637 Algorithmic Arithmetic, Geometry, and Coding Theory 14th International Conference Arithmetic, Geometry, Cryptography and Coding Theory June 3–7, 2013 CIRM, Marseille, France Stéphane Ballet Marc Perret Alexey Zaytsev Editors AmericanMathematicalSociety 637 Algorithmic Arithmetic, Geometry, and Coding Theory 14th International Conference Arithmetic, Geometry, Cryptography and Coding Theory June 3–7, 2013 CIRM, Marseille, France Stéphane Ballet Marc Perret Alexey Zaytsev Editors AmericanMathematicalSociety Providence,RhodeIsland Editorial Board of Contemporary Mathematics Dennis DeTurck, managing editor Michael Loss Kailash Misra Martin J. Strauss 2010 Mathematics Subject Classification. Primary 11G10,11G20, 11G25,11H71, 11Y16, 14G05, 14G15,14Q05, 14Q15, 94B27. Library of Congress Cataloging-in-Publication Data InternationalConference Arithmetic, Geometry, Cryptography and Coding Theory (14th: 2013 : Marseille,France) Algorithmicarithmetic,geometry,andcodingtheory: 14thInternationalConferenceonArith- metic, Geometry, Cryptography, and Coding Theory, June 3-7 2013, CIRM Marseille, France / St´ephaneBallet,MarcPerret,AlexeyZaytsev,editors. pagescm. –(Contemporarymathematics;volume637) Includesbibliographicalreferences. ISBN978-1-4704-1461-0(alk. paper) 1. Coding theory—Congresses. 2. Geometry, Algebraic—Congresses. 3. Cryptography— Congresses. 4. Number theory—Congresses. I. Ballet, St´ephane, 1971–editor. II. Perret, M. (Marc),1963–editor. III.Zaytsev,Alexey(AlexeyI.),1976–editor. IV.Title. QA268.I572013 510(cid:2)–dc23 2014037646 ContemporaryMathematicsISSN:0271-4132(print);ISSN:1098-3627(online) DOI:http://dx.doi.org/10.1090/conm/637 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for useinteachingorresearch. Permissionisgrantedtoquotebriefpassagesfromthispublicationin reviews,providedthecustomaryacknowledgmentofthesourceisgiven. Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink(cid:2) service. Formoreinformation,pleasevisit: http://www.ams.org/rightslink. Sendrequestsfortranslationrightsandlicensedreprintstoreprint-permission@ams.org. Excludedfromtheseprovisionsismaterialforwhichtheauthorholdscopyright. Insuchcases, requestsforpermissiontoreuseorreprintmaterialshouldbeaddresseddirectlytotheauthor(s). Copyrightownershipisindicatedonthecopyrightpage,oronthelowerright-handcornerofthe firstpageofeacharticlewithinproceedingsvolumes. (cid:2)c 2015bytheAmericanMathematicalSociety. Allrightsreserved. TheAmericanMathematicalSocietyretainsallrights exceptthosegrantedtotheUnitedStatesGovernment. PrintedintheUnitedStatesofAmerica. (cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://www.ams.org/ 10987654321 201918171615 Contents Preface v Geometric error correcting codes On products and powers of linear codes under componentwise multiplication Hugues Randriambololona 3 Higher weights of affine Grassmann codes and their duals Mrinmoy Datta and Sudhir R. Ghorpade 79 Algorithmic: special varieties The geometry of efficient arithmetic on elliptic curves David Kohel 95 2–2–2 isogenies between Jacobians of hyperelliptic curves Ivan Boyer 111 Easy scalar decompositions for efficient scalar multiplication on elliptic curves and genus 2 Jacobians Benjamin Smith 127 Algorithmic: point counting A point counting algorithm for cyclic covers of the projective line C´ecile Gonc¸alves 145 Point counting on non-hyperelliptic genus 3 curves with automorphism group Z/2Z using Monsky-Washnitzer cohomology Yih-Dar Shieh 173 Wiman’s and Edge’s sextic attaining Serre’s bound II Motoko Qiu Kawakita 191 Algorithmic: general Genetics of polynomials