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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:
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