Igusa's conjecture on exponential sums modulo $p$ and $p^2$ and the motivic oscillation index

Abstract

We prove the modulo $p$ and modulo $p^2$ cases of Igusa's conjecture on exponential sums. This conjecture predicts specific uniform bounds in the homogeneous polynomial case of exponential sums modulo $p^m$ when $p$ and $m$ vary. We introduce the motivic oscillation index of a polynomial $f$ and prove the stronger, analogue bounds for $m=1,2$ using this index instead of the original bounds. The modulo $p^2$ case of our bounds holds for all polynomials; the modulo $p$ case holds for homogeneous polynomials and under extra conditions also for nonhomogeneous polynomials. We obtain natural lower bounds for the motivic oscillation index by using results of Segers. We also show that, for $p$ big enough, Igusa's local zeta function has a nontrivial pole when there are $\FF_p$-rational singular points on $f=0$. We introduce a new invariant of $f$, the flaw of $f$.