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

By Vladimir Platonov, Andrei Rapinchuk, Rachel Rowen

This milestone paintings at the mathematics conception of linear algebraic teams is now to be had in English for the 1st time. Algebraic teams and quantity conception offers the 1st systematic exposition in mathematical literature of the junction of crew conception, algebraic geometry, and quantity concept. The exposition of the subject is outfitted on a synthesis of tools from algebraic geometry, quantity conception, research, and topology, and the result's a scientific assessment ofalmost all the significant result of the mathematics thought of algebraic teams received so far.

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65 We have reviewed the fundamentals of the theory of semisimple algebraic groups over an algebraically closed field. For semisimple groups defined over an arbitrary field K the theory is more complicated and not as complete. ) However, for semisimple K-split groups the theory can be developed almost in parallel with the case of an algebraically closed field. In particular, for any root system R there exists a simply connected semisimple K-split group G with a maximal K-split torus T C G such that R(T, G) is R.

3). 5 has other applications as well. In particular, the extra structure of "algebra" on a vector space is given by a tensor of type (1,2); hence the LIK-forms of A are in one-to-one correspondence with H 1 ( 3 , A u t ~ ( A ~ )where ), AutL(AL) is the group of L-automorphisms of AL = A @K L. , with the central simple K-algebras of dimension n2 which are split by L. In the above examples the groups of L-automorphisms are groups of Lpoints of algebraic groups (this is always the case when dealing with the automorphism group of a space with a tensor).

The correspondence G --+d X(G) is a contravariant functor from the category A of K-diagonalizable groups split over a finite Galois extension L/K with Galois group 3 and of K-morphisms, to the catego y 23 of finitely and of I'\module homogenerated modules over the group ring r = Z[3] morphisms. 1. @ is a contravariant category equivalence, for which the subcategory A. -modules. 1 is fundamental for the study of algebraic tori, the subject of Voskresenkii's book [3]. This theorem makes it possible to define an algebraic torus by giving the corresponding character module.

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