A bound for the "torsion conductor" of a non-CM elliptic curve

TL;DR

This paper establishes a uniform upper bound for the torsion conductor of semi-stable non-CM elliptic curves over Q, relating it to the curve's conductor with a fifth power bound.

Contribution

It provides a new uniform bound on the torsion conductor for semi-stable non-CM elliptic curves over Q, linking it to the curve's conductor.

Findings

The torsion conductor m_E is bounded above by a constant times the fifth power of the conductor.

The bound applies uniformly to all semi-stable non-CM elliptic curves over Q.

The result advances understanding of Galois representations associated with elliptic curves.

Abstract

Given a non-CM elliptic curve E over Q, define the ``torsion conductor'' m_E to be the smallest positive integer so that the Galois representation on the torsion of E has image Pi^{-1}(Gal(Q(E[m_E])/Q), where Pi denotes the natural projection GL_2(\hat{Z}) onto GL_2(Z/m_E Z). We show that, uniformly for semi-stable non-CM elliptic curves E over Q, m_E is less than a constant times the 5th power of the conductor of E.