Airy functions over local fields

TL;DR

This paper extends classical Airy integrals to non-archimedean local fields, demonstrating their local constancy and moderate growth, with implications for p-adic Lie groups and moduli space topology.

Contribution

It introduces a natural generalization of Airy integrals over p-adic fields and analyzes their properties, expanding their applicability in number theory and geometry.

Findings

p-adic Airy integrals are locally constant

they exhibit moderate growth

similar properties are observed for integrals over p-adic Lie groups

Abstract

Airy integrals are very classical but in recent years they have been generalized to higher dimensions and these generalizations have proved to be very useful in studying the topology of the moduli spaces of curves. We study a natural generalization of these integrals when the ground field is a non-archimedean local field such as the field of p-adic numbers. We prove that the p-adic Airy integrals are locally constant functions of moderate growth and present evidence that the Airy integrals associated to compact p-adic Lie groups also have these properties.