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SUMMARY:Thomas Rüd (MIT)
DTSTART:20221018T203000Z
DTEND:20221018T213000Z
DTSTAMP:20260423T130605Z
UID:MITNT/58
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/MITNT/58/">T
 amagawa numbers for maximal symplectic tori and mass formulae for abelian 
 varieties</a>\nby Thomas Rüd (MIT) as part of MIT number theory seminar\n
 \nLecture held in Room 2-143 in the Simons Building (building 2).\n\nAbstr
 act\nTamagawa numbers are defined as a specific volumes of algebraic group
 \, encapsulating the size of their adelic points modulo rational points. U
 nsurprisingly\, such numbers are intrinsically linked to mass formulae of 
 various kinds. More recently\, Tamagawa numbers of centralizers of element
 s within some algebraic groups also appear in the context of the stable tr
 ace formula.\n\nGekeler's result on a mass formula for elliptic curves def
 ined over $\\mathbb{F}_p$ was extended by Achter-Gordon and then Achter-Al
 tug-Garcia-Gordon to mass formulae for certain principally polarized abeli
 an varieties over finite fields\, using orbital integrals appearing in Lan
 glands-Kottwitz formula. \nGekeler's work uses the analytic class number f
 ormula\, which can be restated as $\\tau_\\mathbb{Q}(\\mathrm{R}_{K/\\math
 bb{Q}}\\mathbb{G}_m)=1$ ($\\mathrm{R}$ denotes the Weil restriction of sca
 lars)\, but in the case of higher-dimensional abelian varieties the tori i
 nvolved are more complicated and their Tamagawa numbers were not known.\n\
 n\nThe work presented aims at showing techniques to compute such numbers a
 s 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 fie
 ld. In particular we will give extensive results for tori splitting over C
 M-fields and the possible range of such Tamagawa numbers.\n
LOCATION:https://researchseminars.org/talk/MITNT/58/
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