On the auto Igusa-zeta function of an Algebraic Curve

TL;DR

This paper introduces the auto-Igusa zeta series for algebraic curves, linking it to smoothness and singularities, and provides explicit formulas for certain singularities, supported by computational evidence using Sage.

Contribution

It defines the auto-Igusa zeta series at a point on a variety, conjectures a closed form for curves with one singularity, and connects the series to motivic integrals and rationality results.

Findings

Explicit formulas for cusp and node singularities

Conjectured closed formula for curves with one singularity

Series often rational, supported by computational evidence

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.