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In this thesis, we use logarithmic methods to study motivic objects. Let R be a complete discrete valuation ring with perfect residue field k, and denote by K its fraction field. We give in chapter 2 a new construction of the motivic Serre invariant of a smooth K-variety and extend it additively to arbitrary K-varieties. The main advantage of this construction is to rely only on resolution of singularities and not on a characteristicnbsp;assumption, asnbsp;previous results. As an application, we give a conditional positive answer to Serre's question on the existence of rational fixed points of a G-action on the affine space, for G a finite l-group. We end the chapter by showing how the logarithmic point of view that we use in our construction leads to a newnbsp;of the motivic nearby cycles with support of Guibert, Loesernbsp;Merle as a motivic volume. In chapter 4 we use the theory of logarithmic geometry to derive a new formula for the motivic zeta function via the volume Poincaré series. More precisely, we show how to compute the volume Poincaré series associated to a generically smooth log smooth R-scheme in terms of its log geometry, more specifically in terms of its associated fan in the sense of Kato. This formula yields a much smaller set of candidate poles for the motivic zeta function and seems especially well suited to tacklenbsp;monodromy conjecture of Halle and Nicaise for Calabi-Yau K-varieties, for which log smooth models appear naturally through the Gross-Siebert programme on mirror symmetry. We end the chapter by showingnbsp;this formula sheds new light on previous results regarding the motivic zeta function of a polynomial nondegenerate with respect tonbsp;Newton polyhedron, and of a polynomial in two variables.

Theses