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Slopes of modular forms and congruences. (English) Zbl 0834.11024

Ann. Inst. Fourier 46, No. 1, 1-32 (1996); addendum ibid. 46, No. 5, 1519 (1996).
Summary: 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.

MSC:

11F33 Congruences for modular and \(p\)-adic modular forms
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References:

[1] [D] , Formes modulaires et représentations l-adiques, in: Séminaire Bourbaki 1968/1969 (Lect. Notes in Math. 179) 139-172, Berlin-Heidelberg-New York, Springer, 1969. · Zbl 0206.49901
[2] [DR] and , 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. · Zbl 0281.14010
[3] [Di] , The refined conjecture of Serre, To appear in the proceedings of a conference on elliptic curves, Hong Kong, December 1993. · Zbl 0853.11031
[4] [E] , The weight in Serre’s conjectures on modular forms, Invent. Math., 109 (1992), 563-594. · Zbl 0777.11013
[5] [GiMe] and , Cycles classes and Riemann-Roch for crystalline cohomology, Duke Math. J., 55 (1987), 501-538. · Zbl 0651.14014
[6] [G] , A tameness criterion for Galois representations associated to modular forms (mod p), Duke Math. J., 61 (1990), 445-517. · Zbl 0743.11030
[7] [I] , 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. · Zbl 0538.14013
[8] [KM] and , Arithmetic Moduli of Elliptic Curves, Princeton, Princeton University Press, 1985. · Zbl 0576.14026
[9] [KMe] and , Some consequences of the Riemann hypothesis for varieties over finite fields, Invent. Math., 23 (1974), 73-77. · Zbl 0275.14011
[10] [MW] and , Class fields of abelian extensions of Q, Invent. Math., 76 (1984), 179-330. · Zbl 0545.12005
[11] [Ri] , 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. · Zbl 0822.11034
[12] [Sc] , Motives for modular forms, Invent. Math., 100 (1990), 419-430. · Zbl 0760.14002
[13] [S] , Groupes Algébriques et Corps de Classes, Paris, Hermann, 1959. · Zbl 0097.35604
[14] [Sh] , Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, Princeton University Press, 1971. · Zbl 0221.10029
[15] [U1] , L-functions of universal elliptic curves over Igusa curves, Amer. J. Math., 112 (1990), 687-712. · Zbl 0731.14013
[16] [U2] , On the Fourier coefficients of modular forms, Ann. Sci. Ec. Norm. Sup., 28 (1995), 129-160. · Zbl 0827.11024
[17] [U3] , On the Fourier coefficients of modular forms II, Math. Annalen, 304 (1996), 363-422. · Zbl 0856.11022
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