Classical forms of weight one in ordinary families

Stubley, Eric

Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 1, pp. 167-217

We develop a new strategy for studying low weight specializations of $p$-adic families of ordinary modular forms. In the elliptic case, we give a new proof of a result of Ghate–Vatsal which states that a Hida family contains infinitely many classical eigenforms of weight one if and only if it has complex multiplication. Our strategy is designed to explicitly avoid use of the related facts that the Galois representation attached to a classical weight one eigenform has finite image, and that classical weight one eigenforms satisfy the Ramanujan conjecture. We indicate how this strategy might be used to prove similar statements in the case of partial weight one Hilbert modular forms, given a suitable development of Hida theory in that setting.

Nous développons une nouvelle stratégie pour étudier les spécialisations en petits poids des familles $p$-adiques de formes modulaires ordinaires. Dans le cas elliptique, nous donnons une nouvelle preuve d’un résultat de Ghate–Vatsal qui énonce qu’une famille de Hida contient une infinité de formes propres classiques de poids un si et seulement si elle est à multiplication complexe. Notre stratégie est conçue afin d’éviter explicitement l’utilisation de certains faits connexes, à savoir que la représentation galoisienne attachée à une forme propre classique de poids un est d’image finie et que les formes propres classiques de poids un satisfont la conjecture de Ramanujan. Nous indiquons comment utiliser cette stratégie pour démontrer des énoncés similaires dans le cas de formes modulaires Hilbert de poids partiel un, en supposant un développement approprié de la théorie de Hida dans ce contexte.

Classification : 11F80, 11F85, 11F11
Keywords: Modular forms, modular forms of weight one, complex multiplication, ordinary families, Galois representations, $p$-adic power series

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