Sorbonne Université CNRS Paris Diderot

Séminaire Théorie des Nombres

Shimura curves and the abc conjecture

Hector Pasten - Pontificia Universidad Católica de Chile

lundi 18 février 2019 à 14:00
Sophie-Germain salle 1016

Elliptic curves over the rationals admit maps from various Shimura curves, and the comparison ratio of the degrees of these maps recovers important information on abc-triples. On the other hand, this ratio can be controlled by the Arakelov height of CM points. This requires a number of tools : zero-density estimates for L-functions, integral models for various objects, Galois representations, and some complex-analytic estimates. The final outcome is an unconditional estimate for the product of p-adic valuations of abc-triples, which lies beyond the reach of existing methods in the context of the abc conjecture such as linear forms in logarithms. Our methods also yield other results. For instance, for totally real fields F of bounded degree, we prove that the Faltings height of modular elliptic curves E over F is bounded linearly on log(modular degree of E) + log(Disc. of F). The logarithmic dependence of the discriminant of F can be seen as evidence towards Vojta’s conjecture on algebraic points of bounded degree.

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