Some new progress on Igusa's conjecture for exponential sums

Let f ∈ Z[x_1,...,x_n] be a non-constant polynomial. Let p be a prime number and m be a positive integer. We associate to f, p, m the exponential sum Ef(p,m):=1/p^(mn) ∑_{x∊(Z/pmZ)n} exp(2πif(x)/p^m). Let σ be a positive real number. Suppose that for each prime number p, there is a positive constant c_p such that |Ef(p,m)|≤c_pp^{-mσ} for all m ≥ 2. Igusa's conjecture for exponential sums predicts that one can take c_p independent of p in the above inequality. This conjecture relates to the existence of a certain adèlic 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 dened by f . This talk is based on recent joint work with Wim Veys and with Raf Cluckers