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Minimal torsion curves in geometric isogeny classes

By Abbey Bourdon

Appears in collection : COUNT - Computations and their Uses in Number Theory / Les calculs et leurs utilisations en théorie des nombres

Let $E / \mathbb{Q}$ be a non-CM elliptic curve and let $\mathcal{E}$ denote the collection of all elliptic curves geometrically isogenous to $E$. That is, for every $E^{\prime} \in \mathcal{E}$, there exists an isogeny $\varphi: E \rightarrow E^{\prime}$ defined over $\overline{\mathbb{Q}}$. Motivated by ties to Serre's Uniformity Conjecture, we will discuss the problem of identifying minimal torsion curves in $\mathcal{E}$, which are elliptic curves $E^{\prime} \in \mathcal{E}$ attaining a point of prime-power order in least possible degree. Using recent classification results of Rouse, Sutherland, and Zureick-Brown, we obtain an answer to this question in many cases, including a complete characterization for points of odd degree.

This is joint work with Nina Ryalls and Lori Watson.

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Citation data

  • DOI 10.24350/CIRM.V.20006603
  • Cite this video Bourdon Abbey (3/2/23). Minimal torsion curves in geometric isogeny classes. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.20006603
  • URL https://dx.doi.org/10.24350/CIRM.V.20006603


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