In 1985, Goldfeld and Hoffstein gave the density formula of quadratic number fields with certain invariants. This formula explains the distribution of the class numbers of quadratic number fields. In 2022, Yukie gave a higher dimensional version of the density theorem of Goldfeld-Hoffstein except for 2-adic case.
In this talk, I will explain a formula for the 2-adic volume of the homogeneous space of a quadratic form. This serves as a key ingredient of the density theorem of the unnormalized Tamagawa numbers of projective special orthogonal groups.

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