Transfer Principles for Bounds of Motivic Exponential Functions

Abstract

We study transfer principles for upper bounds of motivic exponential functions and for linear combinations of such functions, directly generalizing the transfer principles from [7] and [13, Appendix B]. These functions come from rather general oscillatory integrals on local fields, and can be used to describe e.g. Fourier transforms of orbital integrals. One of our techniques consists in reducing to simpler functions where the oscillation only comes from the residue field.2000 Mathematics Subject Classification. Primary 14E18; Secondary 22E50, 40J99.

Uniform analysis on local fields and applications to orbital integrals

“…One branch of analysis on non-Archimedean local fields F deals with functions from (subsets of) F n to C. In this paper, we develop a framework which makes it possible to carry out this kind of analysis uniformly in the local field F , in a similar sense as algebraic geometry works in a field-independent way. This extends the work pursued in [7], [26,Appendix B], [10], and [18,Section 4]. One of our motivations is the program initiated by Hales to reformulate in a field-independent way the entire theory of complex admissible representations of reductive groups over local fields, [21,20].…”

“…By definition of first order formulas, the class of definable conditions is closed under finite boolean combinations and under quantification. Results from [7,12] state that this is partially also true for the new classes of conditions introduced in Notation 1.2.9: Each of them is closed under finite positive boolean combinations and under universal quantification. Note however that unlike definable conditions, they are not closed under negation, and neither under existential quantification.…”

“…This really becomes meaningful only in a parametrized version: Given a C exp -function f in, say, m + n variables, we want to understand how the set X F := {x ∈ F m | f (x, ·) has the desired property} depends on F . Typically, the collection X = (X F ) F is not a definable set; however for most of the properties we will consider, we will prove that X = (X F ) F is what we call a "C exp -locus": X F = {x ∈ F m | g F (x) = 0} for some C exp -function g. Those loci already appeared in [7]; in the present paper, they play a central role.…”

“…The central objects of study are "C exp -functions", as in [12], also called "functions of C exp -class" or "of motivic exponential class". (Those generalize the "motivic exponential functions" of [7,9,10], based on the "motivic constructible exponential functions" of [15].) A C exp -function f is a uniformly given collection of functions f F from, say, F n to C for every local field F .…”

“…In the remainder of Section 1, we recall some more results from [7,12] (in particular about integrability), we introduce one new basic "operation" on C exp -loci (namely eventual behavior; Section 1.4), and we give some first example applications of the locus formalism.…”