Families and dichotomies in the circle method

Abstract

In the eighties, Hooley applied the Grand Riemann Hypothesis, and what practically amounts to the general Langlands reciprocity (automorphy) conjecture, in a fresh new way, over certain families of cubic threefolds.This eventually led to conditional near-optimal bounds for the number $N$ of integral solutions to $x_1^3+\dots+x_6^3 = 0$ in expanding boxes. Elsewhere, building on Hooley's work, we have given new applications of large-sieve hypotheses, the Square-free Sieve Conjecture, and predictions of Random Matrix Theory type, over the same geometric families---for instance, conditional optimal asymptotics for $N$ in a large class of regions, with applications to sums of three cubes.The underlying harmonic analysis---which in rough form goes back to Kloosterman---picks up equally significant contributions from the classical major and minor arcs in the circle method. Here, we mainly provide extended summaries, commentary, and other complementary material, leaving complete traditional accounts to papers available elsewhere.Two central themes of this thesis are families (of arithmetic or analytic objects) and dichotomies (between structure and randomness). We especially consider (mainly in relation to the aforementioned cubic questions) families of regions, weights, point counts, oscillatory integrals, exponential sums, Hasse--Weil $L$-functions, and quadratic equations; and dichotomies for point counts over finite and infinite fields.