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Overconvergent modular forms and the Fontaine-Mazur conjecture. (English) Zbl 1045.11029
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. J.-M. Fontaine and 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 R. Coleman and 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 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 B. Mazur and A. Wiles [Compos. Math. 59, 231–264 (1986; Zbl 0654.12008)], which says that representations attached to ordinary $$p$$-adic modular forms are ordinary.

##### MSC:
 11F33 Congruences for modular and $$p$$-adic modular forms 11F80 Galois representations 11F30 Fourier coefficients of automorphic forms
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