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IGUSA’S CONJECTURE FOR EXPONENTIAL SUMS: OPTIMAL ESTIMATES FOR NONRATIONAL SINGULARITIES

Part of: Exponential sums and character sums Algebraic number theory: local and $p$-adic fields Birational geometry

Published online by Cambridge University Press:  31 July 2019

RAF CLUCKERS Open the ORCID record for RAF CLUCKERS [Opens in a new window] ,
MIRCEA MUSTAŢĂ  and
KIEN HUU NGUYEN Open the ORCID record for KIEN HUU NGUYEN [Opens in a new window]
RAF CLUCKERS
Affiliation:
CNRS, Univ. Lille, UMR 8524 - Laboratoire Paul Painlevé, F-59000 Lille, France KU Leuven, Department of Mathematics, B-3001 Leuven, Belgium; Raf.Cluckers@univ-lille.fr
MIRCEA MUSTAŢĂ
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA; mmustata@umich.edu
KIEN HUU NGUYEN
Affiliation:
KU Leuven, Department of Mathematics, Celestijnenlaan 200B, B-3001 Leuven, Belgium; kien.nguyenhuu@kuleuven.be

Abstract

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We prove an upper bound on the log canonical threshold of a hypersurface that satisfies a certain power condition and use it to prove several generalizations of Igusa’s conjecture on exponential sums, with the log canonical threshold in the exponent of the estimates. We show that this covers optimally all situations of the conjectures for nonrational singularities by comparing the log canonical threshold with a local notion of the motivic oscillation index.

MSC classification

Primary: 11L07: Estimates on exponential sums 11S40: Zeta functions and $L$-functions
Secondary: 11L05: Gauss and Kloosterman sums; generalizations 14E30: Minimal model program (Mori theory, extremal rays)
Type
Research Article
Information
Forum of Mathematics, Pi , Volume 7 , 2019 , e3
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike licence (http://creativecommons.org/licenses/by-nc-sa/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the same Creative Commons licence is included and the original work is properly cited. The written permission of Cambridge University Press must be obtained for commercial re-use.
Copyright
© The Author(s) 2019

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