Motivic Mellin transforms

TL;DR

This paper extends motivic integration by introducing Mellin transforms into the motivic framework, establishing stability, transfer principles, and generalizing previous set-ups for a broader class of functions.

Contribution

It develops a larger class of motivic functions stable under Mellin and Fourier transforms, with new Fubini and change of variables formulas, and provides uniform transfer principles across local fields.

Findings

Motivic Mellin transforms are stable under key operations.

Established transfer principles between different local fields.

Unified framework for $p$-adic and motivic integrals.

Abstract

This work brings Mellin transforms into the realm of motivic integration. The new, larger class of motivic functions is stable under motivic Mellin and Fourier transforms, with general Fubini results and change of variables formulas. It specializes to $p$-adic integrals and $p$-adic Mellin transforms uniformly in $p$, with transfer principles between zero and positive characteristic local fields. In particular, it generalizes previous set-ups of motivic integration with Fubini from among others [16, 17, 18, 29, 9] and simplifies some aspects on the way by using the ideas of [10].