AGCT - Arithmetic, Geometry, Cryptography and Coding Theory / AGCT - Arithmétique, géométrie, cryptographie et théorie des codes 2023

Collection AGCT - Arithmetic, Geometry, Cryptography and Coding Theory / AGCT - Arithmétique, géométrie, cryptographie et théorie des codes 2023

Organizer(s) Anni, Samuele ; Bruin, Nils ; Kohel, David ; Martindale, Chloe

Date(s) 05/06/2023 - 09/06/2023

linked URL https://conferences.cirm-math.fr/2889.html

00:00:00 / 00:00:00

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Counting S4 and S5 extensions satisfying the Hasse norm principle

By Rachel Newton

Let $L/K$ be an extension of number fields. The norm map $N_{L/K} :L^{²}\to K^{²}$ extends to a norm map from the ideles of L to those of $K$. The Hasse norm principle is said to hold for $L/K$ if, for elements of $K^{²}$, being in the image of the idelic norm map is equivalent to being the norm of an element of L^{²}. The frequency of failure of the Hasse norm principle in families of abelian extensions is fairly well understood, thanks to previous work of Christopher Frei, Daniel Loughran and myself, as well as recent work of Peter Koymans and Nick Rome. In this talk, I will focus on the non-abelian setting and discuss joint work with Ila Varma on the statistics of the Hasse norm principle in field extensions with normal closure having Galois group $S_{4}$ or $S_{5}$.

Information about the video

Date of recording 05/06/2023
Date of publication 28/06/2023
Institution CIRM
Licence CC BY NC ND
Language English
Audience Researchers, Graduate Students
Director(s) Jean Petit
Format MP4

Citation data

DOI 10.24350/CIRM.V.20054603
Cite this video Newton, Rachel (05/06/2023). Counting S4 and S5 extensions satisfying the Hasse norm principle. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.20054603
URL https://dx.doi.org/10.24350/CIRM.V.20054603

Domain(s)

Number Theory

Bibliography

FREI, Christopher, LOUGHRAN, Daniel, et NEWTON, Rachel. The Hasse norm principle for abelian extensions. American Journal of Mathematics, 2018, vol. 140, no 6, p. 1639-1685. - https://doi.org/10.1353/ajm.2018.0048
FREI, Christopher, LOUGHRAN, Daniel, et NEWTON, Rachel. Number fields with prescribed norms. Commentarii Mathematici Helvetici, 2022, vol. 97, no 1. - https://doi.org/10.4171/cmh/528
KOYMANS, Peter et ROME, Nick. A note on the Hasse norm principle. arXiv preprint arXiv:2301.10136, 2023. - https://doi.org/10.48550/arXiv.2301.10136
KOYMANS, Peter et ROME, Nick. Weak approximation on the norm one torus. arXiv preprint arXiv:2211.05911, 2022. - https://doi.org/10.48550/arXiv.2211.05911
MACEDO, André et NEWTON, Rachel. Explicit methods for the Hasse norm principle and applications to An and Sn extensions. In : Mathematical Proceedings of the Cambridge Philosophical Society. Cambridge University Press, 2022. p. 489-529. - https://doi.org/10.1017/S0305004121000268
ROME, Nick. The Hasse norm principle for biquadratic extensions. Journal de théorie des nombres de Bordeaux, 2018, vol. 30, no 3, p. 947-964. - https://doi.org/10.5802/jtnb.1058
NEWTON, Rachel et VARMA, Ila. Counting $S_4$ and $S_5$ extensions satisfying the Hasse norm principle. in Progress. -

MSC codes

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