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Non-optimal levels of \(\text{mod }l\) modular representations. (English) Zbl 0847.11025
The main result of the paper is Theorem B. Let \(\rho: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to \text{GL}(2, \overline{\mathbb{F}}_\ell)\) be an irreducible representation which is modular of weight \(k\geq 2\) and level \(M\) prime to \(\ell\). Suppose that \(r\) is square free and prime to \(M\ell\), and that \(\ell> k+ 1\). If \(p(\text{tr } \rho(\text{Frob}_p))^2= (1+ p)^2\text{ det } \rho(\text{Frob}_p)\) for all primes \(p\) dividing \(r\), then \(\rho\) arises from a newform of level divisible by \(r\) in \(S_k(\Gamma_1(M)\cap \Gamma_0(r))\).
The approach to this result follows K. Ribet’s method for raising the levels, developed for \(\Gamma_0(M)\), \(k= 2\), and \(r\) prime in [Invent. Math. 100, 431-476 (1990; Zbl 0773.11039)]. The proof proceeds by induction on the primes which divide the level. The main difficulty arises in the fact that introducing a new prime to the level, one may lose primes not dividing the conductor of \(\rho\). The inductive step is supplied by proving the analogous result for automorphic forms on quaternion algebras over \(\mathbb{Q}\), and using afterwards Jacquet-Langlands correspondence to return to modular forms.
The proof runs in different ways, depending on whether the involved quaternion algebras are definite or not. The indefinite case causes the greatest difficulties. It requires an analogue of a lemma of Ihara concerning the injectivity of a natural homomorphism \(J_1(M)^2\to J_1(M; p)\). The proof is based on the study of Galois actions on the \(\ell\)-adic cohomology of canonical models over \(\mathbb{Q}\) for the Shimura curves. Here some facts on crystalline cohomology become another key ingredient. They are due to G. Faltings and B. Jordan [Isr. J. Math. 90, 1-66 (1995)] and are used to identify a Jordan-Hölder constituent which, locally on \(\ell\), shows a step two filtration. In the situation considered by the authors it turns out that all errors are Eisenstein. This was also the case in Ribet’s paper.
The amount of the techniques involved in the present paper is very impressive. Several delicate points concerning bad reduction of Shimura curves and Jacquet and Langlands correspondence are briefly summarized. This is a valuable help for the readers. A refinement of the theorem has been provided by the same authors in [Duke Math. J. 74, 253-269 (1994; Zbl 0809.11025)]. There they show that it is possible to specify local liftings on the inertia groups at the primes \(p\) for all \(p\neq \ell\).

MSC:
11F33 Congruences for modular and \(p\)-adic modular forms
11G18 Arithmetic aspects of modular and Shimura varieties
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References:
[1] [AL] Atkin, A. Li, W.: Twists of newforms and pseudo-eigenvalues ofW-operators. Invent. Math.48, 221-243 (1978) · Zbl 0377.10017 · doi:10.1007/BF01390245
[2] [BJ] Borel, A. Jacquet, H.: Automorphic forms and automorphic representations. In: Borel, A., Casselman, W. (eds.) Automorphic forms, representations and L-functions. Proc. Symp. Pure Math. vol. 33, part 1, pp 189-202) Providence, RI: Am. Math. Soc. 1979 · Zbl 0414.22020
[3] [B] Boutot, J.-F.: Variétés de Shimura: Le problème de modules en inégale caractéristique. Publ. Math. Univ. Paris, VII6, 43-62 (1979)
[4] [BC] Boutot, J.-F., Carayol, H.: Uniformisation p-adiques des courbes de Shimura; les théorèmes de ?erednik et Drinfeld. Astérisque196-197, 45-159 (1991)
