Slopes of modular forms and congruences
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 θ. 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 θ 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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     title = {Slopes of modular forms and congruences},
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Ulmer, Douglas L. Slopes of modular forms and congruences. Annales de l'Institut Fourier, Volume 46 (1996) no. 1, pp. 1-32. doi : 10.5802/aif.1504. https://aif.centre-mersenne.org/articles/10.5802/aif.1504/

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