Cubic Forms and the Circle Method

Description

The Hardy–Littlewood circle method was invented over a century ago to study integer solutions to special Diophantine equations, but it has since proven to be one of the most successful all-purpose tools available to number theorists. Not only is it capable of handling remarkably general systems of polynomial equations defined over arbitrary global fields, but it can also shed light on the space of rational curves that lie on algebraic varieties.  This book, in which the arithmetic of cubic polynomials takes centre stage, is aimed at bringing beginning graduate students into contact with some of the many facets of the circle method, both classical and modern. This monograph is the winner of the 2021 Ferran Sunyer i Balaguer Prize, a prestigious award for books of expository nature presenting the latest developments in an active area of research in mathematics. Preface Contents 1 Cubic Forms Over Local Fields 1.1 The Hasse Principle 1.2 Systems of Equations Over Finite Fields 1.3 Solubility Over Local Fields 1.4 Local Densities 2 Waring's Problem for Cubes 2.1 Weyl Differencing 2.2 The Asymptotic Formula 2.3 Analysis of the Singular Series 3 Cubic Forms via Weyl Differencing 3.1 Cubic Forms in Many Variables 3.2 The Minor Arcs 3.3 The Major Arcs 4 Norm Forms Over Number Fields 4.1 Background on Algebraic Number Theory 4.2 The Circle Method Over Number Fields 4.3 Singular Integral 4.4 Singular Series 5 Diagonal Cubic Forms Over Function Fields 5.1 Background on Function Fields 5.2 The Circle Method 5.3 The Major Arcs and the Asymptotic Formula 6 Lines on Cubic Hypersurfaces 6.1 Transition to Counting Functions 6.2 Dimension and Irreducibility via the Circle Method 6.3 Singular Locus via the Circle Method References Index