Explicit bounds on common projective torsion points of elliptic curves

TL;DR

This paper establishes explicit bounds on the number of common projective torsion points of elliptic curves over complex numbers, extending previous results with effective bounds under certain reduction conditions.

Contribution

It provides the first effective bounds on common torsion points of elliptic curves with specific reduction assumptions, generalizing Raynaud's methods to bad reduction cases.

Findings

Derived explicit bounds for common torsion points

Extended Raynaud's techniques to bad reduction scenarios

Connected bounds to reduction types of elliptic curves

Abstract

Suppose E_1, E_2 are elliptic curves (over the complex numbers) together with standard double coverings of the projective line identifying a point and its inverse on E_i. Bogomolov, Fu and Tschinkel have asked if the number of common images of torsion points on the elliptic curves under these double coverings is uniformly bounded in the case when the branch loci of the double coverings do not coincide, and recently this was answered affirmatively by various authors, but realistic effective bounds are unknown. In this article we obtain such bounds for common projective torsion points on elliptic curves under some mild extra assumptions on the reduction type of the input data at given primes. The method is based on Raynaud's original groundbreaking work on the Manin-Mumford conjecture. In particular, we generalise several of his results to cases of bad reduction using techniques from logarithmic algebraic geometry.