Ogg's Torsion conjecture: Fifty years later

2023/07/10 by Jennifer S. Balakrishnan, Barry Mazur, Balakrishnan, Jennifer S. +1 · 2 citations
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #History and Theory of Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2307.04752

openalex publication_date 2023/07/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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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Related

• Ogg's conjectures over function fields
• Torsion points on modular curves and Galois theory
• Proof of Serge Lang's Heights Conjecture and an almost optimal Bound for the Torsion of Elliptic Curves
• Torsion subgroups of elliptic curves over quadratic fields and a conjecture of Granville
• Galois theory and torsion points on curves