Héctor H. Pastén Vásquez: Shimura curves and bounds for the abc conjecture

Abstract: I will explain some new connections between the abc conjecture and modular forms. In particular, I will outline a proof of a new unconditional estimate for the abc conjecture, which lies beyond the existing techniques in this context. The proof involves a number of tools such as Shimura curves, CM points, analytic number theory, and Arakelov geometry. It also requires some intermediate results of independent interest, such as bounds for the Manin constant beyond the semi-stable case. If time permits, I will also explain some results towards Szpiro's conjecture over totally real number fields which are compatible with the discriminant term appearing in Vojta's conjecture for algebraic points of bounded degree.

Recording during the meeting "Diophantine Geometry" the May 23, 2018 at the Centre International de Rencontres Mathématiques (Marseille, France)

Filmmaker: Guillaume Hennenfent

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