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Julia Gordon's AbstractsArithmetic motivic integration re-interprets the classical Haar measure on p-adic fields in a geometric way. The main benefit of such an interpretation is that it allows to isolate the dependence on p, so that one can perform integration in a field-independent way, and then ``plug in'' p at the very end.Even though this is not the only achievement of the theory, it will be our main focus. Hence, we begin with a brief review of the p-adics and integration on p-adic manifolds. Lectures 1,2 will be about Haar measure on p-adic fields, Serre-Oesterlé measure on p-adic manifolds, Weil's theorem that relates p-adic volume with counting points over a finite field, and cell decomposition theorem for p-adic integrals. Cell decomposition theorem is the main tool used in the most general construction of the motivic measure. In Lectures 3,4 we will go through the main steps of the construction of arithmetic motivic integrals, and the language needed for it (even though it would not be possible to do the construction in detail). We'll define Pas's language for valued fields, which allows to talk about p-adic sets in a "p-independent" way, and describe a class of sets that would be "measurable". Roughly speaking, these are the sets given by logical formulas in Pas's language. Then we'll talk about Chow motives (because they are needed for the values of the measure), and the steps involved in assigning Chow motives to formulas. Finally, we'll be able to state the comparison theorem that justifies the statement that "motivic integration specializes to p-adic integration for almost all p". Lecture 5 will be devoted to applications of this theory: 1. Rationality of Poincaré series. 2. Some applications to representation theory of p-adic groups, such as orbital integrals, and character calculations. |
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