Dataciencia

Colección SciELO Chile

Shimura curves and the abc conjecture

Indexado

WoS	WOS:001073082200001
Scopus	SCOPUS_ID:85169808648

DOI

10.1016/J.JNT.2023.07.002

Año

2024

Tipo

artículo de investigación

Citas Totales

Autores Afiliación Chile

Instituciones Chile

% Participación
Internacional

Autores
Afiliación Extranjera

Instituciones
Extranjeras

Abstract

In this work we develop a framework that enables the use of Shimura curve parametrizations of elliptic curves to approach the abc conjecture, leading to a number of new unconditional applications over Q and, more generally, totally real number fields. Several results of independent interest are obtained along the way, such as bounds for the Manin constant, a study of the congruence number, extensions of the Ribet-Takahashi formula, and lower bounds for the L2-norm of integral quaternionic modular forms.The methods require a number of tools from Arakelov geometry, analytic number theory, Galois representations, complex-analytic estimates on Shimura curves, automorphic forms, known cases of the Colmez conjecture, and results on generalized Fermat equations.& COPY; 2023 Published by Elsevier Inc.

Revista

Revista	ISSN
Journal Of Number Theory	0022-314X

Métricas Externas

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Disciplinas de Investigación

WOS
Mathematics

Scopus
Sin Disciplinas

SciELO
Sin Disciplinas

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Publicaciones WoS (Ediciones: ISSHP, ISTP, AHCI, SSCI, SCI), Scopus, SciELO Chile.

Colaboración Institucional

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Autores - Afiliación

Ord.	Autor	Género	Institución - País
1	Pasten, Hector	Hombre	Pontificia Universidad Católica de Chile - Chile Facultad de Matemáticas - Chile

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Financiamiento

Fuente
National Science Foundation
Fondo Nacional de Desarrollo Científico y Tecnológico
Comisión Nacional de Investigación Científica y Tecnológica
NSF
Harvard University
Institute for Advanced Study
Schmidt Fellowship
Agencia Nacional de Investigación y Desarrollo
ANID (ex CONICYT) FONDECYT
Benjamin Peirce Fellowship
Kestutis Cesnavicius and Bas Edixhoven

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Agradecimientos

Agradecimiento
*This research was partially supported by a Benjamin Peirce Fellowship (at Harvard) , by a Schmidt Fellowship and the NSF Grant DMS-1128155 (at IAS) , and by ANID (ex CONICYT) FONDECYT Regular grant 1190442.
This research was partially supported by a Benjamin Peirce Fellowship (at Harvard), by a Schmidt Fellowship and the NSF Grant DMS-1128155 (at IAS), and by ANID (ex CONICYT) FONDECYT Regular grant 1190442.This research was partially supported by a Benjamin Peirce Fellowship (at Harvard), by a Schmidt Fellowship and the NSF Grant DMS-1128155 (at IAS), and by ANID (ex CONICYT) FONDECYT Regular grant 1190442. Part of this project was carried out while I was a member at the Institute for Advanced Study in 2015-2016, and then continued at Harvard. I greatly benefited from conversations with Enrico Bombieri, Noam Elkies, Nicholas Katz, Barry Mazur, Peter Sarnak, Richard Taylor, and Shou-Wu Zhang, and I sincerely thank them for generously sharing their ideas and knowledge. In particular, the argument in Paragraph 5.6 originates in an idea of R. Taylor in the case of classical modular curves, and the connection between injectivity radius and Lehmer's conjecture used in Section 8 was pointed out to me by P. Sarnak. Feedback from B. Mazur on the topic of the Manin constant was of great help, and I also thank Kestutis Cesnavicius and Bas Edixhoven for some additional observations on this subject. I learned some of the material relevant for this project by attending the joint IAS-Princeton algebraic number theory seminar during my stay at the IAS, and I thank Christopher Skinner and Richard Taylor for organizing it. Comments from Natalia Garcia-Fritz and Ricardo Menares regarding the presentation of this article, as well as the valuable comments and corrections of the referee, are gratefully acknowledged. I also thank Robert Lemke Oliver and Jesse Thorner for the appendix. Finally, I thank Enrico Bombieri and Natalia Garcia-Fritz for their encouragement, which allowed me to push this project further than I initially had in mind.

Agradecimiento

*This research was partially supported by a Benjamin Peirce Fellowship (at Harvard) , by a Schmidt Fellowship and the NSF Grant DMS-1128155 (at IAS) , and by ANID (ex CONICYT) FONDECYT Regular grant 1190442.

This research was partially supported by a Benjamin Peirce Fellowship (at Harvard), by a Schmidt Fellowship and the NSF Grant DMS-1128155 (at IAS), and by ANID (ex CONICYT) FONDECYT Regular grant 1190442.This research was partially supported by a Benjamin Peirce Fellowship (at Harvard), by a Schmidt Fellowship and the NSF Grant DMS-1128155 (at IAS), and by ANID (ex CONICYT) FONDECYT Regular grant 1190442. Part of this project was carried out while I was a member at the Institute for Advanced Study in 2015-2016, and then continued at Harvard. I greatly benefited from conversations with Enrico Bombieri, Noam Elkies, Nicholas Katz, Barry Mazur, Peter Sarnak, Richard Taylor, and Shou-Wu Zhang, and I sincerely thank them for generously sharing their ideas and knowledge. In particular, the argument in Paragraph 5.6 originates in an idea of R. Taylor in the case of classical modular curves, and the connection between injectivity radius and Lehmer's conjecture used in Section 8 was pointed out to me by P. Sarnak. Feedback from B. Mazur on the topic of the Manin constant was of great help, and I also thank Kestutis Cesnavicius and Bas Edixhoven for some additional observations on this subject. I learned some of the material relevant for this project by attending the joint IAS-Princeton algebraic number theory seminar during my stay at the IAS, and I thank Christopher Skinner and Richard Taylor for organizing it. Comments from Natalia Garcia-Fritz and Ricardo Menares regarding the presentation of this article, as well as the valuable comments and corrections of the referee, are gratefully acknowledged. I also thank Robert Lemke Oliver and Jesse Thorner for the appendix. Finally, I thank Enrico Bombieri and Natalia Garcia-Fritz for their encouragement, which allowed me to push this project further than I initially had in mind.