Algebraic Torus - Arithmetic Invariants

Arithmetic Invariants

In his work on Tamagawa numbers, T. Ono introduced a type of functorial invariants of tori over finite separable extensions of a chosen field k. Such an invariant is a collection of positive real-valued functions f_K on isomorphism classes of tori over K, as K runs over finite separable extensions of k, satisfying three properties:

1. Multiplicativity: Given two tori T₁ and T₂ over K, f_K(T₁ × T₂) = f_K(T₁) f_K(T₂)
2. Restriction: For a finite separable extension L/K, f_L evaluated on an L torus is equal to f_K evaluated on its restriction of scalars to K.
3. Projective triviality: If T is a torus over K whose weight lattice is a projective Galois module, then f_K(T) = 1.

T. Ono showed that the Tamagawa number of a torus over a number field is such an invariant. Furthermore, he showed that it is a quotient of two cohomological invariants, namely the order of the group (sometimes mistakenly called the Picard group of T, although it doesn't classify G_m torsors over T), and the order of the Tate–Shafarevich group.

The notion of invariant given above generalizes naturally to tori over arbitrary base schemes, with functions taking values in more general rings. While the order of the extension group is a general invariant, the other two invariants above do not seem to have interesting analogues outside the realm of fraction fields of one-dimensional domains and their completions.