Effective geometry and arithmetic of curves: an introduction Online CIMPA course Christophe Ritzenthaler Copyright © 2017 Christophe Ritzenthaler RENNES UNIVERSITY Contents 1 Presentation of the course .....................................7 1.1 Content of the course7 1.2 References7 1.3 Notation7 I Effective geometry of curves 2 Affine and projective varieties: a quick review ................. 11 2.1 Affine varieties 11 2.2 Projective varieties 15 2.3 Maps between projective varieties 19 2.4 Bézout theorem 21 3 Elementary properties of curves ............................... 23 3.1 Uniformizers 23 3.1.1 Construction of functions with specific Laurent tails..................... 24 3.2 Maps between curves 26 3.2.1 Dictionary curves/function fields.................................... 26 3.3 Divisors 29 3.4 Differentials 32 3.4.1 Generalities.................................................... 32 3.4.2 Residue....................................................... 34 4 Riemann-Roch and Riemann-Hurwitz .......................... 37 4.1 Proof of Riemann-Roch theorem 37 4.1.1 Répartitions and H1(D) ........................................... 37 4.1.2 Dual of the space of répartitions.................................... 39 4.1.3 The residue map and Serre duality.................................. 40 4.2 Corollaries 41 4.3 Riemann-Hurwitz theorem 42 5 Description of the curves up to genus 5 ........................ 45 5.1 Genus 0 case 45 5.2 Genus 1 case 46 5.3 Genus 2 case 47 5.4 Interlude: canonical map and hyperelliptic curves 48 5.5 Genus 3 case 50 5.6 Genus 4 case 50 5.7 Genus 5 and beyond 51 II Arithmetic of curves and its Jacobian over finite fields 6 Number of points of curves over finite fields .................... 55 6.1 Weil conjectures for curves 57 6.1.1 Rewriting of Z(C=k;T) ............................................. 58 6.1.2 d0 = 1 ......................................................... 59 6.1.3 Functional equation............................................. 60 6.1.4 Riemann hypothesis.............................................. 60 6.2 Maximal number of points 62 6.2.1 General arguments.............................................. 62 6.2.2 Asymptotics.................................................... 65 6.2.3 The cases g = 1 and 2 ............................................. 65 6.3 Codes 66 6.3.1 Definitions...................................................... 66 6.3.2 AG-codes..................................................... 68 6.3.3 Modular codes................................................. 70 7 Jacobian of curves ........................................... 73 7.1 Abelian varieties: algebraic and complex point of view 74 7.2 Jacobians 78 7.3 Application to cryptography 80 7.4 Construction of curves with many points 82 7.4.1 Weil polynomial vs Frobenius characteristic polynomial.................. 82 7.4.2 A construction of maximal curve of genus 3 over F2n .................... 85 III Appendices 8 Using MAGMA and some (open) problems ..................... 91 8.1 Some basic tools: exercises 91 8.1.1 Wording....................................................... 91 8.1.2 Solutions....................................................... 93 8.2 Some (more) open exercises 95 8.2.1 Isomorphisms between hyperelliptic curves........................... 95 8.2.2 Number of points on plane curves.................................. 95 8.2.3 Good correspondences between curves............................. 95 8.2.4 Constraints on the Weil polynomial for curves.......................... 96 8.2.5 Codes from modular curves....................................... 96 8.2.6 Distribution of curves over finite fields................................ 96 8.2.7 Number of points on a genus 4 curve................................ 96 8.2.8 Number of points on a genus 5 curve................................ 97 8.2.9 Non-special divisors on a curve over a finite field....................... 97 8.3 Good models of curves of genus ≤ 5 97 8.3.1 Wordings...................................................... 97 8.4 Isomorphisms-Automorphisms 100 8.5 Exploring the number of points of curves over finite fields 101 8.5.1 Wordings..................................................... 101 8.5.2 Solutions...................................................... 102 Bibliography ................................................ 105 Articles 105 Books 106 Index ....................................................... 