On the auto Igusa-zeta function of an Algebraic Curve - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Conference Papers Year :

On the auto Igusa-zeta function of an Algebraic Curve


We study endomorphisms of complete Noetherian local rings in the context of motivic integration. Using the notion of an auto-arc space, we introduce the (reduced) auto-Igusa zeta series at a point, which appears to measure the degree to which a variety is not smooth that point. We conjecture a closed formula in the case of curves with one singular point, and we provide explicit formulas for this series in the case of the cusp and the node. Using the work of Denef and Loeser, one can show that this series will often be rational. These ideas were obtained through extensive calculations in Sage. Thus, we include a Sage script which was used in these calculations. It computes the affine arc spaces $\nabla_{\mathfrak{n}}X$ provided that $X$ is affine, $\mathfrak{n}$ is a fat point, and the ground field is of characteristic zero. Finally, we show that the auto Poincar\'e series will often be rational as well and connect this to questions concerning new types of motivic integrals.

Dates and versions

hal-01350787 , version 1 (01-08-2016)



Andrew Stout. On the auto Igusa-zeta function of an Algebraic Curve. MEGA'2015 (Special Issue), Jun 2015, Trento, Italy. ⟨hal-01350787⟩
37 View
0 Download



Gmail Facebook Twitter LinkedIn More