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UBC Theses and Dissertations

Tamagawa numbers of symplectic algebraic tori, orbital integrals, and mass formulae for isogeny class of abelian varieties over finite fields Rüd, Thomas


The main goal of this thesis is to provide results to explicitly compute Tamagawa numbers for maximal tori of similitude groups. These numbers correspond to volumes inherently linked to local-global principles, and the ones studied in this thesis have direct applications for evaluations of a mass formula for isogeny classes of principally polarized abelian varieties over finite fields. The polarization yields a symplectic structure on their ℓ-adic cohomology groups, and the torus studied corresponds to the centralizer of the Frobenius element of the abelian variety in the similitude group associated with the polarization. We present many theoretical results, as well as algorithms that were developed to study the the cohomology of any algebraic torus and helped establish the right conjectures leading to the aforementioned results. We also present work and partial results on descent-type problems for orbital integrals, aiming to compare masses of abelian varieties with distinguished type. The work presented aims to establish the proportion of products of elliptic curves within the isogeny class of such a product. We also provide extensive background material, reviewing the most important concepts and results used in the thesis, as well as references.

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