Notes on geometry and arithmetic / Daniel Coray ; translated by John Steinig and Constantin Manoil.

Format:

Book

Author/Creator:

Coray, D. F. (Daniel F.), author.

Contributor:

Steinig, John.

Manoil, Constantin.

Series:

Universitext

Universitext, 0172-5939

Language:

English

French

Subjects (All):

Arithmetical algebraic geometry.

Geometry.

Algebraic Geometry.

Local Subjects:

Algebraic Geometry.

Physical Description:

xii, 181 pages : illustration ; 24 cm.

Place of Publication:

Cham, Switzerland : Springer, [2020]

Summary:

This English translation of Daniel Corays original French textbook Notes de géométrie et darithmétique 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 Hilberts 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.

Contents:

Chapter 1. Diophantus of Alexandria

Chapter 2. Algebraic closure; affine space

Chapter 3. Rational points; finite fields

Chapter 4. Projective varieties; conics and quadrics

Chapter 5. The Nullstellensatz

Chapter 6. Euclidean rings

Chapter 7. Cubic surfaces

Chapter 8. p-adic completions

Chapter 9. The Hasse principle

Chapter 10. Diophantine dimension of fields.

Notes:

Includes bibliographical references (pages 175-176) and index.

ISBN:

9783030437800

3030437809

OCLC:

1192509278