Non-archimedean integrals as limits of complex integrals

TL;DR

This paper demonstrates how non-archimedean integrals can be derived as limits of complex integrals using a non-standard complex field model with both archimedean and non-archimedean norms, establishing a morphism between their forms.

Contribution

It introduces a framework connecting complex and non-archimedean integrals via a non-standard complex field, revealing a natural morphism compatible with integration.

Findings

Established a morphism between archimedean and non-archimedean forms

Connected asymptotics of complex integrals to non-archimedean integrals

Provided a new perspective on limits of complex integrals

Abstract

We explain how non-archimedean integrals considered by Chambert-Loir and Ducros naturally arise in asymptotics of families of complex integrals. To perform this analysis we work over a non-standard model of the field of complex numbers, which is endowed at the same time with an archimedean and a non-archimedean norm. Our main result states the existence of a natural morphism between bicomplexes of archimedean and non-archimedean forms which is compatible with integration.