Alexander Barrios, University of St. Thomas
Authors: Elise Alvarez-Salazar, Alexander Barrios, Calvin Henaku, Summer Soller
2023 AWM Research Symposium
Number Theory at Primarily Undergraduate Institutions [Organized by Bella Tobin and Leah Sturman]

A rational elliptic curve $E$ is said to be good if $N_{E}^{6}<\max\{|c_{4}^{3}|,c_{6}^{2}\}$, where $N_{E}$ is the conductor of $E$ and $c_{4}$ and $c_{6}$ are the invariants associated to a global minimal model of $E$. It is known that for each of the fifteen torsion subgroups $T$ occurring over $\mathbb{Q}$, there are infinitely many good elliptic curves $E$ with $E(\mathbb{Q})_{\text{tors}}\cong T$. In this talk, we consider a generalization of this result by constructing infinitely many isogeny classes of elliptic curves with specified isogeny class degree such that each elliptic curve in the isogeny class is good. This construction is done by first considering sequences of good $abc$ triples and using these to construct parameterized families of isogenous elliptic curves. The study of $abc$ triples is motivated by the $abc$ conjecture, which will be discussed at the start of the talk to motivate the main results. This work began as part of PRiME (Pomona Research in Mathematics Experience), funded by the National Science Foundation (DMS-2113782).

Back to Search Research Symposium Abstracts