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H 1 Shimura curves and bounds for the $abc$ conjecture

Auteurs : Pasten, Hector (Auteur de la Conférence)
CIRM (Editeur )

Résumé : 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.

Codes MSC :
11F11 - Holomorphic modular forms of integral weight
11G05 - Elliptic curves over global fields
11G18 - Arithmetic aspects of modular and Shimura varieties
14G40 - Arithmetic varieties and schemes; Arakelov theory; heights

 Informations sur la Vidéo Réalisateur : Hennenfent, Guillaume Langue : Anglais Date de publication : 29/05/2018 Date de captation : 23/05/2018 Collection : Research talks ; Algebraic and Complex Geometry ; Number Theory Format : MP4 Durée : 01:00:35 Domaine : Algebraic & Complex Geometry ; Number Theory Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2018-05-23_Pasten.mp4 Informations sur la rencontre Nom de la rencontre : Diophantine geometry / ​Géométrie diophantienneOrganisateurs de la rencontre : Bosser, Vincent ; Carrizosa, Maria ; Gaudron, Eric ; Habegger, PhilippDates : 21/05/2018 - 25/05/2018 Année de la rencontre : 2018 URL Congrès : https://conferences.cirm-math.fr/1754.htmlCitation Data DOI : 10.24350/CIRM.V.19408103 Cite this video as: Pasten, Hector (2018). Shimura curves and bounds for the $abc$ conjecture. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19408103 URI : http://dx.doi.org/10.24350/CIRM.V.19408103

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