Slopes of modular forms and congruences

Ulmer, Douglas L.

Annales de l'Institut Fourier, Tome 46 (1996), p. 1-32 / Harvested from Numdam

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Abstract

Le but de cet article est d’établir des congruences entre d’une part certaines formes modulaires paraboliques primitives de niveau $p N$ et de poids plus grand que 2 et d’autre part les formes modulaires de niveau $p N$ 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 $p N$ and weight greater than 2 and on the other hand twists of eigenforms of level $p N$ 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.

Publié le : 1996-01-01

MR 97i:11046a | Zbl 0834.11024

DOI : https://doi.org/10.5802/aif.1504

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     author = {Ulmer, Douglas L.},
     title = {Slopes of modular forms and congruences},
     journal = {Annales de l'Institut Fourier},
     volume = {46},
     year = {1996},
     pages = {1-32},
     doi = {10.5802/aif.1504},
     mrnumber = {97i:11046a},
     zbl = {0834.11024},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1996__46_1_1_0}
}

Ulmer, Douglas L. Slopes of modular forms and congruences. Annales de l'Institut Fourier, Tome 46 (1996) pp. 1-32. doi : 10.5802/aif.1504. http://gdmltest.u-ga.fr/item/AIF_1996__46_1_1_0/

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