Notes on Geometry and Arithmetic (Universitext)

"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." -- prové de l'editor Preface Contents 1 Diophantus of Alexandria 1.1 Pythagorean Triangles 1.2 Cubics 1.3 Diophantus of Alexandria 1.4 An Example from Diophantus Exercises 2 Algebraic Closure; Affine Space 2.1 Algebraic Extensions 2.2 Algebraic Closure 2.3 Affine Space 2.4 Irreducible Components Exercises 3 Rational Points; Finite Fields 3.1 Galois Homomorphisms 3.2 Norm Forms 3.3 Field of Definition 3.4 Finite Fields Exercises 4 Projective Varieties; Conics and Quadrics 4.1 Projective Space 4.2 Morphisms 4.2.1 The Affine Case 4.2.2 The Projective Case 4.3 Springer's Theorem 4.4 Brumer's Theorem 4.5 Choudhry's Lemma Exercises 5 The Nullstellensatz 5.1 Integral Extensions 5.2 The Weak Nullstellensatz 5.3 Hilbert's Nullstellensatz 5.4 Equivalence of Categories 5.5 Local Properties Exercises 6 Euclidean Rings 6.1 Euclidean Norms 6.2 Imaginary Quadratic Fields 6.3 Motzkin's Construction 6.4 Real Quadratic Fields Exercises 7 Cubic Surfaces 7.1 The Space of Cubics 7.2 Unirationality 7.3 Grassmannian of Lines 7.4 Ruled Cubic Surfaces 7.5 The 27 Lines 7.6 Blowing Up 7.7 The Néron–Severi Group Exercises 8 p-Adic Completions 8.1 Valuations 8.2 p-Adic Numbers 8.3 Canonical Representation 8.4 Hensel's Lemma Exercises 9 The Hasse Principle 9.1 The Hasse–Minkowski Theorem 9.2 Counter-Examples 9.3 Affirmative Results Exercises 10 Diophantine Dimension of Fields 10.1 The Ci Property 10.2 Diophantine Dimension of p-Adic Fields 10.3 The Result of Arkhipov and Karatsuba Exercises Solutions to the Exercises Bibliography Index