A geometric view on Iwasawa theory

Adel Betina; Mladen Dimitrov

doi:10.5802/jtnb.1176

Adel Betina ¹; Mladen Dimitrov ²

¹ University of Vienna, Faculty of Mathematics Oskar-Morgenstern-Platz 1 1090 Wien, Austria
² University of Lille, CNRS, UMR 8524 – Laboratoire Paul Painlevé 59000 Lille, France

Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 3.1, pp. 703-731.

Abstract
Résumé

This article extends our study of the geometry of the $p$ -adic eigencurve at a point defined by a weight $1$ cuspform $f$ irregular at $p$ and having complex multiplication, and the implications in Iwasawa and in Hida theories. The novel results include the determination of the Fourier coefficients of certain non-classical $p$ -adic modular forms belonging to the generalized eigenspace of $f$ , in terms of $p$ -adic logarithms of algebraic numbers. We also compute the “mysterious” cross-ratios of the $p$ -ordinary filtrations of the Hida families containing $f$ .

Cet article prolonge notre étude de la géométrie de la courbe $p$ -adique de Hecke en un point défini par une forme modulaire cuspidale $f$ de poids $1$ à multiplication complexe et irrégulière en $p$ , et des implications en théories d’Iwasawa et de Hida. Les nouveaux résultats incluent la détermination des coefficients de Fourier de certaines formes modulaires $p$ -adiques non-classiques appartenant à l’espace propre généralisé de $f$ , en termes de logarithmes $p$ -adiques de nombres algébriques. Nous calculons aussi le « mystérieux » bi-rapport des filtrations $p$ -ordinaires des familles de Hida contenant $f$ .

Received: 2020-03-28
Accepted: 2020-11-04
Published online: 2022-01-26

DOI: 10.5802/jtnb.1176

Classification: 11F33, 11R23, 11F80
Keywords: Hida family, weight one modular form, eigencurve, $p$-adic $L$-function

Author's affiliations:

Adel Betina ¹; Mladen Dimitrov ²

¹ University of Vienna, Faculty of Mathematics Oskar-Morgenstern-Platz 1 1090 Wien, Austria
² University of Lille, CNRS, UMR 8524 – Laboratoire Paul Painlevé 59000 Lille, France

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Adel Betina and Mladen Dimitrov},
     title = {A geometric view on {Iwasawa} theory},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {703--731},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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Adel Betina; Mladen Dimitrov. A geometric view on Iwasawa theory. Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 3.1, pp. 703-731. doi : 10.5802/jtnb.1176. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1176/

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