Adelic algebraic group

The Idelgruppe is defined in mathematics in the context of class field theory and allows a particularly elegant representation of the Artin reciprocity law.

If a global field, ie either an algebraic number field or an algebraic function field over a finite field, the Idelgruppe is the group of units of the noble ring.

Generalizations of the Artin reciprocity law lead to the contexts of automorphic representations of Idelgruppe and Galois representations of ( Langlands program). Specifically operates the absolute Galois group of the algebraic de Rham cohomology of Shimura varieties with values ​​in the Idelgruppe. These representations are Hodge - Tate weights (1,2).

• Algebraic Number Theory