Slopes of modular forms and congruences

Ulmer, Douglas L.

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 $\theta$. 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.

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.

[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

[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 | Zbl

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

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

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

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

[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 | Zbl

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

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

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

[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 | Zbl

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

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

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

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

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

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