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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.
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 root
Irreducible components of rigid spaces
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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 on
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 arithmetic
  • B. Conrad
  • Mathematics
    Journal of the Institute of Mathematics of…
  • 11 July 2006
The theory of generalized elliptic curves gives a moduli-theoretic compactification for modular curves when the level is a unit on the base, and the theory of Drinfeld structures on elliptic curves
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 geometriques
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 immersion
Chow's K/k-image and K/k-trace, and the Lang-Neron theorem
Let K/k be an extension of fields, and assume that it is primary: the algebraic closure of k in K is purely inseparable over k. The most interesting case in practice is when K/k is a regular