Igusa's conjecture on exponential sums and applications

Igusa's conjecture on exponential sums from 1973 is one of the last three large conjectures of Igusa's, on which recently several attack points with partial progress have emerged. It has direct applications to adelic Poisson summation formulas and adelic integrability. The broader context of motivic integration and motivic Fourier transformation have evolved a lot recently, by work by e.g. Cluckers-Loeser and Hrushovksi-Kazhdan. To understand subtle criteria for integrability on the adeles is delicate since there is oscillation, and is key in the context of zeta functions, Poisson summation and functional equations. To study the conjecture further, we suggest developing techniques in non-archimedean geometry, related to resolution of singularities. Previous research on a related conjecture by Igusa, the Monodromy Conjecture, has led to new and unexpected geometrical results of independent interest. We expect a broad impact of  the geometric techniques that will be developed in the project.

Keywords:Igusa's conjecture

Disciplines:Algebra