Integrality of rational points of star quotients X_0(N)^* for N non-squarefree

Abstract: Modular curves X_0(N) parametrise elliptic curves E endowed with a cyclic torsion subgroup of order N, which makes their rational points related to Galois representation associated to torsion points of elliptic curves. A famous result of Mazur implies that X_0(N) has only "trivial" rational points for N>163 prime, and it leads to solving the problem for a general N. For N squarefree, the method used by Mazur (based on formal immersions) cannot work as there should be (under BSD conjecture) no abelian subvariety of the jacobian of X_0(N)^* of Mordell-Weil rank zero. Surprisingly though, when N is not squarefree, there is just enough leeway to allow for such subvarieties to exist. In this talk based on a joint work with Sachi Hashimoto and Timo Keller, I will explain how to make this idea work in practice to prove integrality (i.e. potentially good reduction of elliptic curves associated to non-cuspidal rational points) for almost all non-squarefree levels N, and how some unexpected rational points have been found for some sporadic levels N."

number theory

Audience: researchers in the topic

London number theory seminar

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For a record of talks predating this website see: wwwf.imperial.ac.uk/~buzzard/LNTS/numbtheo_past.html