Igusa integrals and volume asymptotics in analytic and adelic geometry

TL;DR

This paper develops asymptotic formulas for volume calculations in analytic and adelic geometry, linking height functions, Igusa integrals, and geometric invariants, with applications to Tamagawa measures.

Contribution

It introduces a unified approach to relate height volume asymptotics with Igusa integrals and geometric invariants over local and adelic fields.

Findings

Established asymptotic volume formulas for height balls.

Connected Mellin transforms of height functions to Igusa integrals.

Constructed general Tamagawa measures in the adelic setting.

Abstract

We establish asymptotic formulae for volumes of height balls in analytic varieties over local fields and in adelic points of algebraic varieties over number fields, relating the Mellin transforms of height functions to Igusa integrals and to global geometric invariants of the underlying variety. In the adelic setting, this involves the construction of general Tamagawa measures.