Pseudo-reductive groups arise naturally in the study of general smooth linear algebraic groups over non-perfect fields and have many important applications. This self-contained monograph provides a comprehensive treatment of the theory of pseudo-reductive groups and gives their classification in a usable form. The authors present numerous new results and also give a complete exposition of Tits' structure theory of unipotent groups. They prove the conjugacy results (conjugacy of maximal split tori, minimal pseudo-parabolic subgroups, maximal split unipotent subgroups) announced by Armand Borel and Jacques Tits, and also give the Bruhat decomposition, for general smooth connected linear algebraic groups. Researchers and graduate students working in any related area, such as algebraic geometry, algebraic group theory, or number theory, will value this book, as it develops tools likely to be used in tackling other problems.

IntroductionIn this paper, building on work of Wiles [Wi] and of Taylor and Wiles [TW], we will prove the following two theorems (see §2.2). Theorem A. If E /Q is an elliptic curve, then E is modular. Theorem B. If ρ : Gal(Q/Q) → GL 2 (F 5 ) is an irreducible continuous representation with cyclotomic determinant, then ρ is modular.We will first remind the reader of the content of these results and then briefly outline the method of proof.If N is a positive integer, then we let Γ 1 (N ) denote the subgroup of SL 2 (Z) consisting of matrices that modulo N are of the form 1 * 0 1 .The quotient of the upper half plane by Γ 1 (N ), acting by fractional linear transformations, is the complex manifold associated to an affine algebraic curve Y 1 (N ) /C . This curve has a natural model Y 1 (N ) /Q , which for N > 3 is a fine moduli scheme for elliptic curves with a point of exact order N . We will let X 1 (N ) denote the smooth projective curve which contains Y 1 (N ) as a dense Zariski open subset.Recall that a cusp form of weight k ≥ 1 and level N ≥ 1 is a holomorphic function f on the upper half complex plane H such that• for all matrices Key words and phrases. Elliptic curve, Galois representation, modularity. The first author was supported by the CNRS. The second author was partially supported by a grant from the NSF. The third author was partially supported by a grant from the NSF and an AMS Centennial Fellowship, and was working at Rutgers University during much of the research. The fourth author was partially supported by a grant from the NSF and by the Miller Institute for Basic Science. The space S k (N ) of cusp forms of weight k and level N is a finite-dimensional complex vector space. If f ∈ S k (N ), then it has an expansionand we define the L-series of f to befor anywith c ≡ 0 mod N and d ≡ p mod N . The operators T p for p | N can be simultaneously diagonalised on the space S k (N ) and a simultaneous eigenvector is called an eigenform. If f is an eigenform, then the corresponding eigenvalues, a p (f ), are algebraic integers and we haveLet λ be a place of the algebraic closure of Q in C above a rational prime and let Q λ denote the algebraic closure of Q thought of as a Q algebra via λ. If f ∈ S k (N ) is an eigenform, then there is a unique continuous irreducible representation [DS]. Moreover ρ is odd in the sense that det ρ of complex conjugation is −1. Also, ρ f,λ is potentially semi-stable at in the sense of Fontaine. We can choose a conjugate of ρ f,λ which is valued in GL 2 (O Q λ ), and reducing modulo the maximal ideal and semi-simplifying yields a continuous representationwhich, up to isomorphism, does not depend on the choice of conjugate of ρ f,λ . Now suppose that ρ : G Q → GL 2 (Q ) is a continuous representation which is unramified outside finitely many primes and for which the restriction of ρ to a decomposition group at is potentially semi-stable in the sense of Fontaine. To ρ| Gal(Q /Q ) we can associate both a pair of Hodge-Tate numbers and a Weil-Deligne representation of the Weil g...

Pseudo-reductive groups arise naturally in the study of general smooth linear algebraic groups over non-perfect fields and have many important applications. This self-contained monograph provides a comprehensive treatment of the theory of pseudo-reductive groups and gives their classification in a usable form. The authors present numerous new results and also give a complete exposition of Tits' structure theory of unipotent groups. They prove the conjugacy results (conjugacy of maximal split tori, minimal pseudo-parabolic subgroups, maximal split unipotent subgroups) announced by Armand Borel and Jacques Tits, and also give the Bruhat decomposition, of general smooth connected algebraic groups. Researchers and graduate students working in any related area, such as algebraic geometry, algebraic group theory, or number theory, will value this book as it develops tools likely to be used in tackling other problems.

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