同余数、椭圆曲线和从局部到全局的过渡
Congruent numbers, elliptic curves, and the passage from the local to the global
摘要：
古代未解决的全等数问题已被简化为当代算术的主要问题之一：在$\bf Q$上的曲线的数量有限，这些曲线在每个地方都与给定的曲线同构。本文对全等数及其几何特征作了初步的介绍，讨论了导致有限性问题的局部到全局问题，并列举了椭圆曲线算术理论中的一些结果和定理。
英文摘要：
The ancient unsolved problem of congruent numbers has been reduced to one of the major questions of contemporary arithmetic: the finiteness of the number of curves over $\bf Q$ which become isomorphic at every place to a given curve. We give an elementary introduction to congruent numbers and their conjectural characterisation, discuss local-to-global issues leading to the finiteness problem, and list a few results and conjectures in the arithmetic theory of elliptic curves.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
