Algebraic Geometry Seminar

Let f be a non-constant polynomial in n variables with integer coefficients. To each prime number p and positive integer m, one associates an exponential sum E_f(p,m) given as an averaged sum of exp(2\pi if(x)/p^m), where x varies over (Z/p^mZ)^n.

Let t be a positive real number. Suppose that for each prime number p, there is a positive constant c_p such that |E_f(p,m)} is bounded above by c_pp^{-mt} for all m>1. Igusa's conjecture for exponential sums predicts that one can take c_p independent of p in this inequality. This conjecture is related to the existence of a certain adelic Poisson summation formula and the estimation of the major arcs in the Hardy-Littlewood circle method towards the Hasse principle of f In this talk, I will recall Igusa's conjecture for exponential sums and discuss some new progress and open questions relating this conjecture to the singularities of the hypersurface defined by f. This talk is based on recent joint work with Wim Veys and with Raf Cluckers. Speaker(s): Nguyen Huu Kien (KU Leuven)