Tyler Genao : Uniform polynomial bounds on torsion from rational geometric isogeny classes

Posted by Frederick Saia , part of the Number Theory Seminar.

At

April 11, 2025, 1 p.m.

In

636 SEO

Abstract

In 1996, Merel showed that for any elliptic curve $E$ defined over a number field $F$ of degree $d\in\mathbb{Z}^+$, the size of the torsion group of $E$ over $F$ is bounded by a constant $B:=B(d)$ which depends only on $d$, and that conjecturally is in fact a polynomial in $d$. In this talk, I will discuss recent joint work with Abbey Bourdon which shows that $B$ is polynomial in $d$ for torsion from the family $\mathcal{I}_{\mathbb{Q}}$ of elliptic curves which are geometrically isogenous to at least one rational elliptic curve. For torsion from the subfamily $\mathcal{F}_{\mathbb{Q}}$ of elliptic curves with rational $j$-invariant, our results strengthen prior work of Clark and Pollack.