Exponential sums: questions by Denef, Sperber, and Igusa

TL;DR

This paper proves a conjecture on nondegenerate local exponential sums modulo p^m, extending Igusa's conjecture to quasi-homogeneous forms and generalizing results by Katz to this broader context.

Contribution

It completes the proof of a conjecture by Denef and Sperber and extends Igusa's and Katz's results to quasi-homogeneous cases.

Findings

Proved the remaining part of Denef and Sperber's conjecture.

Generalized Igusa's conjecture to quasi-homogeneous forms.

Extended Katz's estimates to quasi-homogeneous exponential sums.

Abstract

We prove the remaining part of the conjecture by Denef and Sperber [Denef, J. and Sperber, S., \textit{Exponential sums mod $p^n$ and {N}ewton polyhedra}, Bull. Belg. Math. Soc., {\bf{suppl.}} (2001) 55-63] on nondegenerate local exponential sums modulo $p^m$. We generalize Igusa's conjecture of the introduction of [Igusa, J., \textit{Lectures on forms of higher degree}, Lect. math. phys., Springer-Verlag, {\bf{59}} (1978)] from the homogeneous to the quasi-homogeneous case and prove the nondegenerate case as well as the modulo $p$ case. We generalize some results by Katz of [Katz, N. M., \textit{Estimates for "singular" exponential sums}, Internat. Math. Res. Notices (1999) no. 16, 875-899] on finite field exponential sums to the quasi-homogeneous case.