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2020 | Buch

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Notes on Geometry and Arithmetic

verfasst von: Daniel Coray

Verlag: Springer International Publishing

Buchreihe : Universitext

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

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Über dieses Buch

This English translation of Daniel Coray’s original French textbook Notes de géométrie et d’arithmétique introduces students to Diophantine geometry. It engages the reader with concrete and interesting problems using the language of classical geometry, setting aside all but the most essential ideas from algebraic geometry and commutative algebra. Readers are invited to discover rational points on varieties through an appealing ‘hands on’ approach that offers a pathway toward active research in arithmetic geometry. Along the way, the reader encounters the state of the art on solving certain classes of polynomial equations with beautiful geometric realizations, and travels a unique ascent towards variations on the Hasse Principle.

Highlighting the importance of Diophantus of Alexandria as a precursor to the study of arithmetic over the rational numbers, this textbook introduces basic notions with an emphasis on Hilbert’s Nullstellensatz over an arbitrary field. A digression on Euclidian rings is followed by a thorough study of the arithmetic theory of cubic surfaces. Subsequent chapters are devoted to p-adic fields, the Hasse principle, and the subtle notion of Diophantine dimension of fields. All chapters contain exercises, with hints or complete solutions.

Notes on Geometry and Arithmetic will appeal to a wide readership, ranging from graduate students through to researchers. Assuming only a basic background in abstract algebra and number theory, the text uses Diophantine questions to motivate readers seeking an accessible pathway into arithmetic geometry.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Diophantus of Alexandria

Abstract

Diophantus is like an island in the history of mathematics. He lived in Alexandria around 250 C.E. Nobody before him had ever tackled a study of arithmetic over the field of rational numbers. It was 1,300 years before Western mathematicians became interested in this type of problem (Bombelli, Viète, Bachet, Fermat), … on reading Diophantus to be precise. He also introduced new methods and a special symbol to express an unknown, which makes him an essential precursor of algebraic notation.

Daniel Coray

Chapter 2. Algebraic Closure; Affine Space

Abstract

In general, Diophantus’ methods do not use any specific property of rational numbers. They are therefore applicable independently of the base field. This is why they can quite easily be interpreted in terms of algebraic geometry, by adding if necessary rudiments of Galois theory. In this chapter, we introduce the algebraic and geometric concepts that seem best adapted to the arithmetic context.

Daniel Coray

Chapter 3. Rational Points; Finite Fields

Abstract

Finite fields play an essential role in the study of rational solutions of equations. In this chapter we study Galois actions, in order to define correctly the arithmetic notions of point and k-rational variety, which we illustrate in particular in the case of finite fields. We also prove the Chevalley–Warning Theorem on the diophantine dimension of finite fields.

Daniel Coray

Chapter 4. Projective Varieties; Conics and Quadrics

Abstract

The arithmetic study of varieties defined by homogenous polynomials leads to the identification of proportional solutions, which means that we are working in a projective setting. Arithmetic properties of projective varieties are strongly dependent on their geometry. The case of conics serves as a first illustration. Then we shall prove Springer’s and Brumer’s theorems on algebraic points on quadrics and intersections of quadrics.

Daniel Coray

Chapter 5. The Nullstellensatz

Abstract

Hilbert’s Zeros Theorem plays an essential role in algebraic geometry. It allows one to better define morphisms, as the Nullstellensatz implies that the affine algebraic k-varieties form a category equivalent to that of reduced k-algebras of finite type. The most important consequence is the unity of algebra and geometry. Since the field k is arbitrary, this leads to numerous arithmetic applications.

Daniel Coray

Chapter 6. Euclidean Rings

Abstract

We have seen the important role that factorial rings play in algebraic geometry (see in particular the exercises in the previous chapter). It is natural to be interested in Euclid’s division algorithm too, which has given rise to some impressive works. On formalizing the notion of the Euclidean ring, unexpected algorithms, revealed by new methods, have recently been discovered. There are also connections to several old unsolved conjectures.

Daniel Coray

Chapter 7. Cubic Surfaces

Abstract

Projective varieties of degree 3 are an important topic in arithmetic. This includes genus 1 plane curves, which continue to generate research (theory of elliptic curves: computation of the rank of the Mordell–Weil group, the study of the Tate–Shafarevich group, and the Birch and Swinnerton-Dyer conjecture). For smooth cubic hypersurfaces of dimension 3, a difficult theorem (Clemens & Griffiths, 1972) states that they are never k-rational (in the sense of definition 5.5.11), even if k is algebraically closed. They are nevertheless k-unirational, which means that one can parametrize their points, even if the base field is not algebraically closed, but not in an injective manner.

Daniel Coray

Chapter 8. p-Adic Completions

Abstract

The field of p-adic numbers was introduced by Hensel at the beginning of the twentieth century. This remarkable idea greatly simplifies computations involving congruences, and is also of considerable theoretical interest, preparing the way for powerful generalizations.

Daniel Coray

Chapter 9. The Hasse Principle

Abstract

The Hasse principle asks the natural question: if a polynomial equation has non-trivial solutions in R and in Q _p for every prime p, can one deduce that it also has solutions in Q? For quadratic forms, the answer is encouraging, but for more general situations this is only a “principle”, which may be verified or not.

Daniel Coray

Chapter 10. Diophantine Dimension of Fields

Abstract

Some Diophantine questions depend mainly on the geometry of the algebraic varieties involved, but others depend strongly on the field over which these varieties are defined. Diophantine dimension is a property of this nature.

Daniel Coray

Backmatter

Titel: Notes on Geometry and Arithmetic
verfasst von: Daniel Coray
Verlag: Springer International Publishing
Electronic ISBN: 978-3-030-43781-7
Print ISBN: 978-3-030-43780-0
DOI: https://doi.org/10.1007/978-3-030-43781-7