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On the conductors of mod $$\ell$$ Galois representations coming from modular forms. (English) Zbl 0674.10024
Let $$\rho$$ : Gal($${\bar {\mathbb{Q}}}/{\mathbb{Q}})\to GL_ 2({\mathbb{F}})$$ be a continuous odd, absolutely irreducible representation (with $${\mathbb{F}}^ a$$finite field of characteristic $$\ell)$$ which arises as the reduction mod $$\ell$$ of the $$\ell$$-adic representation attached to a newform g. Let N($$\rho)$$ be the conductor of $$\rho$$, and N(g) the level of g. The author proves that there is a Dirichlet character $$\chi_ 0$$ of $$\ell$$-power order, and of conductor dividing the product of primes $$p\neq \ell$$ that divided N(g), such that if $$\chi$$ is any other such Dirichlet character then $$N(g\otimes \chi_ 0)$$ divides N(g$$\otimes \chi)$$ and if $$p^ s | | N(g\otimes \chi_ 0)$$ for a prime $$p\neq \ell$$ then $$p^ s | | N(\rho)$$ unless $$s=1$$ or 2. In these cases the author determines the nature of the local component $$\pi_ p$$ of the automorphic representation $$\pi$$ of $$GL_ 2({\mathbb{A}})$$ associated to g.
As the author points out, his result says that the problem of reducing the level at a prime $$p\neq \ell$$ in the Serre conjecture [see J.-P. Serre, Duke Math. J. 54, 179-230 (1987; Zbl 0641.10026)] reduces to a purely local problem except in the special cases cited above.
Reviewer: S.Kamienny

##### MSC:
 11F80 Galois representations 11F11 Holomorphic modular forms of integral weight 22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
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