Hida families, $p$-adic heights, and derivatives

[Familles de Hida, hauteurs $p$-adiques, et dérivées]

Arnold, Trevor

Annales de l'Institut Fourier, Tome 60 (2010), p. 2275-2299 / Harvested from Numdam

Résumé

Cet article concerne l’arithmétique de certaines familles $p$-adiques de formes modulaires elliptiques. En utilisant une formule de Rubin, on examine quelques aspects de la théorie d’Iwasawa pour les objets du titre, dont trois affirment la non-trivialité d’un système d’Euler, d’un régulateur $p$-adique, et de la dérivée d’une fonction $L$ $p$-adique. En particulier, on étudie des conditions suffisantes pour que la première conjecture soit vraie et on démontre que, sous des hypothèses supplémentaires, la première conjecture implique que les deux dernières conjectures sont équivalentes.

Abstract

This paper concerns the arithmetic of certain $p$-adic families of elliptic modular forms. We relate, using a formula of Rubin, some Iwasawa-theoretic aspects of the three items in the title of this paper. In particular, we examine several conjectures, three of which assert the non-triviality of an Euler system, a $p$-adic regulator, and the derivative of a $p$-adic $L$-function. We investigate sufficient conditions for the first conjecture to hold and show that, under additional assumptions, the first conjecture implies the equivalence of the last two.

Détail

Publié le : 2010-01-01

MR 2791658 | Zbl 1259.11099

DOI : https://doi.org/10.5802/aif.2584
Classification:  11R23,  11G40,  11F11,  11S25
Mots clés: théorie d’Iwasawa, famille de Hida, hauteur $p$-adique, fonction $L$ $p$-adique

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