Sobre los teoremas de Shafarevich y Siegel

TL;DR

The paper introduces a novel proof of Shafarevich's theorem on the finiteness of elliptic curves with good reduction outside a finite set of primes, providing new insights into diophantine finiteness theorems.

Contribution

It offers a new proof approach that avoids diophantine approximation and transcendence theory, aligning more with Faltings' ideas in Mordell's conjecture.

Findings

New proof of Shafarevich's theorem established.

Provides a new perspective on diophantine finiteness theorems.

Connects Shafarevich's theorem to Siegel's theorem on $S$-unit equations.

Abstract

Presentaremos una nueva demostraci\'on del teorema de Shafarevich sobre finitud de curvas el\'ipticas con buena reducci\'on fuera de un conjunto finito de primos dado. Esto da un nuevo punto de entrada a teoremas fundamentales de finitud diofantina tales como el teorema de Siegel sobre la ecuaci\'on $S$-unidad. Nuestro argumento est\'a libre de aproximaci\'on diofantina o teor\'ia de trascendencia, y se acerca m\'as a las ideas de Faltings en su demostraci\'on de la conjetura de Mordell. -- We present a new proof of Shafarevich's theorem on finiteness of elliptic curves with good reduction outside a given finite set of primes. This gives a new entry point to fundamental diophantine finiteness theorems such as Siegel's theorem on the $S$-unit equation. Our proof is free from diophantine approximation or transcendence theory, and it is closer to the ideas of Faltings in his proof of Mordell's conjecture .