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

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

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

References

Aizenbud, A. and Avni, N., ‘Representation growth and rational singularities of the moduli space of local systems’, Invent. Math. 204 (2016), 245–316.Google Scholar
Birkar, C., Cascini, P., Hacon, C. D. and McKernan, J., ‘Existence of minimal models for varieties of log general type’, J. Amer. Math. Soc. 23 (2010), 405–468.Google Scholar
Blum, H., ‘On divisors computing mld’s and lct’s’. Preprint, 2016, arXiv:1605.09662.Google Scholar
Castryck, W. and Nguyen, K. H., ‘New bounds for exponential sums with a non-degenerate phase polynomial’, J. Math. Pures Appl. (9) (2019), doi:10.1016/j.matpur.2019.01.015.Google Scholar
Chambille, S. and Nguyen, K. H., Proof of Cluckers-Veys’s conjecture on exponential sums for polynomials with log canonical threshold at most a half, Int. Math. Res. Not. IMRN 2019, rnz036, doi:10.1093/imrn/rnz036.Google Scholar
Cluckers, R., ‘Igusa and Denef-Sperber conjectures on nondegenerate p-adic exponential sums’, Duke Math. J. 141 (2008), 205–216.Google Scholar
Cluckers, R., ‘Igusa’s Conjecture on exponential sums modulo p and p 2 and the motivic oscillation index’, Int. Math. Res. Not. IMRN (4) (2008), article ID rnm118, 20 pages.Google Scholar
Cluckers, R., ‘Exponential sums: questions by Denef, Sperber, and Igusa’, Trans. Amer. Math. Soc. 362 (2010), 3745–3756.Google Scholar
Cluckers, R. and Mustaţă, M., ‘An invariant detecting rational singularities via the log canonical threshold’. Preprint, 2019, arXiv:1901.08111.Google Scholar
Cluckers, R. and Veys, W., ‘Bounds for log-canonical thresholds and p-adic exponential sums’, Amer. J. Math. 138 (2016), 61–80.Google Scholar
Cochrane, T., ‘Bounds on complete exponential sums’, inAnalytic Number Theory, Vol. 1 (Allerton Park, IL, 1995), Progress in Mathematics, 138 (Birkhäuser Boston, Boston, MA, 1996), 211–224.Google Scholar
Denef, J., ‘Local zeta functions and Euler characteristics’, Duke Math. J. 63 (1991), 713–721.Google Scholar
Denef, J., ‘Report on Igusa’s local zeta function, Séminaire Bourbaki, Vol. 1990/91’, Astérisque 201–203 (1991), Exp. No. 741, 359–386 (1992).Google Scholar
Denef, J. and Hoornaert, K., ‘Newton polyhedra and Igusa’s local zeta function’, J. Number Theory 89 (2001), 31–64.Google Scholar
Denef, J. and Sperber, S., ‘Exponential sums mod p n and Newton polyhedra’, Bull. Belg. Math. Soc. Simon Stevin 2001 55–63. suppl.Google Scholar
Denef, J. and Veys, W., ‘On the holomorphy conjecture for Igusa’s local zeta function’, Proc. Amer. Math. Soc. 123 (1995), 2981–2988.Google Scholar
Hironaka, H., ‘Resolution of singularities of an algebraic variety over a field of characteristic zero. I’, Ann. of Math. 79(2) (1964), 109–203.Google Scholar
Hoornaert, K., ‘Newton polyhedra, unstable faces and the poles of Igusa’s local zeta function’, Trans. Amer. Math. Soc. 356 (2004), 1751–1779.Google Scholar
Igusa, J., ‘On certain representations of semi-simple algebraic groups and the arithmetic of the corresponding invariants. I’, Invent. Math. 12 (1971), 62–94.Google Scholar
Igusa, J., ‘On the arithmetic of Pfaffians’, Nagoya Math. J. 47 (1972), 169–198.Google Scholar
Igusa, J., ‘Complex powers and asymptotic expansions I’, J. Reine Angew. Math. 268/269 (1974), 110–130.Google Scholar
