Integrability of oscillatory functions on local fields: transfer principles

TL;DR

This paper establishes transfer principles for the integrability of oscillatory functions over local fields, showing that integrability depends only on the residue field's isomorphism class for large primes, with applications to Harish-Chandra characters.

Contribution

It introduces transfer principles for integrability and related properties of motivic exponential functions across different local fields, linking geometric loci to integrability conditions.

Findings

Integrability over $Q_p^n$ implies integrability over $F_p ((t))^n$ for large $p$.

Integrability depends only on the residue field's isomorphism class for sufficiently large characteristic.

Transfer principles for boundedness and local integrability are established.

Abstract

For oscillatory functions on local fields coming from motivic exponential functions, we show that integrability over $Q_p^n$ implies integrability over $F_p ((t))^n$ for large $p$, and vice versa. More generally, the integrability only depends on the isomorphism class of the residue field of the local field, once the characteristic of the residue field is large enough. This principle yields general local integrability results for Harish-Chandra characters in positive characteristic as we show in other work. Transfer principles for related conditions such as boundedness and local integrability are also obtained. The proofs rely on a thorough study of loci of integrability, to which we give a geometric meaning by relating them to zero loci of functions of a specific kind.