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79
Ringtheoretic properties of certain Hecke algebras
 Ann. of Math
, 1995
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Galois representations with conjectural connections to arithmetic cohomology
 Duke Math. J
"... In this paper we extend a conjecture of A. Ash and W. Sinnott relating niveau 1 Galois representations to the mod p cohomology of congruence subgroups of SLn(Z) to include Galois representations of higher niveau. We then present computational evidence for our conjecture in the case n = 3 in the form ..."
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Cited by 44 (18 self)
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In this paper we extend a conjecture of A. Ash and W. Sinnott relating niveau 1 Galois representations to the mod p cohomology of congruence subgroups of SLn(Z) to include Galois representations of higher niveau. We then present computational evidence for our conjecture in the case n = 3 in the form of threedimensional Galois representations which appear to correspond to cohomology eigenclasses as predicted by the conjecture. Our examples include Galois representations with nontrivial weight and level, as well as irreducible threedimensional representations that are in no obvious way related to lowerdimensional representations. In addition, we prove that certain symmetric square representations are actually attached to cohomology eigenclasses predicted by the conjecture. 1.
Localglobal compatibility in the padic Langlands programme for GL2/Q
, 2010
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Construction of some families of 2dimensional crystalline representations
, 2004
"... Abstract. We construct explicitly some analytic families of étale (ϕ, Ɣ)modules, which give rise to analytic families of 2dimensional crystalline representations. As an application of our constructions, we verify some conjectures of Breuil on the reduction modulo p of those representations, and ex ..."
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Cited by 30 (4 self)
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Abstract. We construct explicitly some analytic families of étale (ϕ, Ɣ)modules, which give rise to analytic families of 2dimensional crystalline representations. As an application of our constructions, we verify some conjectures of Breuil on the reduction modulo p of those representations, and extend some results (of Deligne, Edixhoven, Fontaine and Serre) on the representations arising from modular forms. Mathematics Subject Classification (2000): 11F80, 11F33, 11F85, 14F30
Galois representations modulo p and cohomology of Hilbert modular varieties
 MR MR2172950 (2006k:11100
"... Abstract. The aim of this paper is to extend some arithmetic results on elliptic modular forms to the case of Hilbert modular forms. Among these results let’s mention: − the control of the image of the Galois representation modulo p [37][35], − Hida’s congruence criterion outside an explicit set of ..."
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Cited by 29 (2 self)
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Abstract. The aim of this paper is to extend some arithmetic results on elliptic modular forms to the case of Hilbert modular forms. Among these results let’s mention: − the control of the image of the Galois representation modulo p [37][35], − Hida’s congruence criterion outside an explicit set of primes p [21], − the freeness of the integral cohomology of the Hilbert modular variety over certain local components of the Hecke algebra and the Gorenstein property of these local algebras [30][16]. We study the arithmetic of the Hilbert modular forms by studying their modulo p Galois representations and our main tool is the action of the inertia groups at the primes above p. In order to determine this action, we compute the HodgeTate (resp. the FontaineLaffaille) weights of the padic (resp. the modulo p) étale cohomology of the Hilbert modular variety. The cohomological part of our paper is inspired by the work of Mokrane, Polo and Tilouine [31, 33] on the cohomology of the Siegel modular varieties and builds upon the geometric constructions of [10, 11]. Contents
Shimura varieties and motives
, 1993
"... Deligne has expressed the hope that a Shimura variety whose weight is defined over Q is the moduli variety for a family of motives. Here we prove that this is the case for “most ” Shimura varieties. As a consequence, for these Shimura varieties, we obtain an explicit interpretation of the canonical ..."
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Cited by 26 (6 self)
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Deligne has expressed the hope that a Shimura variety whose weight is defined over Q is the moduli variety for a family of motives. Here we prove that this is the case for “most ” Shimura varieties. As a consequence, for these Shimura varieties, we obtain an explicit interpretation of the canonical model and a modular description of its points in any field containing the reflex field. Moreover, when we assume the existence of a sufficiently good theory of motives in mixed characteristic, we are able to obtain a description of the points on the Shimura variety modulo a prime of good reduction.
Motivic structures in noncommutative geometry. Available at arXiv:1003.3210
 the Proceedings of the ICM
, 2010
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Comparison of integral structures on spaces of modular forms of weight two, and computation of spaces of forms mod 2 of weight one
, 2008
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Constructing elements in Shafarevich–Tate groups of modular motives
 in Number theory and algebraic geometry, London Mathematical Society Lecture Notes, Volume 303
, 2003
"... We study Shafarevich–Tate groups of motives attached to modular forms on Γ0(N) of weight> 2. We deduce a criterion for the existence of nontrivial elements of these Shafarevich–Tate groups, and give 16 examples in which a strong form of the Beilinson–Bloch conjecture would imply the existence of ..."
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Cited by 17 (3 self)
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We study Shafarevich–Tate groups of motives attached to modular forms on Γ0(N) of weight> 2. We deduce a criterion for the existence of nontrivial elements of these Shafarevich–Tate groups, and give 16 examples in which a strong form of the Beilinson–Bloch conjecture would imply the existence of such elements. We also use modular symbols and observations about Tamagawa numbers to compute nontrivial conjectural lower bounds on the orders of the Shafarevich–Tate groups of modular motives of low level and weight ≤ 12. Our methods build upon the idea of visibility due to Cremona and Mazur, but in the context of motives rather than abelian varieties. 1