Congruent numbers, elliptic curves, and the passage from the local to the global: an update

TL;DR

This paper reviews recent advances in the arithmetic of elliptic curves, focusing on the congruent number problem and illustrating the transition from local to global properties in number theory.

Contribution

It provides an update on recent progress in understanding the relationship between congruent numbers and elliptic curves, emphasizing the local-global passage.

Findings

New results on the congruent number problem

Insights into the local-global principle for elliptic curves

Advances in the arithmetic of elliptic curves

Abstract

This update to my article on Congruent numbers, elliptic curves, and the passage from the local to the global, which appeared in Resonance, December 2009, pp. 1183--1205 (https://www.ias.ac.in/describe/article/reso/014/12/1183-1205) and was posted here as arXiv:0704.3783, covers a few recent advances in the arithmetic of elliptic curves with special reference to the congruent number problem.