no. 1
Slopes of modular forms and congruences
Annales de l'Institut Fourier, Tome 46 (1996) no. 1, pp. 1-32.

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.

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.

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     title = {Slopes of modular forms and congruences},
     journal = {Annales de l'Institut Fourier},
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     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {46},
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     year = {1996},
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Ulmer, Douglas L. Slopes of modular forms and congruences. Annales de l'Institut Fourier, Tome 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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