 227, 263, 163, 197, 211, 223, 269, 271, 359. Keywords: Rational points; Curves of arbitrary genus or genus =6 1 over global fields; arithmetic aspects of modular and Shimura varieties 2020 Mathematics subject classification: 14G05 (primary); 11G30; 11G18 1 Introduction + The curve X0 (N) is a quotient of the modular curve X0(N) by the Atkin-Lehner involution wN (also called the Fricke + involution). The non-cuspidal points of X0 (N) classify unordered pairs of elliptic curves together with a cyclic isogeny + of degree N between them, where the Atkin-Lehner involution wN sends an isogeny to its dual.
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