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Explore advanced research in algebraic number theory through this 29-minute conference talk examining vertical coincidences of elliptic curves defined over number fields. Delve into the mathematical analysis of when the extension F(E[p^k])/F generated by coordinates of p^k-torsion points equals F(E[p^{k+1}]) for an elliptic curve E/F over a number field F. Learn about the classification work by Daniels and Lozano-Robledo for the case when F=ℚ, then discover new results extending this analysis to general number fields F and additional possible coincidences that emerge in this broader mathematical context. Gain insights into the intersection of elliptic curve theory, Galois theory, and arithmetic geometry as presented during the thematic meeting on "Arithmetic, Geometry, Cryptography and Coding Theory" at the Centre International de Rencontres Mathématiques in Marseille, France.
