Ogg 挠点猜想：五十年后
Ogg's Torsion conjecture: Fifty years later
AI总结：
本文综述了 Ogg 挠点猜想及其在过去五十年中推动模曲线有理点研究发展的重要影响与后续成果。
摘要：
Andrew Ogg 的数学观点激发了一系列日益广泛的成果与猜想。他的成果与猜想为本领域指明了富有成果的转折点，他的影响对我们所有人而言都是一份珍贵的礼物。Ogg 著名的挠点猜想——就模曲线而言——可以转述为：模曲线上参数化椭圆曲线挠点的有理点存在，当且仅当存在良好的几何理由使其存在。我们综述了 Ogg 挠点猜想以及过去五十年间我们对模曲线上有理点理解的后续发展。
英文摘要：
Andrew Ogg's mathematical viewpoint has inspired an increasingly broad array of results and conjectures. His results and conjectures have earmarked fruitful turning points in our subject, and his influence has been such a gift to all of us. Ogg's celebrated Torsion Conjecture -- as it relates to modular curves -- can be paraphrased as saying that rational points (on the modular curves that parametrize torsion points on elliptic curves) exist if and only if there is a good geometric reason for them to exist. We give a survey of Ogg's Torsion Conjecture and the subsequent developments in our understanding of rational points on modular curves over the last fifty years.
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