Ogg's conjectures over function fields

TL;DR

This paper reviews the function field analogues of Ogg's conjectures on torsion points of elliptic curves, discussing their current status, proof methods, and potential for future generalizations.

Contribution

It provides a comprehensive overview of the status and methods related to the function field versions of Ogg's conjectures, highlighting differences from the number field case.

Findings

Some function field analogues have been proved using adapted methods.

Differences and technical challenges arise compared to the classical case.

Potential new directions for generalizations are identified.

Abstract

In the early 1970s, Andrew Ogg made several conjectures about the rational torsion points of elliptic curves over $\mathbb{Q}$ and the Jacobians of modular curves. These conjectures were proved shortly after by Barry Mazur as a consequence of his fundamental study of the arithmetic properties of modular curves and Hecke algebras. In this paper, we review the function field analogues of Ogg's conjectures, their current status, and the methods that have been applied to prove some of these conjectures. The methods are based on the ideas of Mazur and Ogg, but there are interesting differences and technical complications that arise in the function field setting, as well as intriguing possible new directions for generalizations.