Motivic volume of families of polarized rigid-analytic tori

TL;DR

This paper constructs a moduli space of polarized analytic tori over non-Archimedean fields, showing its relation to algebraic tori, and proves the motivic volume of certain families of Abelian varieties vanishes, supporting conjectures in tropical geometry.

Contribution

It establishes a new connection between moduli spaces of polarized tori and algebraic tori, and proves the vanishing of motivic volume for specific Abelian families.

Findings

Moduli space is in definable rigid subanalytic bijection with a PGL_N-bundle over a polyhedral domain.

Motivic volume of certain non-Archimedean semi-algebraic families of Abelian varieties vanishes.

Supports conjectural geometric interpretations in tropical refined multiplicities.

Abstract

Let $k$ be a non-Archimedean rational valued field. We construct the moduli space of linearly rigidified polarized analytic tori over $k$ that admit rigid-analytic uniformization by an algebraic torus and observe that it is in definable rigid subanalytic bijection with a $PGL_N$-bundle over a polyhedral domain in an algebraic torus. We use this observation to prove that the Hrushovski-Kazhdan motivic volume of a non-Archimedean semi-algebraic family of Abelian varieties admitting such a uniformization fibrewise vanishes. This question is motivated by the conjectural geometric interpretation of tropical refined multiplicities of Block and Goetsche proposed by Nicaise, Payne and Schroeter.