over local fields Jordi Gua`rdia and Enric Nart 207 iii iv CONTENTS Explicit algebraic geometry Explicit equations of optimal curves of genus 3 over certain finite fields with three parameters Ekaterina Alekseenko and Alexey Zaytsev 245 Smooth embeddings for the Suzuki and Ree curves Abdulla Eid and Iwan Duursma 251 Arithmetic geometry Uniform distribution of zeroes of L-functions of modular forms Alexey Zykin 295 A survey on class field theory for varieties Alexander Schmidt 301 Preface The14thAGCTconference(Arithmetic,Geometry,Cryptography,andCoding Theory) tookplace at CIRM (Centre International de Rencontres Math´ematiques) in Marseille, France, on June 3–7, 2013. This international conference has been a major event in the area of arithmetic geometry and its applications for more than 25 years, 77 participants attended this year. We thank all of them for creating a stimulating research environment. The topics of the talks extended from algebraic number theory to diophantine geometry, curves and abelian varieties over finite fields from the theoretical or the algorithmic point of view, and applications to error-correcting codes. We especially thank the speakers Ekaterina Alekseenko, Nurdagul Ambar, Alp Bassa, Peter Beelen, Jean-Robert Belliard, Ivan Boyer, Niels Bruin, Florian Caullery, Claus Diem, Virgile Ducet, Iwan Duursma, Sudhir Ghorpade, C´ecile Gon¸calves,E´mmanuelHallouin,SafiaHaloui,JohanPeterHansen,MasaakiHomma, Grigory Kabatiansky, Motoko Kawakita, David Kohel, Dmitry Kubrak, Gilles Lachaud,KristinLauter,WinnieLi,EnricNart,FerruhOzbudak,LaurentPoinsot, HuguesRandriambololona,ChristopheRitzenthaler,DamienRobert,KarlRo¨kaeus, Robert Rolland, Sergey Rybakov, Alexander Schmidt, Jeroen Sijsling, Benjamin Smith, Patrick Sol´e, and Milakulo Tukumuli for their lectures. The editors would like to thank the anonymous referees and the staff of CIRM (Olivia Barbarroux, Muriel Milton and Laure Stefanini) for their remarkable pro- fessionalism. v Geometric error correcting codes ContemporaryMathematics Volume637,2015 http://dx.doi.org/10.1090/conm/637/12749 On products and powers of linear codes under componentwise multiplication Hugues Randriambololona Abstract. Inthistextwedeveloptheformalismforproductsandpowersof linear codes under componentwise multiplication. As an expanded version of theauthor’stalkatAGCT-14,focusisputmostlyonbasicpropertiesandde- scriptivestatementsthatcouldotherwiseprobablynotfitinaregularresearch paper. On the other hand, more advanced results and applications are only quickly mentioned with references to the literature. We also point out a few openproblems. Ourpresentationalternatesbetweentwopointsofview,whichthetheory intertwines in an essential way: that of combinatorial coding, and that of algebraicgeometry. In appendices that can be read independently, we investigate topics in multilinear algebra over finite fields, notably we establish a criterion for a symmetric multilinear map to admit a symmetric algorithm, or equivalently, forasymmetrictensortodecomposeasasumofelementarysymmetrictensors. Contents 1. Introduction Basic definitions Link with tensor constructions Rank functions Geometric aspects 2. Basic structural results and miscellaneous properties Support Decomposable codes Repeated columns Extension of scalars Monotonicity Stable structure Adjunction properties Symmetries and automorphisms 3. Estimates involving the dual distance 4. Pure bounds The generalized fundamental functions An upper bound: Singleton Lower bounds for q large: AG codes (cid:3)c2015 American Mathematical Society 3