Overview
- Offers readers a ‘hands on’ introduction to Diophantine geometry
- Progresses from introductory to advanced topics using the language of classical geometry
- Assumes only modest prerequisites in abstract algebra and number theory
- Contains numerous exercises throughout
Part of the book series: Universitext (UTX)
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Table of contents (10 chapters)
Keywords
About this book
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.
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Bibliographic Information
Book Title: Notes on Geometry and Arithmetic
Authors: Daniel Coray
Translated by: Constantin Manoil, John Steinig
Series Title: Universitext
DOI: https://doi.org/10.1007/978-3-030-43781-7
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Switzerland AG 2020
Softcover ISBN: 978-3-030-43780-0Published: 07 July 2020
eBook ISBN: 978-3-030-43781-7Published: 06 July 2020
Series ISSN: 0172-5939
Series E-ISSN: 2191-6675
Edition Number: 1
Number of Pages: XII, 181
Number of Illustrations: 118 b/w illustrations, 3 illustrations in colour
Topics: Algebraic Geometry