Slopes of modular forms and congruences

Ulmer, Douglas L.

Annales de l'Institut Fourier, Volume 46 (1996) no. 1, pp. 1-32.

Our aim in this paper is to prove congruences between on the one hand certain eigenforms of level $pN$ and weight greater than 2 and on the other hand twists of eigenforms of level $pN$ and weight 2. One knows a priori that such congruences exist; the novelty here is that we determine the character of the form of weight 2 and the twist in terms of the slope of the higher weight form, i.e., in terms of the valuation of its eigenvalue for $U_p$. Curiously, we also find a relation between the leading terms of the $p$-adic expansions of the eigenvalues for $U_p$ of the two forms. This allows us to determine the restriction to the decomposition group at $p$ of the Galois representation modulo $p$ attached to the higher weight form.

Le but de cet article est d’établir des congruences entre d’une part certaines formes modulaires paraboliques primitives de niveau $pN$ et de poids plus grand que 2 et d’autre part les formes modulaires de niveau $pN$ et de poids 2, tordues par une puissance de l’opérateur $\theta$. On sait a priori qu’il y a de telles congruences; la nouveauté ici est qu’on peut lire le caractère de la forme de poids 2 et la puissance de $\theta$ sur la pente de la forme de poids supérieur, i.e., sur la valuation de sa valeur propre pour l’opérateur $U_p$. Curieusement, on trouve aussi un lien entre les termes dominants des développements $p$-adiques des valeurs propres de $U_p$ sur les deux formes. À partir de ceci, on détermine la restriction à un sous-groupe de décomposition en $p$ de la représentation galoisienne attachée à la forme de poids plus grand que 2.

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