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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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