J.-M. Fontaine and B. Mazur, Geometric Galois representations, in Elliptic Curves, Modular Forms, and Fermat 's Last Theorem, International Press, Cambridge, 1995, 41--78.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a prime # #= 2, and consider an odd two dimensional Galois representation # : GK where E is either a finite field of characteristic # or a finite extension of Q # . Assume that the restrictions of # to the inertia groups at the primes of K above # are potentially semistable in the sense of [FM]. The representation # is called modular if it is associated to a Hilbert modular form on GL 2 (K) as is explained, for example, in [W1] and [W2] Fontaine and Mazur [FM] conjectured that this is always the case. Significant progress on this conjecture was achieved [W3] by proving particular ....

.... Assume that the restrictions of # to the inertia groups at the primes of K above # are potentially semistable in the sense of [FM] The representation # is called modular if it is associated to a Hilbert modular form on GL 2 (K) as is explained, for example, in [W1] and [W2] Fontaine and Mazur [FM] conjectured that this is always the case. Significant progress on this conjecture was achieved [W3] by proving particular instances of the following lifting conjecture : Conjecture 1.1. Suppose that # is odd and that the residual representation # attached to # is modular. Then # itself is ....

J.-M. Fontaine and B. Mazur, Geometric Galois representations, in Elliptic Curves, Modular Forms, and Fermat 's Last Theorem, International Press, Cambridge, 1995, 41--78.

....relating the size of a certain module measuring congruences between modular forms to that of a certain Galois cohomology group. This was carried out in [Wi] and [T W] in the context of modular forms of weight 2, where it was used to prove results in the direction of the Fontaine Mazur conjecture [F M]. While it was no surprise that the method could be generalized to higher weight modular forms and that the resulting formula would be related to the Bloch Kato conjecture, there remained many technical details to verify in order to accomplish this. In particular, the very formulation of the ....

....of Ribet, Wiles and others ( Ri] D T] Wi] T W] Di1] we verify that these hypotheses are satisfied by the set # of liftings of # associated to newforms of weight k, level prime to # and character #. We thus obtain the following result in the direction of the Fontaine Mazur conjecture [F M]: Theorem 1. Suppose # : GQ # AutK# V # is continuous, two dimensional, unramified outside a finite set of primes and has crystalline restriction to GQ # of Hodge Tate type (0, d) with 0 d # 1. If # is modular and has absolutely irreducible restriction to GL , then # is modular. We ....

J.-M. Fontaine and B. Mazur, Geometric Galois representations, in: Elliptic curves, modular forms and Fermat's last theorem, International Press, 1995, pp. 41--78.

....ned above) if and only if for some , E; arises from an eigenform. In fact the Shimura Taniyama conjecture can be generalized to a conjecture that every adic representation, satisfying suitable local conditions, arises from a modular form. Such a conjecture was proposed by Fontaine and Mazur [FM]. Galois groups and modular forms Viewed in this way, the Shimura Taniyama conjecture becomes part of a much larger picture: the emerging, partly conjectural and partly proven correspondence between certain modular forms and two dimensional representations of G . This correspondence, which ....

....slightly the notion of adic representation 11 that was introduced before) Given a holomorphic newform f one can attach to f a system of adic representations, following Eichler, Shimura, Deligne and Serre. These adic representations are called modular. The Fontaine Mazur conjecture (see [FM]) predicts if is an odd, irreducible, adic representation whose restriction to the decomposition group at is well enough behaved, then is modular. The restriction on the behaviour of the representation on the decomposition group at is essential in this conjecture; it is not true that ....

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J.-M. Fontaine and B. Mazur, Geometric Galois representations, in

....prime 6= 2, and consider an odd two dimensional Galois representation : GK GL 2 (E) where E is either a nite eld of characteristic or a nite extension of . Assume that the restrictions of to the inertia groups at the primes of K above are potentially semistable in the sense of [FM]. The representation is called modular if it is associated to a Hilbert modular form on GL 2 (K) as is explained, for example, in [W1] and [W2] Fontaine and Mazur [FM] conjectured that this is always the case. Signi cant progress on this conjecture was achieved [W3] by proving particular ....

.... Assume that the restrictions of to the inertia groups at the primes of K above are potentially semistable in the sense of [FM] The representation is called modular if it is associated to a Hilbert modular form on GL 2 (K) as is explained, for example, in [W1] and [W2] Fontaine and Mazur [FM] conjectured that this is always the case. Signi cant progress on this conjecture was achieved [W3] by proving particular instances of the following lifting conjecture : Conjecture 1.1 Suppose that is odd and that the residual representation attached to is modular. Then itself is ....

J.-M. Fontaine, B. Mazur, Geometric Galois representations, in Elliptic curves, modular forms, and Fermat's Last theorem, International Press, Cambridge, 1995, 41-78.

....relating the size of a certain module measuring congruences between modular forms to that of a certain Galois cohomology group. This was carried out in [Wi] and [T W] in the context of modular forms of weight 2, where it was used to prove results in the direction of the Fontaine Mazur conjecture [F M]. While it was no surprise that the method could be generalized to higher weight modular forms and that the resulting formula would be related to the Bloch Kato conjecture, there remained many technical details to verify in order to accomplish this. In particular, the very formulation of the ....

