The $p$-adic zeta function of a plane curve singularity

TL;DR

This paper computes the local p-adic zeta function of a plane curve singularity using toric modifications, leveraging compatibility to simplify calculations without adapting to algebraic settings.

Contribution

It introduces a method to compute the local p-adic zeta function via toric modifications, avoiding the need for algebraic adaptation and Denef's formula.

Findings

Explicit computation of the local p-adic zeta function for plane curve singularities.

Demonstrates the effectiveness of toric modifications in p-adic analysis.

Simplifies calculations by using compatibility and analytic change of variables.

Abstract

Using toric modifications and some compatibility we compute the local $p$-adic zeta function of a plane curve singularity. Thanks to the compatibility, we can work over the analytic change of variables formula for $p$-adic integrals, hence avoid adapting to the algebraic setting and Denef's formula.