Tamagawa numbers for maximal symplectic tori and mass formulae for abelian varieties

Abstract: Tamagawa numbers are defined as a specific volumes of algebraic group, encapsulating the size of their adelic points modulo rational points. Unsurprisingly, such numbers are intrinsically linked to mass formulae of various kinds. More recently, Tamagawa numbers of centralizers of elements within some algebraic groups also appear in the context of the stable trace formula.

Gekeler's result on a mass formula for elliptic curves defined over $\mathbb{F}_p$ was extended by Achter-Gordon and then Achter-Altug-Garcia-Gordon to mass formulae for certain principally polarized abelian varieties over finite fields, using orbital integrals appearing in Langlands-Kottwitz formula. Gekeler's work uses the analytic class number formula, which can be restated as $\tau_\mathbb{Q}(\mathrm{R}_{K/\mathbb{Q}}\mathbb{G}_m)=1$ ($\mathrm{R}$ denotes the Weil restriction of scalars), but in the case of higher-dimensional abelian varieties the tori involved are more complicated and their Tamagawa numbers were not known.

The work presented aims at showing techniques to compute such numbers as well as many general results on Tamagawa numbers of a vast class of tori, which includes maximal tori of $\mathrm{GSp}_{2n}$ over any global field. In particular we will give extensive results for tori splitting over CM-fields and the possible range of such Tamagawa numbers.

algebraic geometrynumber theory

Audience: researchers in the topic

( paper )

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MIT number theory seminar

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