Integration in algebraically closed valued fields with sections

Abstract

We construct Hrushovski-Kazhdan style motivic integration in certain expansions of ACVF. Such an expansion is typically obtained by adding a full section or a cross-section from the RV-sort into the VF-sort and some (arbitrary) extra structure in the RV-sort. The construction of integration, that is, the inverse of the lifting map L, is rather straightforward. What is a bit surprising is that the kernel of L is still generated by one element, exactly as in the case of integration in ACVF. The overall construction is more or less parallel to the main construction of [10], as presented in [19,20]. As an application, we show uniform rationality of Igusa zeta functions for non-archimedean local fields with unbounded ramification degrees.Corollary 3.4. If U ⊆ RV n is a definable subset such that vrv(U ) is a singleton then U is L RV -definable.

Generalized Euler characteristic in power-bounded T-convex valued fields

“…From here on, our discussion will be of an increasingly formal nature. Many statements are exact copies of those in [Yin10, Yin11, Yin13b] and often the same proofs work, provided that the auxiliary results are replaced by the corresponding ones obtained above. For the reader’s convenience, we will write down all the details.…”

“…We reiterate [Yin13b, Convention 2.32] here, with a different terminology, since this trivial-looking convention is actually quite crucial for understanding the discussion below, especially the parts that involve special bijections. For any set , let The natural bijection is called the regularization of .…”

“…For a description of the ideas and the main results of the Hrushovski–Kazhdan style integration theory, we refer the reader to the original introduction in [HK06] and also the introductions in [Yin11, Yin13b]. There is also a quite comprehensive introduction to the same material in [HL15] and, more importantly, a specialized version that relates the Hrushovski–Kazhdan style integration to the geometry and topology of Milnor fibers over the complex field.…”