Superficies elípticas y el décimo problema de Hilbert

Pasten, Hector; Pasten, Hector

doi:10.15517/rmta.v30i1.52266

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Revista de Matemática Teoría y Aplicaciones

Print version ISSN 1409-2433

Rev. Mat vol.30 n.1 San José Jan./Jun. 2023

http://dx.doi.org/10.15517/rmta.v30i1.52266

Artículo

Superficies elípticas y el décimo problema de Hilbert

Elliptic surfaces and Hilbert’s tenth problem

Hector Pasten¹

^¹Pontificia Universidad Católica de Chile, Departamento de Matemáticas, Facultad de Matemáticas, Santiago, Chile; hector.pasten@mat.uc.cl

Resumen

Es sabido que se obtendría una solución negativa al decimo problema de Hilbert para el anillo de enteros OF de un campo de números F si Z fuera diofantino en OF . Denef y Lipshitz conjeturaron que esto ultimo ocurre para todo F. En esta nota se demuestra que la conjetura de Denef y Lipshitz es consecuencia de una conocida conjetura sobre superficies

elípticas.

Palabras clave: Decimo problema de Hilbert; anillos de enteros; superficies elípticas; curvas elípticas.

Abstract

A negative solution to Hilbert’s tenth problem for the ring of integers OF of a number field F would follow if Z were Diophantine in OF . Denef and Lipshitz conjectured that the latter occurs for every number field F. In this note we show that the conjecture of Denef and Lipshitz is a consequence of a well-known conjecture on elliptic surfaces.

Keywords: Hilbert’s tenth problem; rings of integers; elliptic surfaces; elliptic curves.

Mathematics Subject Classification: Primario: 11U05; Secundario: 14J27, 11G05.

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Agradecimientos

Agradezco a Barry Mazur por comentarios en una version previa, y a Cecilia Salgado por responder varias dudas. Además, agradezco profundamente los comentarios de los tres revisores anónimos.

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Recibido: 31 de Agosto de 2022; Revisado: 06 de Diciembre de 2022; Aprobado: 23 de Noviembre de 2022

Este es un artículo publicado en acceso abierto bajo una licencia Creative Commons