Directions in Number Theory: Proceedings of the 2014 WIN3 Workshop PDF
Preview Directions in Number Theory: Proceedings of the 2014 WIN3 Workshop
Association for Women in Mathematics Series Ellen E. Eischen Ling Long Rachel Pries Katherine E. Stange Editors Directions in Number Theory Proceedings of the 2014 WIN3 Workshop Association for Women in Mathematics Series Volume 3 Moreinformationaboutthisseriesathttp://www.springer.com/series/13764 Association for Women in Mathematics Series Focusingonthegroundbreakingworkofwomeninmathematicspast,present,and future,Springer’sAssociationforWomeninMathematicsSeriespresentsthelatest research and proceedings of conferences worldwide organized by the Association forWomeninMathematics(AWM).Allworksarepeer-reviewedtomeetthehighest standardsofscientificliterature,whilepresentingtopicsatthecuttingedgeofpure andappliedmathematics.Sinceitsinceptionin1971,TheAssociationforWomenin Mathematicshasbeenanon-profitorganizationdesignedtohelpencouragewomen and girls to study and pursue active careers in mathematics and the mathematical sciencesandtopromoteequalopportunityandequaltreatmentofwomenandgirls inthemathematicalsciences.Currently,theorganizationrepresentsmorethan3000 members and 200 institutions constituting a broad spectrum of the mathematical community,intheUnitedStatesandaroundtheworld. Ellen E. Eischen (cid:129) Ling Long (cid:129) Rachel Pries Katherine E. Stange Editors Directions in Number Theory Proceedings of the 2014 WIN3 Workshop 123 Editors EllenE.Eischen LingLong DepartmentofMathematics DepartmentofMathematics UniversityofOregon LouisianaStateUniversity Eugene,OR,USA BatonRouge,LA,USA RachelPries KatherineE.Stange DepartmentofMathematics DepartmentofMathematics ColoradoStateUniversity UniversityofColorado FortCollins,CO,USA Boulder,CO,USA ISSN2364-5733 ISSN2364-5741 (electronic) AssociationforWomeninMathematicsSeries ISBN978-3-319-30974-3 ISBN978-3-319-30976-7 (eBook) DOI10.1007/978-3-319-30976-7 LibraryofCongressControlNumber:2016940911 ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland Preface This volume is a compilation of research and survey papers in number theory, written by members of the Women in Numbers (WIN) network, principally by the collaborative research groups formed at Women in Numbers 3, a conference at the BanffInternationalResearchStationinBanff,Alberta,onApril21–25,2014. The WIN conference series began in 2008, with the aim of strengthening the researchcareersoffemalenumbertheorists.Theseriesintroducedanovelresearch- mentorship model: women at all career stages, from graduate students to senior members of the community, joined forces to work in focused research groups on cutting-edgeprojectsdesignedandledbyexperiencedresearchers.Thismodelhad tremendoussuccess,branchingoutnotonlytoWINE(WomeninNumbersEurope) but also to Algebraic Combinatorixx, WIT (Women in Topology), and others. The Association for Women in Mathematics (AWM), funded by the National Science Foundation, isnowsupportingthisresearch-mentorship modelundertheumbrella oftheResearchCollaborationConferencesforWomeninitiative. The goals for Women In Numbers 3 were to establish ambitious new collabo- rations between women in number theory, to train junior participants about topics of current importance, and to continue to build a vibrant community of women in number theory. The majority of the week was devoted to research activities. Beforetheconference,theparticipantswereorganizedintonineprojectgroupsby research interest and asked to learn background for their project topics. This led tomoreproductiveon-siteresearchconversationsandthegroupswereabletoshare preliminaryresultsonthelastday.Theworkshopalsoincludedalectureseriesabout arithmeticofcurves,includingellipticcurves,modularcurves,andShimuracurves. Forty-two women attended the WIN3 workshop, which was organized by the lastthreeeditorsofthisvolume.Thisincluded15seniorandmid-levelfaculty,15 junior faculty and postdocs, and 12 graduate students. This volume is the fourth proceedings to come out of the WIN conference series. It is also the first in the seriespublishedbySpringerforAWM. The editors invited WIN3 research groups and members of the larger WIN3 communitytosubmitarticlesin2014.Afterathoroughrefereeprocessbyexternal experts, we accepted 10 papers for the volume. One interesting attribute of the v vi Preface collection is the interplay between deep theory and intricate computation. The papers span a wide range of research areas: arithmetic geometry, analytic number theory, algebraic number theory, and applications to coding and cryptography. In thispreface,wepointoutafewconnectionsbetweenthepapers. A major theme of the volume is the study of rational points on varieties via cohomologicalmethods.Threepapersonthisthemeareaboutrationalpointsover numberfields.ThepaperInsufficiencyoftheBrauer-Maninobstructionforrational points on Enriques surfaces (Balestrieri et al.) is about the failure of the Hasse principleforsurfaces.InthepaperShadowlinesinthearithmeticofellipticcurves (Balakrishnan et al.), the authors use information about analytic ranks and Tate- Shafarevich groups to develop an algorithm for computing anticyclotomic p-adic