Motivic volume of families of polarized rigid-analytic tori

2018/05/13 by Dmitry Sustretov, Sustretov, Dmitry
Mathematics · Engineering · #Geometric and Algebraic Topology #Advanced Numerical Analysis Techniques #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.1805.04942

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.

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