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Through a cascade of generalizations, we develop a theory of motivicintegration which works uniformly in all non-archimedean local fields ofcharacteristic zero, overcoming some of the difficulties related toramification and small residue field characteristics. We define a class offunctions, called functions of motivic exponential class, which we show to bestable under integration and under Fourier transformation, extending resultsand definitions from previous papers. We prove uniform results related torationality and to various kinds of loci. A key ingredient is a refined form ofDenef-Pas quantifier elimination which allows us to understand definable setsin the value group and in the valued field.

Integration of functions of motivic exponential class, uniform in all non-archimedean local fields of characteristic zero