heights and shadow lines cast by rational points on elliptic curves over imaginary quadratic fields. In the paper Galois action on the homology of Fermat curves (Davis et al.), the authors use topology and the étale fundamental group to study obstructionsforpointsonFermatcurvesdefinedovercyclotomicfields. ThepaperZetafunctionsofaclassofArtin-Schreiercurveswithmanyautomor- phisms over finite fields (Bouw et al.) is a bridge between several of the disparate topics. It fits in the vein of studying rational points via cohomological methods, becausethe`-adiccohomologyprovidesinformationaboutpointsoncurvesdefined over finite fields. It connects to the topic of applications to coding theory and cryptography, because the class of Artin-Schreier curves produces large families of supersingular curves useful for error-correcting codes. Similarly, the paper Hypergeometric series, truncated hypergeometric series, and Gaussian hypergeo- metric functions (Deines et al.) draws together several topics. The hypergeometric varieties are higher-dimensional analogues of Legendre curves and the authors obtaininformationaboutthenumberofpointsonthesevarietiesdefinedoverfinite fields.Thispaperalsoconnectstothemoreanalyticpapersinthevolume. There are two other papers with an analytic and geometric focus. The paper A generalization of S. Zhang’s local Gross-Zagier formula for GL2 (Maurischat) is about Hecke operators and contains a fundamental lemma for some relative trace formulae. The paper p-adic q-expansion principles on unitary Shimura varieties (Caraianietal.)hasresultsaboutvanishingtheoremsforp-adicautomorphicforms onunitarygroupsofarbitrarysignature. The final three papers are about applications of algebraic number theory. The paper Kneser-Hecke-operators for codes over finite chain rings (Feaver et al.) is aboutthetaseriesforlatticesforcodesoverfinitefieldsandananalogueforHecke operatorsinthiscontext.InRing-LWEcryptographyforthenumbertheorist(Elias etal.),theauthorsgiveasurveyaboutattacksontheringandpolynomiallearning with errors problems and discuss connections with open problems about algebraic numberfields.Finally,thevolumeendswithasurveyaboutarithmeticstatisticsin algebraic number theory, Asymptotics for number fields and class groups (Wood). ThissurveyisanextendedversionofWood’slecturenotesfortheArizonaWinter Schoolin2014,onthetopicofcountingnumberfieldsandthedistributionofclass groups. Preface vii viii Preface Acknowledgments It was a pleasure to work with BIRS to organize the WIN3 conference and with Springer to prepare this volume. We would like to thank the following sponsoringorganizationsfortheirgenerousfinancialsupportoftheworkshop:Banff International Research Station, Clay Math Institute, Microsoft Research, Pacific Institute for the Mathematical Sciences, and the Number Theory Foundation. We wouldalsoliketothankthemanyreferees,whoseintelligenceandefforthelpedthe authorsimprovethepapersforthisvolume. WINEditorialCommittee: December2015 EllenEischen,UniversityofOregon LingLong,LouisianaStateUniversity RachelPries,ColoradoStateUniversity KatherineE.Stange,UniversityofColoradoBoulder Workshop ParticipantsandAffiliationsatthe Timeofthe Workshop: JenniferBalakrishnan,UniversityofOxford,UnitedKingdom JenniferBerg,UniversityofTexasatAustin,USA IreneBouw,UniversitätUlm,Germany AlinaBucur,UniversityofCaliforniaSanDiego,USA MirelaCiperiani,UniversityofTexasatAustin,USA AlinaCarmenCojocaru,UniversityofIllinoisatChicago,USA RachelDavis,PurdueUniversity,USA AlysonDeines,UniversityofWashington,USA EllenEischen,UniversityofNorthCarolinaatChapelHill,USA YaraElias,McGillUniversity,Canada AmyFeaver,UniversityofColoradoBoulder,USA JessicaFintzen,HarvardUniversity,USA JennyFuselier,HighPointUniversity,USA BonitaGraham,WesleyanUniversity,USA AnnaHaensch,MaxPlanckInstituteforMathematics,Germany WeiHo,ColumbiaUniversity,USA MatildeLalín,UniversitédeMontrèal,Canada JaclynLang,UniversityofCaliforniaatLosAngeles,USA KristinLauter,MicrosoftResearch,USA JingboLiu,WesleyanUniversity,USA LingLong,LouisianaStateUniversity,USA BethMalmskog,ColoradoCollege,USA MichelleManes,UniversityofHawai‘iatMa¯noa,USA ElenaMantovan,CaliforniaInstituteofTechnology,USA Preface ix BahareMirza,McGillUniversity,Canada GabrieleNebe,RWTHAachen,Germany RachelNewton,UniversityofLeiden,Netherlands EkinOzman,UniversityofTexasatAustin,USA JenniferPark,McGillUniversity,Canada LillianPierce,HausdorffCenterforMathematics,Bonn,Germany RachelPries,ColoradoStateUniversity,USA RenateScheidler,UniversityofCalgary,Canada PadmavathiSrinivasan,MassachusettsInstituteofTechnology,USA KatherineE.Stange,UniversityofColoradoBoulder,USA VesnaStojanoska,UniversityofIllinoisatUrbana-Champaign,USA HollySwisher,OregonStateUniversity,USA Fang-TingTu,NationalCenterofTheoreticalSciencesinTaiwan,Taiwan IlaVarma,PrincetonUniversity,USA ChristelleVincent,StanfordUniversity,USA BiancaViray,BrownUniversity,USA KirstenWickelgren,GeorgiaInstituteofTechnology,USA Workshop Website http://www.birs.ca/events/2014/5-day-workshops/14w5009 Eugene,OR,USA EllenE.Eischen BatonRouge,LA,USA LingLong FortCollins,CO,USA RachelPries Boulder,CO,USA KatherineE.Stange May2016
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