108 1. Presentation of the course 1.1 Content of the course 1.2 References The references for Chapter 2 will be [42, Chap.I] for a fast overview and [38, chap.I, chap.II], [30, Chap.II] or [44, chap.1, 2, 3] for a more exhaustive understanding. Although we are interested in effective aspects, the algorithms behind the systematic computations would bring us too far. We refer to [25] for an introduction to the topic. Chapter 3 follows [42, Chap.II] and partly [41] for the properties of the residue. Chapter 4 is a mix of [41] and [34]. For the models of curves in Chapter 5, there are information in [20], [28], [34] and [33] and part of the underlying theory is also contained in [30, Chap.IV]. The proof of Weil conjecture in Chapter 6 and the consequences for maximal curves are inspired by [8] and [3]. The application to codes is in [46]. Chapter 7 is an overview and the interested reader will be able to learn much more from various sources. For the complex theory [24, chap.IV], [26] and [22] give a deeper and deeper path into the theory. For the general point of view, [24, chap.V] provides a first overview at the general theory whereas [24, chap.VII] focuses on Jacobians. The application to cryptography can be found in various sources, for instance in [23]. The final application to construction of curves takes some arguments from [24, chap.V] and then from [13]. 1.3 Notation In the rest of the course (and unless specified) we will use the following notation • k a perfect field (i.e. all its finite extensions are separable) of characteristic p equal to 0 or a prime. • for varieties V=k, we will write P 2 V instead of P 2 V(k¯). IEffective geometry of curves 2 Affine and projective varieties: a quick re- view ................................ 11 2.1 Affine varieties 2.2 Projective varieties 2.3 Maps between projective varieties 2.4 Bézout theorem 3 Elementary properties of curves ...... 23 3.1 Uniformizers 3.2 Maps between curves 3.3 Divisors 3.4 Differentials 4 Riemann-Roch and Riemann-Hurwitz . 37 4.1 Proof of Riemann-Roch theorem 4.2 Corollaries 4.3 Riemann-Hurwitz theorem 5 Description of the curves up to genus 5 45 5.1 Genus 0 case 5.2 Genus 1 case 5.3 Genus 2 case 5.4 Interlude: canonical map and hyperelliptic curves 5.5 Genus 3 case 5.6 Genus 4 case 5.7 Genus 5 and beyond 2. Affine and projective varieties: a quick review 2.1 Affine varieties Definition 2.1.1 For every n > 0, we define the affine space of dimension n as n n A = f(x1;:::;xn) 2 k g and its k-rational points as n n A (k) = f(x1;:::;xn) 2 k g: n n s s s If P = (a1;:::;an) 2 A = A (k), for every s 2 Gal(k=k), we define P = (a1 ;:::;an ). We see n n that A (k) = fP 2 A ;s.t. P = Ps 8s 2 Gal(k=k)g. This point of view is often useful because it allows to see an arithmetic problem (i.e. over k) as a geometric problem (i.e. over k) plus a Galois action. Let now Rn = k[X1;:::;Xn] and I ⊂ Rn be an ideal. Definition 2.1.2 The affine algebraic set associated to I is the set of points n V(I) = fP 2 A s.t. f (P) = 0 8 f 2 Ig: If I is defined over k (i.e. I can be generated by polynomials with coefficients in k) then the n set of k-rational points of V(I) is the set V(I)(k) = V(I)\A (k) = fP 2 V(I) s.t. P = Ps 8s 2 Gal(k=k)g. The noetherian property of Rn shows that we can assume that I is finitely generated by a set n of polynomials f1;:::; fm. Hence V(I) = fP 2 A s.t. fi(P) = 0 81 ≤ i ≤ mg: We will also denote V = V(( f1;:::; fm)) as the set defined by V : f1 = 0;:::; fm = 0. The action of s 2 Gal(k=k) on elements of Rn (induced by the action on each coefficients) defines an action on the algebraic sets: if V = V(I) for I = ( f1;:::; fm) an ideal of Rn one defines s s s s V = V(( f1 ;:::; fm )). One can prove that V = V for all s if and only if V = V(J) with J defined over k. 12 Chapter 2. Affine and projective varieties: a quick review n Example 2.1 V((1)) = /0and V((0)) = A . Let k = R and let us draw some pictures of algebraic sets in A2 and A3. By the way, the picture behind the title of this chapter is also (the real points of an) algebraic set given by a quartic equation in A3 and called a Kümmer surface. Figure 2.1: V : Y − X2 = 0 Figure 2.2: V : X2Y + XY 4 − X4 −Y 4 = 0 Figure 2.3: V : X2 +Y 2 − Z2 = 0 Figure 2.4: V : Y − X2 = 0;Y = 2 p The last picture defines two points (± 2;2). Note that although V is defined over Q, it has no Q- n rational points. More generally, a point (a1;:::;an) 2 A is defined by I = (X1 − a1;:::;Xn − an). n n 2 Let us fix n ≥ 1 and look at the Q-rational points of Vn : x + y − 1 = 0 in A . Proving that Vn(Q) = f(1;0);(0;1)g when n > 1 is odd and Vn(Q) = f(±1;0);(0;±1)g when n > 2 is even is equivalent to prove Fermat last theorem. One can therefore guess that statements over non- algebraically closed fields are often deep. Note that the underlying idea is that there is also a strong control of the geometry on the arithmetic. It can for instance be proved computing only a geometric invariant of Vn (its genus, see Definition 3.4.5) that its set of Q-rational points must be finite for all n > 4 (consequence of Mordell conjecture proved by Faltings).