Overconvergent modular forms and the Fontaine-Mazur conjecture
Let \(\rho: G_{\mathbb{Q}}\to \text{GL}_m(\mathbb{Q}_p)\) be a continuous irreducible representation of \(G_{\mathbb{Q}}= \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\), which is unramified outside finitely many primes and whose restriction to a decomposition group at \(p\) is potentially semistable. \textit{J.-M. Fontaine} and \textit{B. Mazur} [Elliptic curves, modular forms, and Fermat's last theorem, Ser. Number Theory 1, 41--78 (1995; Zbl 0839.14011)] made the following conjectures: (i) \(\rho\) appears (up to a twist) as a subquotient in the cohomology of a finite type \(\mathbb{Q}\)-scheme (i.e. \(\rho\) is ``geometric''), (ii) Let \(E/\mathbb{Q}_p\) be a finite extension, and let \(\rho: G_{\mathbb{Q}}\to \text{GL}_2(E)\) be a continuous, odd, irreducible representation, unramified outside finitely many primes, whose restriction to a decomposition group at \(p\) is potentially semistable when regarded as a \(\mathbb{Q}_p\)-representation. Then, up to a twist, \(\rho\) arises from a modular form. The author proves that, apart from a certain exceptional case, the Fontaine-Mazur conjecture is true for representations coming from finite slope, overconvergent eigenforms of weight \(k\neq 1\) (Theorem 6.6). One of the key ingredients in the proof is the theory of the ``eigencurve'' of \textit{R. Coleman} and \textit{B. Mazur} [Galois representations in arithmetic algebraic geometry, Lond. Math. Soc. Lect. Note Ser. 254, 1--113 (1998; Zbl 0932.11030)]. As an applicaton of this theorem (more precisely, the technical result Theorem 6.3), he answers a question of \textit{F. Q. Gouvêa} [Arithmetic of \(p\)-adic modular forms, Lectures Notes in Mathematics 1304, Springer-Verlag, Berlin (1988; Zbl 0641.10024)], who asked to what extent the \(u_p\)-eigenvalue of an overconvergent modular form \(f\) is determined by its Fourier coefficients \(a_n(f)\) with \(pX_n\). As a second application of Theorem 6.3 he recovers the result of \textit{B. Mazur} and \textit{A. Wiles} [Compos. Math. 59, 231--264 (1986; Zbl 0654.12008)], which says that representations attached to ordinary \(p\)-adic modular forms are ordinary.
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