Cubic Forms and the Circle Method

verfasst von: Prof. Tim Browning

Verlag: Springer International Publishing

Buchreihe : Progress in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Cubic Forms Over Local Fields

Abstract

A fundamental theme in mathematics concerns the solubility of Diophantine equations over the integers. The study of Diophantine equations is both ancient and difficult, having commanded attention since the time of the ancient Greeks. Hilbert’s 10th problem asks for an algorithm that is capable of checking the solubility in integers of arbitrary Diophantine equations with integer coefficients. We now know that this is an impossible dream in full generality, through work of Matiyasevich (Dokl Akad Nauk SSSR 191, 279–282, 1970) in mathematical logic, leaving us with the task of finding classes of Diophantine equations where we can prove that an algorithm exists. More than a century has passed since Hilbert presented his problems, but we have only achieved full success in degrees at most 2, or for polynomials in one variable.

Tim Browning

Chapter 2. Waring’s Problem for Cubes

Abstract

The analytic Hardy–Littlewood circle method was originally developed to treat additive problems in number theory, typically concerning the representation of large positive integers as a sum of integers drawn from a particular set, such as the set of primes, squares or cubes. All of the examples in this book concern cubic polynomials and the goal of this chapter is to illustrate the genesis of the circle method through one of the most famous additive problems.

Tim Browning

Chapter 3. Cubic Forms via Weyl Differencing

Abstract

In Sect. 1.1, we discussed the solubility of cubic forms, as part of our discussion of the Hasse principle. The main goal of this chapter will be to show how the circle method can address this question for arbitrary non-singular cubic forms, by which we mean that the only solution to the system of equations ∇f(x) = 0 in \({\overline {\mathbb {Q}}}^n\) is the vector x = 0. By adapting the methods from the preceding chapter, we will be able to furnish a proof of the following result, which works under the assumption that there are sufficiently many variables.

Tim Browning

Chapter 4. Norm Forms Over Number Fields

Abstract

Up to this point our applications of the circle method have all involved polynomials defined over \(\mathbb {Q}\). Part of the appeal of the circle method comes from the robust way in which its workings can be reengineered to handle problems over other fields of arithmetic interest. In this chapter we shall illustrate how it can be adapted to study polynomials defined over arbitrary number fields.

Tim Browning

Chapter 5. Diagonal Cubic Forms Over Function Fields

Abstract

In this chapter we discuss cubic Diophantine equations over the function field \(\mathbb {F}_q(t)\), where \(\mathbb {F}_q\) is a finite field with q elements. Any homogeneous cubic polynomial in at least 10 variables over \(\mathbb {F}_q(t)\) admits a non-trivial zero over \(\mathbb {F}_q(t)\), by Theorem 1.8.

Tim Browning

Chapter 6. Lines on Cubic Hypersurfaces

Abstract

A key part of higher-dimensional algebraic geometry involves uncovering the connection between the geometry of algebraic varieties and the geometry of the moduli spaces of genus 0 curves (of fixed degree) that are contained in them. In general, these spaces are very difficult to access and remain largely elusive.

Tim Browning

Backmatter

Titel: Cubic Forms and the Circle Method
verfasst von: Prof. Tim Browning
Verlag: Springer International Publishing
Electronic ISBN: 978-3-030-86872-7
Print ISBN: 978-3-030-86871-0
DOI: https://doi.org/10.1007/978-3-030-86872-7

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

Chapter 1. Cubic Forms Over Local Fields

Chapter 2. Waring’s Problem for Cubes

Chapter 3. Cubic Forms via Weyl Differencing

Chapter 4. Norm Forms Over Number Fields

Chapter 5. Diagonal Cubic Forms Over Function Fields

Chapter 6. Lines on Cubic Hypersurfaces

Backmatter

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