nakamura@gaia.math.wani.osaka-u.ac.jp watanabe@math.wani.osaka-u.ac.jp

Submission: 2004, Jun 11

The ratio of the Tamagawa measure and a certain invariant measure on the group GL_n(D_A) is computed, where D_A is the adele of a division algebra D over a global field. An explicit formula of the ratio is described in terms of the special values of the zeta function of D. This formula yields (1) an explicit lower bound of the Hermite--Rankin constant of D and (2) an explicit asymptotic behavior of the distribution of rational points on Brauer--Severi variety.

2000 Mathematics Subject Classification: 11R52

Keywords and Phrases: Tamagawa measure, Hermite constant, Brauer-Severi variety

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