Towards strong uniformity for isogenies of prime degree

TL;DR

This paper investigates uniform bounds on prime degree isogenies of elliptic curves over number fields, providing explicit divisibility conditions and extending uniform boundedness results for torsion points over unramified extensions.

Contribution

It establishes explicit divisibility conditions for prime degrees of isogenies under certain signatures, leading to a uniform bound on torsion points over unramified extensions.

Findings

Explicit divisibility conditions for prime degree isogenies

A uniform bound on torsion points over unramified extensions

Extension of Merel's uniform boundedness result

Abstract

Let $E$ be an elliptic curve over a number field $k$ of degree $d$ that admits a $k$-rational isogeny of prime degree $p$. We study the question of finding a uniform bound on $p$ that depends only on $d$, and obtain, under a certain condition on the signature of the isogeny, such a uniform bound by explicitly constructing nonzero integers that $p$ must divide. As a corollary we find a uniform bound on torsion points defined over unramified extensions of the base field, generalising Merel's Uniform Boundedness result for torsion.