On the number of elliptic curves with prescribed isogeny or torsion group over number fields of prime degree

TL;DR

This paper investigates the quantity of elliptic curves over prime degree number fields that possess specific torsion subgroups or isogenies, providing counts and classifications for these curves.

Contribution

It offers new counts and classifications of elliptic curves with prescribed torsion or isogeny properties over number fields of prime degree.

Findings

Quantifies the number of such elliptic curves for given properties.

Provides classification results for elliptic curves with prescribed torsion or isogeny.

Establishes bounds or exact counts in certain cases.

Abstract

Let $p$ be a prime and $K$ a number field of degree $p$. We count the number of elliptic curves, up to $\bar{K}$-isomorphism, having a prescribed property, where this property is either that the curve contains a fixed torsion group as a subgroup, or that it has an isogeny of prescribed degree.