Algebraic Groups and Number Theory - download pdf or read online

By Vladimir Platonov, Andrei Rapinchuk, Rachel Rowen

ISBN-10: 0125581807

ISBN-13: 9780125581806

This milestone paintings at the mathematics idea of linear algebraic teams is now on hand in English for the 1st time. Algebraic teams and quantity concept presents the 1st systematic exposition in mathematical literature of the junction of workforce concept, algebraic geometry, and quantity conception. The exposition of the subject is equipped on a synthesis of equipment from algebraic geometry, quantity thought, research, and topology, and the result's a scientific review ofalmost the entire significant result of the mathematics idea of algebraic teams got so far.

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In several instances it is convenient to view an algebraic group G as a Zariski-closed subset not only of GL,(R) but of the matrix algebra M,(R) as well. This can always be achieved by increasing n (called the degree of G) by 1. Indeed, it suffices to realize GL,(R) itself as a closed subset of Mn+l(R). The desired embedding is given by In this book we shall study algebraic groups defined over a subfield K of R, usually either an algebraic number field or its completion. In this regard, recall that an algebraic group G c GL,(R) is said to be defined over K(or simply a K-group) if a, the ideal of the coordinate ring A of GL,(R) consisting of those polynomials that vanish on G, is generated by a~ = a n AK, where AK = K[x11, .

Then the image of G i in Mnd(K) under Q is defined by the equations (where 0 in the last equation denotes the zero matrix in Md(K)). 3) in GLn(R). Then GI is the desired algebraic K-group. Note that G = RLIK(G) is independent of the choice of the base L/K (up to K-isomorphism). -algebra L@K K. Note that L@K K 2 Kd, the embedding of L in Kd obtained by x H (01 (x), . . ,o ~ ( x ) ) where , 01,. . ,ad are the distinct embeddings of L in K over K . 4) GI e! Gul x . . x Godl where Gut is the subgroup of GLn(R) determined by the equations from a 2 , which is obtained by applying ~i to all polynomials in a ~ .

If X is an affine K-variety or an algebraic K-group, then cp is a bijection. Let us give a rough sketch of the main aspects of the proof (cf. Serre [l], ~oskresenskii[3, Ch. 31). , to be independent of the choice of Y in its class of K-isomorphic L/K-forms and of the choice of L-isomorphism f : X + Y), and injective. This part of the proof is formal and holds for much more general situations. The proof of Chapter 2. 2. Classification of K-forms using Galois cohomology the surjectivity of cp requires a more subtle line of reasoning and is based on the construction of twisting, which we have already encountered.

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Algebraic Groups and Number Theory by Vladimir Platonov, Andrei Rapinchuk, Rachel Rowen

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