A New Generating Function for Calculating the Igusa Local Zeta Function

TL;DR

This paper introduces a novel generating function approach to compute Igusa local zeta functions, expanding explicit calculations to broader classes of quadratic polynomials over p-adic fields.

Contribution

A new generating function method is developed for calculating Igusa local zeta functions, enabling explicit results for more quadratic polynomials over p-adic fields.

Findings

Explicit local zeta functions for quadratic polynomials over p-adic fields with odd p.

Identification of actual poles among candidate poles for quadratic forms.

Expansion of calculable quadratic polynomials using the new generating function approach.

Abstract

A new method is devised for calculating the Igusa local zeta function $Z_f$ of a polynomial $f(x_1,\dots,x_n)$ over a $p$-adic field. This involves a new kind of generating function $G_f$ that is the projective limit of a family of generating functions, and contains more data than $Z_f$. This $G_f$ resides in an algebra whose structure is naturally compatible with operations on the underlying polynomials, facilitating calculation of local zeta functions. This new technique is used to expand significantly the set of quadratic polynomials whose local zeta functions have been calculated explicitly. Local zeta functions for arbitrary quadratic polynomials over $p$-adic fields with $p$ odd are presented, as well as for polynomials over unramified $2$-adic fields of the form $Q+L$ where $Q$ is a quadratic form and $L$ is a linear form where $Q$ and $L$ have disjoint variables. For a quadratic form over an arbitrary $p$-adic field with odd $p$, this new technique makes clear precisely which of the three candidate poles are actual poles.