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Integral Points on Modular Curves, Singular Moduli and Conductor-Discriminant Inequality

Abstract : This thesis discusses three topics, so it includes three parts. In the first part, we study S-integral points on the modular curve X0(p). Bilushowed that, using Baker’s method, they can be effectively bounded in terms of p, the base field and the set of places S. Sha made this result explicit, but the bound he obtained is double exponential in p. We drastically improve upon the result of Sha, obtaining a simple exponential bound. This is done using a very explicit version of the Chevalley-Weil principle based on the work of Liu and Lorenzini. Our bound is not only sharper than that of Sha, but is also explicit in all parameters. In the second part, we consider singular moduli. For a fixed singular modulus a, we give an effective upper bound of norm of x - a for another singular modulus x with large discriminant. In the third part, we give a relation between Artin conductors of a Weierstrass model Y and the ones of two given Weierstrass models Y1,Y2. With this relation, we know that the conductor-discriminant inequality holds for Y if it holds for Y1 and Y2.
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Yulin Cai. Integral Points on Modular Curves, Singular Moduli and Conductor-Discriminant Inequality. Number Theory [math.NT]. Université de Bordeaux, 2020. English. ⟨NNT : 2020BORD0098⟩. ⟨tel-02952884⟩

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