Algebraic Geometry of Curves Andrew Kobin Fall 2016 Contents Contents Contents 0 Introduction 1 0.1 Geometry and Number Theory . .2 0.2 Rational Curves . .4 0.3 The p-adic Numbers . .6 1 Algebraic Geometry 9 1.1 Affine and Projective Space . .9 1.2 Morphisms of Affine Varieties . 15 1.3 Morphisms of Projective Varieties . 19 1.4 Products of Varieties . 21 1.5 Blowing Up . 22 1.6 Dimension of Varieties . 24 1.7 Complete Varieties . 26 1.8 Tangent Space . 27 1.9 Local Parameters . 32 2 Curves 33 2.1 Divisors . 34 2.2 Morphisms Between Curves . 36 2.3 Linear Equivalence . 38 2.4 Differentials . 40 2.5 The Riemann-Hurwitz Formula . 42 2.6 The Riemann-Roch Theorem . 43 2.7 The Canonical Map . 45 2.8 B´ezout'sTheorem . 45 2.9 Rational Points of Conics . 47 3 Elliptic Curves 52 3.1 Weierstrass Equations . 53 3.2 Moduli Spaces . 55 3.3 The Group Law . 56 3.4 The Jacobian . 58 4 Rational Points on Elliptic Curves 61 4.1 Isogenies . 62 4.2 The Dual Isogeny . 65 4.3 The Weil Conjectures . 67 4.4 Elliptic Curves over Local Fields . 69 4.5 Jacobians of Hyperelliptic Curves . 75 5 The Mordell-Weil Theorem 77 5.1 Some Galois Cohomology . 77 5.2 Selmer and Tate-Shafarevich Groups . 80 5.3 Twists, Covers and Principal Homogeneous Spaces . 84 i Contents Contents 5.4 Descent . 89 5.5 Heights . 97 6 Elliptic Curves and Complex Analysis 99 6.1 Elliptic Functions . 99 6.2 Elliptic Curves . 105 6.3 The Classical Jacobian . 110 6.4 Jacobians of Higher Genus Curves . 115 7 Complex Multiplication 117 7.1 Classical Complex Multiplication . 117 7.2 Torsion and Rational Points . 120 7.3 Class Field Theory with Elliptic Curves . 123 ii 0 Introduction 0 Introduction These notes come from a course in algebraic geometry and elliptic curves taught by Dr. Lloyd West at the University of Virginia in Fall 2016. The first part of the notes are a survey of the main concepts in algebraic geometry, with an emphasis on curves (i.e. varieties of dimension 1). Key topics include: Affine and projective varieties Dimension Singular and nonsingular points and tangent spaces Morphisms between varieties Intersection theory Divisors Genus The Riemann-Hurwitz theorem and Riemann-Roch theorem Jacobian of a curve The main algebraic geometry reference used is Shafarevich's Basic Algebraic Geometry 1. The second part of the course covers the basic results in the arithmetic geometry of elliptic curves, including: Abelian varieties and isogenies Models over local and global fields Moduli Reduction mod p Zeta functions Statement of the Weil conjectures for curves Heights Descent (`ala Fermat) Hasse's local-global principle Torsors and Galois actions Galois cohomology in degrees 0, 1 and 2 Selmer and Tate-Shafarevich groups Additional topics include the application of elliptic curves to cryptography, higher genus curves and L-functions. The main text used is Silverman's Arithmetic of Elliptic Curves. 1 0.1 Geometry and Number Theory 0 Introduction 0.1 Geometry and Number Theory Consider the following questions: Question 1. Describe the set of all right triangles with integer sides. Question 2. A rational number n is said to be congruent if there exists a rational right triangle with area n. Which rational numbers n are congruent? We will see that Question 1 is easy to answer, while Question 2 is still unsolved. The fundamental difference lies in the geometry of each situation. Definition. We say (a; b; c) 2 Z3 is a pythagorean triple if a2 + b2 = c2. For example, (3; 4; 5) and (5; 12; 13) are pythagorean triples. Notice that multiplying any pythagorean triple by an integer n 2 Z yields another pythagorean triple (in particular, there are infinitely many pythagorean triples), so we may assume a; b; c are coprime. Such a triple is called a primitive pythagorean triple. Theorem 0.1.1. Denote the set of all primitive pythagorean triples by Π. Then there is a bijection Π $ f(x; y) 2 Q2 j x2 + y2 = 1g. Proof. It is easy to check that the assignments a b (a; b; c) 7−! ; c c a b (a; b; c) 7−! (x; y) = ; with a; b; c coprime c c exhibit the desired bijection. Thus