Number Theory Seminar

Title: Conditionally around the square-root barrier for cubes

Abstract: In 1986, Hooley applied (what practically amounts to) the general Langlands reciprocity (modularity) conjecture and GRH in a fresh new way, over certain families of cubic 3-folds. This eventually led to conditional near-optimal bounds for the number N of integral solutions to x₁³+...+x₆³ = 0 in expanding boxes. Building on Hooley's work, I will sketch new applications of large-sieve hypotheses, the Square-free Sieve Conjecture, and predictions of Random Matrix Theory type, over the same geometric families - e.g. conditional optimal asymptotics for N in a large class of regions, with applications to sums of three cubes (a project suggested by Amit Ghosh and Peter Sarnak). The underlying harmonic analysis - which goes back to Kloosterman - picks up equally significant contributions from both the classical major and minor arcs.