....Here # # f , defined in 6.4, is a generalization of the congruence ideal of Hida and Wiles; it can also be viewed as measuring the failure of the pairing on M f to be perfect on # f . Another consequence of Theorem 0. 2, is the following result in the direction of Fontaine Mazur conjecture [F M]. Theorem 0.3. Theorem 7.12) Suppose # : G# # GL 2 (K # ) is a continuous geometric representation whose restriction to G # is ramified and crystalline and its associated Dieudonne module has filtration length less than # 1. If its residual representation is modular and absolutely ....

[Article contains additional citation context not shown here]

J.-M. Fontaine and B. Mazur, Geometric Galois representations, in: Elliptic curves, modular forms and Fermat's last theorem, International Press, 1995, pp. 41--78.

....#= 2, and consider an odd two dimensional Galois representation # : GK # GL 2 (E) where E is either a finite field of characteristic # or a finite extension of Q # . Assume that the restrictions of # to the inertia groups at the primes of K above # are potentially semistable in the sense of [FM]. The representation # is called modular if it is associated to a Hilbert modular form on GL 2 (K) as is explained, for example, in [W1] and [W2] Fontaine and Mazur [FM] conjectured that this is always the case. Significant progress on this conjecture was achieved [W3] by proving particular ....

.... Assume that the restrictions of # to the inertia groups at the primes of K above # are potentially semistable in the sense of [FM] The representation # is called modular if it is associated to a Hilbert modular form on GL 2 (K) as is explained, for example, in [W1] and [W2] Fontaine and Mazur [FM] conjectured that this is always the case. Significant progress on this conjecture was achieved [W3] by proving particular instances of the following lifting conjecture : Conjecture 1.1. Suppose that # is odd and that the residual representation # attached to # is modular. Then # itself is ....

J.-M. Fontaine and B. Mazur, Geometric Galois representations, in Elliptic Curves, Modular Forms, and Fermat 's Last Theorem, International Press, Cambridge, 1995, 41--78.

.... it can be seen as partial confirmation of a conjecture of Fontaine and Mazur which asserts that if # : GQ # GL n (Q p ) is a continuous irreducible representation ramified at only finitely many primes and such that the image of the inertia group at p is finite, then the image of # is finite (see [FM]) Our theorem verifies this conjecture in the case that n = 2, p # 5 and the reduction (# mod #) is modular, irreducible and takes Frob p to an element with distinct eigenvalues. In our opinion the most serious assumption here is that (# mod #) should be modular, but we remind the reader that ....

J.-M. Fontaine and B. Mazur, Geometric Galois representations, in Elliptic Curves, Modular Forms and Fermat 's Last Theorem (J.Coates and S.-T.Yau eds.), International Press, Cambridge, MA, 1995.

....the fact that ZA3 is rigid (Lemma 3. 7) means that H 3 (ZA3 ; Q ) admits a two dimensional Galois representation; Fontaine and Mazur conjecture (roughly speaking) that all irreducible odd 2 dimensional Galois representations coming from geometry should be modular, up to a Tate twist see [FM] conjecture 3 for the precise statement relevant here. Further, there is a relationship between ZA3 and a pencil of elliptic curves, which also leads us to expect ZA3 to be modular. The two dimensional Galois representation associated to f by Deligne [D] to a new form f of weight k, is a piece ....

Fontaine, J-M., Mazur, B.: Geometric Galois representations. Elliptic curves, modular forms, & Fermat's last theorem (Hong Kong, 1993), 41--78,

....2. ae is modular; 3. aej G Q( p ( Gamma1) l Gamma1) 2 ) is absolutely irreducible; 4. ae(Frob l ) has two distinct eigenvalues. Then ae has a finite image and the strong Artin conjecture is true for ae. This theorem also provides some evidence for a conjecture of Fontaine and Mazur (see [10]) that any continuous l adic representation of G Q which is ramified at only finitely many primes and is finitely ramified at l (i.e. the image of I l is a finite group) has finite image. As far as we are aware the only previous evidence for this conjecture was in the case of one dimensional ....

J.-M. Fontaine and B. Mazur, Geometric Galois representations, in `Elliptic curves, modular forms and Fermat's last theorem' (J.Coates and S.-T.Yau eds.), International Press, 1995. ICOSAHEDRAL GALOIS REPRESENTATIONS 347

....a by product, we are able to give a description of all special points on these Shimura curves, see proposition 3.14. Another spin off of this work is the construction, using fibres of the Mumford Shimura families, of irreducible p adic Galois representations which are geometric in the sense of [FM95]. We will give this construction in section 5. By a conjecture of Fontaine and Mazur, these representations should come from geometry , but it seems difficult to decide whether this is the case. The author hopes to pose a challenging problem in 5.11. Representations coming from geometry should ....

....hence that ae(G F ) ae G(Q p ) is open for the p adic topology. 5 A conjecture of Fontaine and Mazur The ideas in this paper, especially those from section 1, can be used to construct Galois representation which might be interesting in view of the conjecture of Fontaine and Mazur formulated in [FM95], Conjecture 1. Loosely speaking, this conjecture says that an irreducible geometric representation of the absolute Galois group of a number field should come from algebraic geometry . It turns out that one can construct geometric Galois representations in the following way. If X=S is an ....

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J.-M. Fontaine and B. Mazur. Geometric Galois representations. In J. Coates and S. T. Yau, editors, Elliptic curves, modular forms, & Fermat's last theorem, Series in Number Theory I, pages 41--78. International Press Inc., 1995.

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