Explicit Tamagawa numbers for certain algebraic tori over number fields

Abstract

Given a number field extension K / k K/k with an intermediate field K + K^+ fixed by a central element of Gal ⁡ ( K / k ) \operatorname {Gal}(K/k) of prime order p p , there exists an algebraic torus over k k whose rational points are elements of K × K^\times sent to k × k^\times by the norm map N K / K + N_{K/K^+} . The goal is to compute the Tamagawa number such a torus explicitly via Ono’s formula that expresses it as a ratio of cohomological invariants. A fairly complete and detailed description of the cohomology of the character lattice of such a torus is given when K / k K/k is Galois. Partial results including the numerator of Ono’s formula are given when the extension is not Galois, or more generally when the torus is defined by an étale algebra. We also present tools developed in SageMath for this purpose, allowing us to build and compute the cohomology and explore the local-global principles for such an algebraic torus. Particular attention is given to the case when [ K : K + ] = 2 [K:K^+]=2 and K K is a CM-field. This case corresponds to maximal tori in G S p 2 n \mathrm {GSp}_{2n} , and most examples will be in that setting. This is motivated by the application to abelian varieties over finite fields and the Hasse principle for bilinear forms.