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On the modularity of elliptic curves over Q
In this paper, building on work of Wiles [Wi] and of Wiles and one of us (R.T.) [TW], we will prove the following two theorems (see §2.2). Theorem A. If E/Q is an elliptic curve, then E is modular.Expand
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Pseudo-reductive Groups
Preface to the second edition Introduction Terminology, conventions, and notation Part I. Constructions, Examples, and Structure Theory: 1. Overview of pseudo-reductivity 2. Root groups and rootExpand
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Modularity of Certain Potentially Barsotti-Tate Galois Representations
where is the reduction of p. These results were subject to hypotheses on the local behavior of p at X, i.e., the restriction of p to a decomposition group at X, and to irreducibility hypotheses onExpand
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Irreducible components of rigid spaces
Cet article donne les fondements de la théorie globale des composantes irréductibles d’espaces analytiques rigides sur un corps complet k. Nous prouvons l’excellence d’anneaux locaux sur les espacesExpand
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Complex Multiplication and Lifting Problems
Introduction Algebraic theory of complex multiplication CM lifting over a discrete valuation ring CM lifting of $p$-divisible groups CM lifting of abelian varieties up to isogeny Some arithmeticExpand
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1.1. Motivation. In [DR], Deligne and Rapoport developed the theory of generalized elliptic curves over arbitrary schemes and they proved that various moduli stacks for (ample) “level-N” structuresExpand
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1.1. Motivation. Let E be an elliptic curve over a p-adic integer ring R, and assume that E has supersingular reduction. Consider the 2-dimensional Fp-vector space of characteristic-0 geometricExpand
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Relative ampleness in rigid geometry
Nous developpons une theorie analytique rigide de l'amplitude relative pour les fibres en droites et notons quelques applications a la descente fidelement plate de morphismes et d'objets geometriquesExpand
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Several approaches to non-archimedean geometry
and it rests upon versions of the inverse and implicit function theorems that can be proved for convergent power series over k by adapting the traditional proofs over R and C. Serre's HarvardExpand
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We provide a proof of Nagata’s compactification theorem: any separated map of finite type between quasi-compact and quasi-separated schemes (e.g., noetherian schemes) factors as an open immersionExpand
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