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ON A CONJECTURE OF IGUSA

  • Ben Lichtin (a1)

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.

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[1]Adolphson, A. and Sperber, S., Exponential sums and Newton polyhedra: cohomology and estimates. Ann. of Math. (2) 130 (1989), 367406.
[2]Cluckers, R., Igusa and Denef–Sperber conjectures on nondegenerate $p$-adic exponential sums. Duke Math. J. 141 (2008), 205216.
[3]Cluckers, R., Igusa’s conjecture on exponential sums modulo $p$ and $p^2$ and the motivic oscillation index. Int. Math. Res. Not. IMRN 2008 (2008), doi:10.1093/imrn/rnm118.
[4]Cluckers, R., Exponential sums: questions by Denef, Sperber, and Igusa. Trans. Amer. Math. Soc. 362 (2010), 37453756.
[5]Cochrane, T., Bounds on complete exponential sums. In Analytic Number Theory, Vol. 1 (Progress in Mathematics 138), Birkhäuser (1996), 211–224.
[6]Denef, J., On the degree of Igusa’s local zeta function. Amer. J. Math. 109 (1987), 9911008.
[7]Denef, J. and Sperber, S., Exponential sums mod $p^n$ and Newton polyhedra. Bull. Belg. Math. Soc. Simon Stevin (2001), 5563 (suppl.).
[8]Heath-Brown, R. and Konyagin, S., New bounds for Gauss sums derived from $k$th powers, and for Heibronn’s exponential sum. Q. J. Math. 51 (2000), 221235.
[9]Igusa, J.-I., Forms of Higher Degree (Tata Institute Lectures 59), Springer (1978).
[10]Katz, N., Estimates for “singular” exponential sums. Int. Math. Res. Not. IMRN 16 (1999), 875899.
[11]Koblitz, N., $p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions (Graduate Texts in Mathematics 58), Springer (1984).
[12]Wright, J., Exponential sums and polynomial congruences in two variables: the quasi-homogeneous case. Preprint, 2012, arXiv:1202.2686.
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ON A CONJECTURE OF IGUSA

  • Ben Lichtin (a1)

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