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On modular representations of \(\text{Gal}(\overline{\mathbb Q}/\mathbb Q)\) arising from modular forms. (English) Zbl 0773.11039
In this paper the author proves a conjecture of Serre on the level of an irreducible modular Galois representation \(\rho:\text{Gal}(\overline{\mathbb Q}/\mathbb Q)\to\text{GL}(2,\mathbb F)\), where \(\mathbb F\) is a finite field of odd characteristic \(\ell\). Followed by an idea of G. Frey [Ann. Univ. Sarav., Ser. Math. 1, 1–40 (1986; Zbl 0586.10010)], the main result of this paper has the remarkable application that the Taniyama-Shimura-Weil conjecture (i.e. every elliptic curve over \(\mathbb Q\) is modular) implies Fermat’s Last Theorem.
The representation \(\rho\) is said to be modular of level \(N\) if it arises from a weight-2 newform of level dividing \(N\) and trivial “Nebentypus character”. We say that \(\rho\) is “finite at \(p\)” if there is a finite flat \(\mathbb F\)-vector space scheme \(H\) over \(\mathbb Z_ p\) for which the action of \(\text{Gal}(\overline{\mathbb Q}_ p/\mathbb Q_ p)\) on the \(\mathbb F\)- vector space \(H(\overline{\mathbb Q}_ p)\) gives \(\rho_ p\), where \(\rho_ p\) is the restriction of \(\rho\) to the decomposition group \(\text{Gal}(\overline{\mathbb Q}_ p/\mathbb Q_ p)\) of \(\text{Gal}(\overline{\mathbb Q}/\mathbb Q)\). If \(\ell\neq p\), this means simply that \(\rho\) is unramified at \(p\). J.-P. Serre conjectured [Contemp. Math. 67, 263–268 (1987; Zbl 0629.14016)] that if \(\rho\) is modular of level \(N\) and if \(\rho\) is finite at a prime \(p\) which divides \(N\) exactly once, then \(\rho\) is also modular of level \(N/p\). Mazur proved this conjecture in the case of \(p\not\equiv 1\pmod\ell\). The main theorem of this paper asserts that Serre’s conjecture is true whenever \(N\) is not divisible by \(\ell\).
Besides Mazur’s techniques, the paper makes use of results of Néron models of Jacobians (due to Raynaud) and of the bad reduction of classical modular curves (Deligne–Rapoport) and Shimura curves (Cherednik–Drinfel’d). Particularly, the author developed a beautiful interchange principle – analogous to the Jacquet-Langlands correspondence – which compares certain data obtained from Shimura curves in characteristic \(p\) to corresponding data obtained from certain modular curves in characteristic \(q\neq p\).

11F80 Galois representations
11G18 Arithmetic aspects of modular and Shimura varieties
11G05 Elliptic curves over global fields
11S37 Langlands-Weil conjectures, nonabelian class field theory
14G35 Modular and Shimura varieties
Full Text: DOI EuDML
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