Rationality of Igusa local zeta functions in positive characteristic

Establish the rationality, as a function of the variable t, of the Igusa local zeta function associated with a finite-type scheme over a non-Archimedean local field of positive characteristic.

Background

The paper studies Igusa local zeta functions as a tool for obtaining uniform estimates for the number of points of schemes over finite rings. In characteristic zero, rationality follows from resolution of singularities and permits the use of poles to analyze point-counting asymptotics.

For local fields of positive characteristic, the authors explain that the general rationality statement is unavailable because resolution of singularities is not known in that setting. They note that rationality is nevertheless obtained for sufficiently large residue-field characteristic through transfer arguments, leaving the unrestricted positive-characteristic case unresolved.