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Junyan Xu : Bounds for certain families of character sums: how to obtain strong bounds with more exceptions from weak bounds with fewer exceptions

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I will first introduce some generalities about exponential sums, in particular that square-root cancellation is expected for many algebraic character sums over the rational points of an algebraic variety over a finite field. I will then set the stage for my work: we consider a family of exponential sums, which in our case is parameterized by the rational points of a variety (the parameter space). Our task is to obtain a good bound on the number of exceptional ("bad") parameters for which square-root cancellations fail. Following an idea of Michael Larsen, we consider even moments of the family of exponential sums. If the summands are of certain product form, a transformation can be applied to produce another family of exponential sums (of the same type). If the summands are products of multiplicative characters composed with certain polynomial functions, a weak bound can then be applied to the character sums in this family (with few bad parameters), yielding bounds for the moments. We know from the theory of l-adic sheaves that the parameter space for the original family have a stratification by smooth varieties, which is uniform in some sense as long as the degrees of the characters and polynomials are bounded. Moreover, on each stratum the character sum behave in certain uniform way, so that we can talk about good and bad strata. The bounds on moments yield bounds on dimensions of bad strata, which in turn yield bounds on the number of bad parameters (in any box) of the original family. Though not optimal, the bounds already imply nontrivial Burgess bounds for forms, in joint work with Lillian Pierce.

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