Slopes of modular forms and congruences
Annales de l'Institut Fourier, Tome 46 (1996) no. 1, p. 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.
@article{AIF_1996__46_1_1_0,
     author = {Ulmer, Douglas L.},
     title = {Slopes of modular forms and congruences},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {46},
     number = {1},
     year = {1996},
     pages = {1-32},
     doi = {10.5802/aif.1504},
     zbl = {0834.11024},
     mrnumber = {97i:11046a},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1996__46_1_1_0}
}
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. http://www.numdam.org/item/AIF_1996__46_1_1_0/

[D] P. Deligne, Formes modulaires et représentations l-adiques, in: Séminaire Bourbaki 1968/1969 (Lect. Notes in Math. 179) 139-172, Berlin-Heidelberg-New York, Springer, 1969. | Numdam | Zbl 0206.49901

[DR] P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques, in : W. Kuyk and P. Deligne (Eds.) Modular Functions of One Variable II (Lect. Notes in Math. 349) 143-316, Berlin-Heidelberg-New York, Springer, 1973. | MR 49 #2762 | Zbl 0281.14010

[Di] F. Diamond, The refined conjecture of Serre, To appear in the proceedings of a conference on elliptic curves, Hong Kong, December 1993. | Zbl 0853.11031

[E] B. Edixhoven, The weight in Serre's conjectures on modular forms, Invent. Math., 109 (1992), 563-594. | MR 93h:11124 | Zbl 0777.11013

[GiMe] H. Gillet and W. Messing, Cycles classes and Riemann-Roch for crystalline cohomology, Duke Math. J., 55 (1987), 501-538. | MR 89c:14025 | Zbl 0651.14014

[G] B.H. Gross, A tameness criterion for Galois representations associated to modular forms (mod p), Duke Math. J., 61 (1990), 445-517. | MR 91i:11060 | Zbl 0743.11030

[I] L. Illusie, Finiteness, duality, and Künneth theorems in the cohomology of the deRham Witt complex, in : M. Raynaud and T. Shiota (eds.) Algebraic Geometry Tokyo-Kyoto (Lect. Notes in Math. 1016) 20-72, Berlin-Heidelberg-New York, Springer, 1982. | MR 85m:14033 | Zbl 0538.14013

[KM] N. Katz and B. Mazur, Arithmetic Moduli of Elliptic Curves, Princeton, Princeton University Press, 1985. | MR 86i:11024 | Zbl 0576.14026

[KMe] N. Katz and W. Messing, Some consequences of the Riemann hypothesis for varieties over finite fields, Invent. Math., 23 (1974), 73-77. | MR 48 #11117 | Zbl 0275.14011

[MW] B. Mazur and A. Wiles, Class fields of abelian extensions of Q, Invent. Math., 76 (1984), 179-330. | MR 85m:11069 | Zbl 0545.12005

[Ri] K. Ribet, Report on mod l representations of Gal(Q/Q), In : U. Jannsen, S. Kleiman, J.-P. Serre (eds.), Motives (Proceedings of Symposia in Pure Mathematics 55, part 2, 639-676, Providence, American Mathematical Society, 1994. | MR 95d:11056 | Zbl 0822.11034

[Sc] A. J. Scholl, Motives for modular forms, Invent. Math., 100 (1990), 419-430. | MR 91e:11054 | Zbl 0760.14002

[S] J.-P. Serre, Groupes Algébriques et Corps de Classes, Paris, Hermann, 1959. | MR 21 #1973 | Zbl 0097.35604

[Sh] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, Princeton University Press, 1971. | Zbl 0221.10029

[U1] D. L. Ulmer, L-functions of universal elliptic curves over Igusa curves, Amer. J. Math., 112 (1990), 687-712. | MR 91j:11050 | Zbl 0731.14013

[U2] D. L. Ulmer, On the Fourier coefficients of modular forms, Ann. Sci. Ec. Norm. Sup., 28 (1995), 129-160. | Numdam | MR 95k:11066 | Zbl 0827.11024

[U3] D. L. Ulmer, On the Fourier coefficients of modular forms II, Math. Annalen, 304 (1996), 363-422. | MR 96j:11062 | Zbl 0856.11022