# On the auto Igusa-zeta function of an Algebraic Curve

Abstract : 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.
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Conference papers

https://hal.inria.fr/hal-01350787
Contributor : Alain Monteil <>
Submitted on : Monday, August 1, 2016 - 5:04:13 PM
Last modification on : Tuesday, August 13, 2019 - 1:42:01 PM

### Identifiers

• HAL Id : hal-01350787, version 1
• ARXIV : 1406.6083

### Citation

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

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