Ogg's Torsion conjecture: Fifty years later
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.
Keywords
Cite
@article{arxiv.2307.04752,
title = {Ogg's Torsion conjecture: Fifty years later},
author = {Jennifer S. Balakrishnan and Barry Mazur},
journal= {arXiv preprint arXiv:2307.04752},
year = {2024}
}
Comments
This text is an expanded version of a 45-minute lecture that B.M. gave at the IAS, on the occasion of "Talks Celebrating the Ogg Professorship in Mathematics - October 13, 2022." Appendix A, by Netan Dogra, gives a complete characterization of all quadratic points on Bring's curve