MATH Seminar
| Title: Minimal and nilpotent images of Galois for elliptic curves |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Jeremy Rouse of Wake Forest University |
| Contact: Santiago Arango, santiago.arango@emory.edu |
| Date: 2025-02-25 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: If $K$ is a number field and $E/K$ an elliptic curve, then for every positive integer $n$, there is a Galois representation $\rho_{E,n} : G_{K} \to {\rm GL}_{2}(\mathbb{Z}/n\mathbb{Z})$. If $K = \mathbb{Q}$, $\det \circ \rho_{E,n} : G_{\mathbb{Q}} \to (\mathbb{Z}/n\mathbb{Z})^{\times}$ is surjective. We say that a subgroup $H$ of ${\rm GL}_{2}(\mathbb{Z}/n\mathbb{Z})$ is \emph{minimal} if $\det : H \to (\mathbb{Z}/n\mathbb{Z})^{\times}$ is surjective. We show that essentially the only way for the image of $\rho_{E,n}$ to be minimal is for $n$ to be a power of $2$, and that minimal subgroups of ${\rm GL}_{2}(\mathbb{Z}/2^{k} \mathbb{Z})$ are plentiful.\\ \\ The question of minimality is connected with the question of when the Galois group of $\mathbb{Q}(E[n])/\mathbb{Q}$ is a nilpotent group. In 2016, Lozano-Robledo and Gonz\'alez-Jim\'enez showed that if $E/\mathbb{Q}$ is an elliptic curve and ${\rm Gal}(\mathbb{Q}(E[n])/\mathbb{Q})$ is abelian, then $n \in \{ 2,3,4,6, 8\}$. We show that, subject to a positive answer to Serre's uniformity question, if $E/\mathbb{Q}$ is a non-CM elliptic curve and ${\rm Gal}(\mathbb{Q}(E[n])/\mathbb{Q})$ is nilpotent, then $n \in \{ 2^{k}, 3, 5, 6, 7, 15, 21 \}$.\\ \\ All of the work in this talk is joint with Harris Daniels. |
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