Effective geometry and arithmetic of curves: an introduction Online CIMPA course Christophe Ritzenthaler Copyright©2017ChristopheRitzenthaler RENNES UNIVERSITY Contents 1 Presentation of the course ..................................... 7 1.1 Content of the course 7 1.2 References 7 1.3 Notation 7 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 ConstructionoffunctionswithspecificLaurenttails . . . . . . . . . . . . . . . . . . . . . 24 3.2 Maps between curves 26 3.2.1 Dictionarycurves/functionfields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 RépartitionsandH1(D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.1.2 Dualofthespaceofrépartitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.1.3 TheresiduemapandSerreduality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 RewritingofZ(C/k,T) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 6.1.2 δ =1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 0 6.1.3 Functionalequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6.1.4 Riemannhypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6.2 Maximal number of points 62 6.2.1 Generalarguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 6.2.2 Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.2.3 Thecasesg=1and2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.3 Codes 66 6.3.1 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 6.3.2 AG-codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 6.3.3 Modularcodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 WeilpolynomialvsFrobeniuscharacteristicpolynomial . . . . . . . . . . . . . . . . . . 82 7.4.2 Aconstructionofmaximalcurveofgenus3overF2n . . . . . . . . . . . . . . . . . . . . 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 Isomorphismsbetweenhyperellipticcurves . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 8.2.2 Numberofpointsonplanecurves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 8.2.3 Goodcorrespondencesbetweencurves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 8.2.4 ConstraintsontheWeilpolynomialforcurves . . . . . . . . . . . . . . . . . . . . . . . . . . 96 8.2.5 Codesfrommodularcurves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 8.2.6 Distributionofcurvesoverfinitefields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 8.2.7 Numberofpointsonagenus4curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 8.2.8 Numberofpointsonagenus5curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 8.2.9 Non-specialdivisorsonacurveoverafinitefield . . . . . . . . . . . . . . . . . . . . . . . 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 ThereferencesforChapter2willbe[42,Chap.I]forafastoverviewand[38,chap.I,chap.II],[30, Chap.II]or[44,chap.1,2,3]foramoreexhaustiveunderstanding. Althoughweareinterestedin effectiveaspects,thealgorithmsbehindthesystematiccomputationswouldbringustoofar. We referto[25]foranintroductiontothetopic. Chapter3follows[42,Chap.II]andpartly[41]forthepropertiesoftheresidue. Chapter4isa mixof[41]and[34]. ForthemodelsofcurvesinChapter5,thereareinformationin[20],[28], [34]and[33]andpartoftheunderlyingtheoryisalsocontainedin[30,Chap.IV]. TheproofofWeilconjectureinChapter6andtheconsequencesformaximalcurvesareinspired by [8] and [3]. The application to codes is in [46]. Chapter 7 is an overview and the interested readerwillbeabletolearnmuchmorefromvarioussources. Forthecomplextheory[24,chap.IV], [26] and [22] give a deeper and deeper path into the theory. For the general point of view, [24, chap.V]providesafirstoverviewatthegeneraltheorywhereas[24,chap.VII]focusesonJacobians. Theapplicationtocryptographycanbefoundinvarioussources,forinstancein[23]. Thefinal applicationtoconstructionofcurvestakessomeargumentsfrom[24,chap.V]andthenfrom[13]. 1.3 Notation Intherestofthecourse(andunlessspecified)wewillusethefollowingnotation • kaperfectfield(i.e. allitsfiniteextensionsareseparable)ofcharacteristic pequalto0ora prime. • forvarietiesV/k,wewillwriteP∈V insteadofP∈V(k¯). I Effective geometry of curves 2 Affine and projective varieties: a quick re- view ................................ 11 2.1 Affinevarieties 2.2 Projectivevarieties 2.3 Mapsbetweenprojectivevarieties 2.4 Bézouttheorem 3 Elementary properties of curves ...... 23 3.1 Uniformizers 3.2 Mapsbetweencurves 3.3 Divisors 3.4 Differentials 4 Riemann-Roch and Riemann-Hurwitz . 37 4.1 ProofofRiemann-Rochtheorem 4.2 Corollaries 4.3 Riemann-Hurwitztheorem 5 Description of the curves up to genus 5 45 5.1 Genus0case 5.2 Genus1case 5.3 Genus2case 5.4 Interlude: canonicalmapandhyperellipticcurves 5.5 Genus3case 5.6 Genus4case 5.7 Genus5andbeyond