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arXiv · 2302.08350

Towards strong uniformity for isogenies of prime degree

Barinder S. Banwait·Maarten Derickx
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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.

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Barinder S. Banwait, Maarten Derickx. 2023-02-16. Towards strong uniformity for isogenies of prime degree. https://arxiv.org/abs/2302.08350

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