F or a more detail insight into this attractive and promising. There is a short chapter constructing general restricted product of groups, giving them their topology and measures, then applying to obtain adeles and ideles groups, plus approximation theorems and others properties, and class group. Normal subgroup structure of groups of rational points of algebraic groups pages 509569 download pdf. On the failure of the gorenstein property for hecke algebras of prime weight kilford, l. Denote by gk the group of krational points of g and by ga, k or ga the corresponding adele group. Adeles and algebraic groups progress in mathematics softcover reprint of the original 1st ed. This barcode number lets you verify that youre getting exactly the right version or edition of a. Fundamental hermite constants of linear algebraic groups watanabe, takao, journal of the mathematical society of japan, 2003. The chapter defines the adele group ga of a linear algebraic group g over a. We establish conditions on the group g, related to the structure of its borel groups, under which k and l have isomorphic adele rings.
These notes have been supplemented by an extended bibliography, and by takashi onos brief survey. Bondarenko 1 mathematical notes of the academy of sciences of the ussr volume 50, pages 1008 1010 1991 cite this article. Adeles and algebraic groups progress in mathematics. Strong approximation for algebraic groups library msri. Linear algebraic groups s if q c, every affine algebraic group g can be viewed as a complex lie group. Hecke algebraisomorphisms and adelic points on algebraic. The group glnaf is the group of finite adelic points of gln. Harder, minkowskische reduktionstheorie uber funktionenkorpern invent. We refer to 2 for the theory of linear algebraic groups.
In particular, it allows one to associate to an algebraic matric group g defined over a finite number field k a locally compact group g, the adele group of g, in. This volume contains the original lecture notes presented by a. Adeles in this chapter we introduce adele groups, a concept which plays a central. Hecke algebra isomorphisms and adelic points on algebraic groups. Some finiteness properties of adele groups over number. One of the primary research tools for the arithmetic theory of algebraic groups is adele groups, whose properties are studied in chapter 5. These notes have been supplemented by an extended bibliography, and by takashi onos brief survey of subsequent research. Structure of the adele group of a product of algebraic groups a. Borel, some finiteness properties of adele groups over number fields publ. To prove this, one determines separately the abstract structures of the additive and multiplicative groups of the adele ring ak and sees that they. Let g denote a linear algebraic group over qand k and l two number. Weil in which the concept of adeles was first introduced, in conjunction with various aspects of c.
A short list of references among the answers, with comments, hoping it will be of some use. We establish conditions on the group g, related to the structure of its borel groups, under which the existence of a group isomorphism gak,f. Structure of the adele group of a product of algebraic groups. Strong approximation in algebraic groups and homogeneous spaces.