Chromatic Selmer groups and arithmetic invariants of elliptic curves

Florian Ito Sprung

doi:10.5802/jtnb.1190

Florian Ito Sprung ¹

¹ School of Mathematical and Statistical Sciences Arizona State University Tempe, AZ 85287-1804, USA

Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 3.2, pp. 1103-1114.

Abstract
Résumé

Chromatic Selmer groups are modified Selmer groups with local information for supersingular primes $p$ . We sketch their role in establishing the $p$ -primary part of the Birch–Swinnerton-Dyer formula in Sections 2–5, and then study the growth of the Mordell–Weil rank along the $ℤ_{p}^{2}$ -extension of a quadratic imaginary number field in which $p$ splits in Section 6.

Les groupes de Selmer chromatiques sont des modifications des groupes de Selmer, qui contiennent des informations locales pour les nombres premiers $p$ supersinguliers. Dans les sections 2–5, on esquisse leur rôle dans la démonstration de la partie $p$ -primaire de la formule de Birch et Swinnerton-Dyer, et ensuite, dans la section 6, on étudie la croissance du rang de Mordell–Weil le long de la $ℤ_{p}^{2}$ -extension d’un corps quadratique imaginaire dans lequel $p$ est décomposé.

Received: 2019-12-12
Revised: 2020-06-09
Accepted: 2020-07-28
Published online: 2022-01-26

DOI: 10.5802/jtnb.1190

Classification: 11G40, 11R23, 14H52
Keywords: Elliptic curves, Selmer group, Mordell–Weil rank

Author's affiliations:

Florian Ito Sprung ¹

¹ School of Mathematical and Statistical Sciences Arizona State University Tempe, AZ 85287-1804, USA

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Florian Ito Sprung},
     title = {Chromatic {Selmer} groups and arithmetic invariants of elliptic curves},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {1103--1114},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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     doi = {10.5802/jtnb.1190},
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Florian Ito Sprung. Chromatic Selmer groups and arithmetic invariants of elliptic curves. Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 3.2, pp. 1103-1114. doi : 10.5802/jtnb.1190. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1190/

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