Minimal Torsion Curves in Geometric Isogeny Classes

Let $E/\mathbb{Q}$ be an elliptic curve. By Mordell's 1922 theorem, the points on $E$ with coordinates in $\mathbb{Q}$ form a finitely generated abelian group. In particular, the torsion subgroup of $E(\mathbb{Q})$ is a finite abelian group, and the groups that occur as $E(\mathbb{Q})_{\text{tors}}$ are known due to work of Mazur in 1977. The past decade has seen a renewed interest in studying torsion points on rational elliptic curves: from identifying the torsion subgroups that can arise on $E/\mathbb{Q}$ under base extension to a field of higher degree to a near complete classification of the image of $\ell$-adic Galois representations associated to elliptic curves over $\mathbb{Q}$. In this talk, we will discuss recent results which leverage this knowledge to begin to understand a new class of elliptic curves, namely, those geometrically isogenous to an elliptic curve defined over $\mathbb{Q}$. Motivated by the problem of producing low degree points on modular curves, we seek to characterize the elliptic curves within a fixed geometric isogeny class producing a point of prime-power order in least possible degree.

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