Igusa, J., ‘On a certain Poisson formula’, Nagoya Math. J. 53 (1974), 211–233.Google Scholar
Igusa, J., ‘Complex powers and asymptotic expansions II’, J. Reine Angew. Math. 278/279 (1975), 307–321.Google Scholar
Igusa, J., ‘A Poisson formula and exponential sums’, J. Fac. Sci. Univ. Tokyo Sect. IAMath. 23 (1976), 223–244.Google Scholar
Igusa, J., ‘Criteria for the validity of a certain Poisson formula’, inAlgebraic Number Theory (Kyoto Internat. Sympos., Res. Inst. Math. Sci., Univ. Kyoto, Kyoto, 1976) (Japan Soc. Promotion Sci., Tokyo, 1977), 43–65.Google Scholar
Igusa, J., Lectures on Forms of Higher Degree (notes by S. Raghavan), Lectures on Mathematics and Physics, 59 (Tata institute of fundamental research, Springer-Verlag, 1978).Google Scholar
Igusa, J., An Introduction to the Theory of Local Zeta Functions, AMS/IP Studies in Advanced Mathematics, 14 (American Mathematical Society, Providence, RI, 2000), International Press, Cambridge, MA.Google Scholar
Katz, N., ‘Sums of Betti numbers in arbitrary characteristic’, Finite Fields Appl. 7 (2001), 29–44.Google Scholar
Kollár, J., ‘Singularities of pairs’, inAlgebraic Geometry—Santa Cruz 1995, Proceedings of Symposia in Pure Mathematics, 62 (American Mathematical Society, Providence, RI, 1997), 221–287.Google Scholar
Kollár, J. and Mori, S., Birational Geometry of Algebraic Varieties, Cambridge Tracts in Mathematics, 134 With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original, Cambridge University Press, Cambridge, 1998.Google Scholar
Laeremans, A. and Veys, W., ‘On the poles of maximal order of the topological zeta function’, Bull. Lond. Math. Soc. 31 (1999), 441–449.Google Scholar
Lichtin, B., ‘On a conjecture of Igusa’, Mathematika 59 (2013), 399–425.Google Scholar
Lichtin, B., ‘On a conjecture of Igusa II’, Amer. J. Math. 138 (2016), 201–249.Google Scholar
Matsumura, H., Commutative Ring Theory, 2nd edn, Cambridge Studies in Advanced Mathematics, 8 Translated from the Japanese by M. Reid, Cambridge University Press, Cambridge, 1989.Google Scholar
Mustaţă, M., ‘IMPANGA lecture notes on log canonical thresholds (notes by Tomasz Szemberg)’, inContributions to Algebraic Geometry, EMS Series of Congress Reports (European Mathematical Society, Zürich, 2012), 407–442.Google Scholar
Nguyen, K. H., ‘Uniform rationality of Poincaré series of $p$ -adic equivalence relations and Igusa’s conjecture on exponential sums’, PhD thesis, University of Lille (2018), arXiv:1903.06738.Google Scholar
Nicaise, J. and Xu, C., ‘Poles of maximal order of motivic zeta functions’, Duke Math. J. 165 (2016), 217–243.Google Scholar
Rojas-León, A., ‘Estimates for singular multiplicative character sums’, Int. Math. Res. Not. IMRN 20 (2005), 1221–1234.Google Scholar
Saito, M., ‘On b-function, spectrum and rational singularity’, Math. Ann. 295 (1993), 51–74.Google Scholar
Veys, W., ‘On the log canonical threshold and numerical data of a resolution in dimension $2$ ’. Preprint, 2018, arXiv:1810.11062.Google Scholar
Veys, W. and Zúñiga-Galindo, W. A., ‘Zeta functions for analytic mappings, log-principalization of ideals, and Newton polyhedra’, Trans. Amer. Math. Soc. 360 (2008), 2205–2227.Google Scholar
Wright, J., ‘Exponential sums and polynomial congruences in two variables: the quasi-homogeneous case’, Preprint, 2012, arXiv:1202.2686v1.Google Scholar