STNB2024(37th edition)
Coincidence of division fields of an elliptic curve
Presenters
Zoé Yvon
Abstract
Let $E/F$ be an elliptic curve over a number field $F$. For a positive integer $n$, the extension $F(E[n])/F$ generated by the coordinates of the $n$-torsion points, is finite and Galois. For $p$ a prime and $k\geq1$, we consider when the coincidence $F(E[p^k]) = F(E[p^{k+1}])$ holds. Daniels and Lozano-Robledo showed that, for $F = \mathbb{Q}$, the equality occurs only for $(p^k,p^{k+1})=(2,4)$. In this talk, we will describe similarly results over a general number field $F$.
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