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ON A CONJECTURE OF IGUSA

Part of: Local theory Exponential sums and character sums Forms and linear algebraic groups Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  07 February 2013

Ben Lichtin
Ben Lichtin*
Affiliation:
49 Boardman St., Rochester, NY 14607, U.S.A. (email: lichtin@math.rochester.edu)
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Abstract

In his Tata Lecture Notes, Igusa conjectured the validity of a strong uniformity in the decay of complete exponential sums modulo powers of a prime number and determined by a homogeneous polynomial. This was proved for non-degenerate forms by Denef–Sperber and then by Cluckers for weighted homogeneous non-degenerate forms. In a recent preprint, Wright has proved this for degenerate binary forms. We give a different proof of Wright’s result that seems to be simpler and relies upon basic estimates for exponential sums mod $p$as well as a type of resolution of singularities with good reduction in the sense of Denef.

MSC classification

Secondary: 11L07: Estimates on exponential sums 11E95: $p$-adic theory 11E76: Forms of degree higher than two 14B05: Singularities 14G10: Zeta-functions and related questions (Birch-Swinnerton-Dyer conjecture)
Type
Research Article
Information
Mathematika , Volume 59 , Issue 2 , July 2013 , pp. 399 - 425
Copyright
Copyright © 2013 University College London 

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References

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