[5] [C] Carayol, H.: Sur les représentations Galoisiennes modulol attachées aux formes modulaires. Duke Math. J.59, 785-801 (1989) · Zbl 0703.11027 · doi:10.1215/S0012-7094-89-05937-1
[6] [De] Deligne, P. Travaux de Shimura. In: Sémin. Bourbaki exposi 389 (1970-1971). In: Lect. Notes Math., vol. 244, pp. 123-165. Berlin, Heidelberg New York Springer 1971
[7] [DR] Deligne, P., Rapoport, M.: Les schémas de modules de courbes elliptiques. In: Deligne, P., Kuyk, W. (eds.) Modular functions of one variable II. (Lect. Notes Math., vol. 349, pp. 143-316) Berlin Heidelberg New York: Springer 1973
[8] [Di] Diamond, F.: Congruence primes for cusp forms of weightk?2. Astérisque196-197 202-215 (1991)
[9] [F] Faltings, G.: Crystalline cohomology and p-adic Galois representations. In: Algebraic Analysis, Geometry and Number Theory. Proc. JAMI Inaugural Conference, pp. 25-79. Baltimore: Johns-Hopkins University Press 1989 · Zbl 0805.14008
[10] [FC] Faltings, G. Chai, C.-L.: Degeneration of abelian varieties. (Ergeb. Math. Grenzgeb. 3. Folge, Bd. 22) Berlin Heidelberg New York: Springer 1990 · Zbl 0744.14031
[11] [FJ] Faltings, G., Jordan, B.: Crystalline cohomology and GL(2, ?). (Preprint)
[12] [FL] Fontaine, J.-M., Lafaille, G.: Construction de représentations p-adiques. Ann. Sci. Ec. Norm. Supér.15, 547-608 (1982)
[13] [FM] Fontaine, J.-M., Messing, W.: p-adic periods and p-adic etale cohomology. Contemp. Math.67, 179-207 (1987)
[14] [G] Gelbart, S.: Automorphic forms on adele groups. (Ann. Math. Stud., 83) vol. Princeton: Princeton University Press 1975 · Zbl 0329.10018
[15] [H] Huppert, B.: Endliche Gruppen I. Grundlehren Math. Wiss., Bol. 134) Berlin Heidelberg New York: Springer 1983
[16] [Hi] Hida, H.: On p-adic Hecke algebras for GL2 over totally real fields. Ann. Math.128, 295-384 (1988) · Zbl 0658.10034 · doi:10.2307/1971444
[17] [JaLa] Jacquet, H., Langlands, R.: Automorphic forms on GL2 (Lect. Notes Math., vol. Berlin Heidelberg New York. Springer 1970 · Zbl 0236.12010
[18] [JoLi] Jordan, B., Livné, R.: Conjecture ?Epsilon? for weightk>2. Bull. Am. Math. Soc.21, 51-56 (1989) · Zbl 0675.10020 · doi:10.1090/S0273-0979-1989-15758-3
[19] [L] Livné, R.: On the conductors of moll representations coming from modular forms. Number Theory31, 133-141 (1989) · Zbl 0674.10024 · doi:10.1016/0022-314X(89)90015-2
[20] [M] Mumford, D.: Abelian varieties. Oxford: University Press 1970 · Zbl 0223.14022
[21] [R1] Ribet, K.: Congruence relations between modular forms. In: Ciesielski, Z., Olech, C. (eds.) Proc. Int. Cong. of Mathematicians 1983, pp. 503-514. Warsaw: PWN 1984
[22] [R2] Ribet, K.: On modular representations of 462-2 arising from modular forms. Invent. Math.100, 431-476 (1990) · Zbl 0773.11039 · doi:10.1007/BF01231195
[23] [R3] Ribet, K.: Report on modl representations of Gal(?/?) arising from modular forms. In: Proc. of the motives conference, Seattle 1991, (to appear)
[24] [S1] Serre, J-P.: Arbres, amalgames, SL2. Astérisque46 (1977)
[25] [S2] Serre, J-P.: Sur les représentations modulaires de degré 2 de 462-4. Duke Math. J.54, 179-230 (1987) · Zbl 0641.10026 · doi:10.1215/S0012-7094-87-05413-5
[26] [Sh] Shimura, G.: Introduction to the arithmetic theory of, automorphic functions. Publ. Math. Soc. Japan, vol. 11) Tokyo: Iwanami Shoten 1971 · Zbl 0221.10029
[27] [T] Taylor, R.: On Galois representations associated to Hilbert modular forms. Invent. Math.98, 265-280 (1989) · Zbl 0705.11031 · doi:10.1007/BF01388853
[28] [W] Wiles, A.: On ordinary ?-adic representations associated to modular forms. Invent. Math.94, 529-573 (1988) · Zbl 0664.10013 · doi:10.1007/BF01394275
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