the problem of rational triangles and pythagorean triples reduces to studying the rational points of the unit circle in the xy-plane. Definition. Let k be a field and fix a polynomial f 2 k[x; y] which is irreducible over the ¯ algebraic closure k. Then the curve associated to f is a functor C = Cf given by C : Fieldsk −! Sets 2 K=k 7−! Ck(K) := f(x; y) 2 K j f(x; y) = 0g: For a field extension K=k, the set C(K) is called the K-rational points of the curve C. 2 2 In this language, Question 1 reads, \What is #Cf (Q) when f = x + y − 1"? Example 0.1.2. Let f = x2 + y2 − 1 and consider the geometric objects defined by C(K) = Cf (K) for K = R and K = Q. C(R) = S1 ⊆ R2 C(C), the Riemann sphere in C2 2 0.1 Geometry and Number Theory 0 Introduction Also note that since f 2 Q[x; y], we can view f as a polynomial with coefficients in any finite field Fq, and consequently the Fq-rational points C(Fq) are defined. Next, fix the point (−1; 0) on C(K) for any field K and consider the line L : x = 0. slope = t (−1; 0) L 2 2 Theorem 0.1.3. Let k be any field and C = Cf the curve defined by f = x + y − 1. Then there is a bijection C(k) r f(−1; 0)g −! L(k) (x; y) 7−! (0; χ(x; y)) ( (t); φ(t)) 7−! t where χ, and φ are rational functions, i.e. χ 2 k(x; y) and ; φ 2 k(t). Proof. The rational functions y 1 − t2 2t χ(x; y) = ; (t) = and φ(t) = x + 1 1 + t2 1 + t2 exhibit the bijection. Theorems 0.1.1 and 0.1.3 answer Question 1: the set of all primitive pythagorean triples is completely described by the line L given by x = 0, and this description holds over any field k. For Question 2, we must understand the set of congruent numbers over a field k. For n 2 Q, define the set 3 2 2 2 1 Cn(k) = (a; b; c) 2 k : a + b = c and 2 ab = n : Definition. We say n is congruent over k if Cn(k) is nonempty. In particular, Question 2 reduces to deciding when Cn(Q) is nonempty. Notice that we may assume n is a squarefree integer. Above, we parametrized the circle S1 by a line L. Here, we parametrize Cn(k) with a zero set of a different polynomial. Define a bijection 2 2 3 Cn(k) −! En(k) := f(x; y) 2 k j y = x − nx; y 6= 0g nb 2n2 (a; b; c) 7−! ; c − a c − a x2 − n2 2nx x2 + n2 ; ; 7−! (x; y): y y y The set En(k) is called an elliptic curve over k. 3 0.2 Rational Curves 0 Introduction Example 0.1.4. The elliptic curve defined by y2 = x3 − 25x over R is shown below, with some points of En(Q) highlighted. 0.2 Rational Curves Let C be a plane curve defined by an irreducible polynomial f 2 k[x; y]. Definition. We say C is unirational over k if there exist nonconstant rational functions ; φ 2 k(t) such that f( (t); φ(t)) = 0 for all t. Definition. We say C is rational over k if it is unirational and there exists a rational function χ 2 k(x; y) such that (χ(x; y)) = x and φ(χ(x; y)) = y for all x; y 2 k, with the possible exception of finitely many points. 1 2 2 Example 0.2.1. By Theorem 0.1.3, the circle S = Cf for f = x +y −1 is rational over Q. This example illustrates the idea that a curve is rational if it has a `rational parametrization' by a line. In general, the notions of unirationality and rationality are equivalent for curves (this is not true for higher dimensional varieties): Theorem 0.2.2 (L¨uroth). A curve C over k is unirational if and only if it is rational. To prove L¨uroth'stheorem, we formulate the statement in terms of field theory. ¯ Definition. Let f 2 k[x; y] be an irreducible polynomial over k and let C = Cf be the associated plane curve. Then a rational function on C is an equivalence class of functions p ¯ p1 u(x; y) 2 k(x; y), with u = , p; q 2 k[x; y] and f q over k, where we say u1 = and q - q1 p2 u2 = are equivalent if f divides p1q2 − p2q1. q2 4 0.2 Rational Curves 0 Introduction 1 2 2 Example 0.2.3. On the circle S = Cf , f = x + y − 1, the functions y 1 − x u (x; y) = and u = 1 1 + x 2 y are equivalent, so they define a common rational function on C. Definition. The set of rational functions on C with coefficients in k is called the function field of C, denoted k(C). Lemma 0.2.4. k(C) is a field.