AGCT 2025 - Arithmetic, Geometry, Cryptography and Coding Theory / AGCT 2025 - Arithmétique, Géométrie, Cryptographie et Théorie des Codes

Collection AGCT 2025 - Arithmetic, Geometry, Cryptography and Coding Theory / AGCT 2025 - Arithmétique, Géométrie, Cryptographie et Théorie des Codes

Organizer(s) Aubry, Yves ; Pazuki, Fabien ; Salgado, Cecilia

Date(s) 09/06/2025 - 13/06/2025

linked URL https://conferences.cirm-math.fr/3343.html

00:00:00 / 00:00:00

6 6

Vertical coincidences of an elliptic curve defined over a number field

Let $E/F$ be an elliptic curve over a number field $F$. For a prime $p$, the extension $F(E[p^k])/F$ generated by the coordinates of the $p^k$-torsion points, is finite and Galois. We consider when the coincidence $F(E[p^k])=F(E[p^{k+1}])$ holds. Daniels and Lozano-Robledo classified such coincidences when $F=\mathbb{Q}$. In this talk, we will describe some results over a general number field $F$ and give additional possible coincidencesin this larger case.

Information about the video

Date of recording 09/06/2025
Date of publication 03/07/2025
Institution CIRM
Licence CC BY NC ND
Language English
Audience Researchers, Graduate Students
Director(s) Guillaume Hennenfent
Format MP4

Citation data

DOI 10.24350/CIRM.V.20363703
Cite this video Yvon, Zoé (09/06/2025). Vertical coincidences of an elliptic curve defined over a number field. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.20363703
URL https://dx.doi.org/10.24350/CIRM.V.20363703

Domain(s)

Number Theory

Bibliography

DANIELS, Harris B. et LOZANO-ROBLEDO, Álvaro. Coincidences of division fields. , Annales de l'Institut Fourier. 2023. p. 163-202. - https://doi.org/10.5802/aif.3520
GONZÁLEZ–JIMÉNEZ, Enrique et LOZANO-ROBLEDO, Álvaro. Elliptic curves with abelian division fields. Mathematische Zeitschrift, 2016, vol. 283, p. 835-859. - https://doi.org/10.1007/s00209-016-1623-z
ROUSE, Jeremy et ZUREICK-BROWN, David. Elliptic curves over ℚ and 2-adic images of Galois. Research in Number Theory, 2015, vol. 1, no 1, p. 12. - https://doi.org/10.1007/s40993-015-0013-7
SERRE, Jean-Pierre. Propriétés galoisiennes des points d'ordre ﬁni des courbes elliptiques. Invent. math, 1971, vol. 15, p. 259-331. - https://doi.org/10.1007/BF01405086

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