Superficies el\'ipticas y el d\'ecimo problema de Hilbert

TL;DR

This paper explores the connection between Hilbert's tenth problem for number fields and conjectures on elliptic surfaces, showing that a well-known elliptic surface conjecture implies Denef and Lipshitz's conjecture.

Contribution

It demonstrates that Denef and Lipshitz's conjecture on Diophantine definability in number fields follows from a well-known conjecture on elliptic surfaces.

Findings

The conjecture of Denef and Lipshitz is a consequence of a known conjecture on elliptic surfaces.

A negative solution to Hilbert's tenth problem would follow if $bZ$ were Diophantine in $O_F$.

The paper establishes a link between number theory and algebraic geometry through elliptic surface conjectures.

Abstract

A negative solution to Hilbert's tenth problem for the ring of integers $O_F$ of a number field $F$ would follow if $\mathbb{Z}$ were Diophantine in $O_F$. 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. -- Es sabido que se obtendr\'ia una soluci\'on negativa al d\'ecimo problema de Hilbert para el anillo de enteros $O_F$ de un campo de n\'umeros $F$ si $\mathbb{Z}$ fuera diofantino en $O_F$. 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 el\'ipticas.