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International Mathematics Research Notices (2008) Vol. 2008 : article ID rnm118, 20 pages, doi:10.1093/imrn/rnm118 published on January 11, 2008
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The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org.

Igusa's Conjecture on Exponential Sums Modulo p and p2 and the Motivic Oscillation Index

Raf Cluckers

Katholieke Universiteit Leuven, Departement wiskunde, Celestijnenlaan 200B, B-3001 Leuven, Belgium

Correspondence: Correspondence to be sent to: cluckers{at}ens.fr

We prove the modulo p and modulo p2 cases of Igusa's conjecture on exponential sums. This conjecture predicts specific uniform bounds in the homogeneous polynomial case of exponential sums modulo pm 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 p2 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 Fp-rational singular points on f = 0. We introduce a new invariant of f, the flaw of f.

Present address: École Normale Supérieure, Département de mathématiques et applications, 45 rue d'Ulm, 75230 Paris Cedex 05, France.


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