A New Generating Function for Calculating the Igusa Local Zeta Function

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.