A geometric view on Iwasawa theory
Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 3.1, pp. 703-731.

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:
Accepted:
Published online:
DOI: 10.5802/jtnb.1176
Classification: 11F33, 11R23, 11F80
Keywords: Hida family, weight one modular form, eigencurve, $p$-adic $L$-function
Betina, Adel 1; Dimitrov, Mladen 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
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Betina, Adel; Dimitrov, Mladen. 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. http://archive.numdam.org/articles/10.5802/jtnb.1176/

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