On the Tamagawa Number of Algebraic Tori

Abstract

Introduction 1. Extensions and cross sections 1.1. Extensions 1.2. Cross sections 2. Nakayama's duality 2.1. Preliminaries 2.2. Nakayama's duality 2.3. Some sequences 3. The numbers T(T), h(T) and i(T) 3.1. Functorial properties 3.2. The number r(T) 3.3. The number h(T) 3.4. The number i(T) 4. The numbers T(E), h(E) and i(E) 4.1. A function f (E) 4.2. The number T(E) 4.3. Proof of (Al), (A2) 4.4. Relation among r(E), h(E) and i(E) 4.5. Cyclic splitting field 5. Main theorem 6. Examples 6.1. The sequence (N) 6.2. Construction of T with T(T) E Z 6.3. An example of semi-simple group G with z(G) E Z References