TheAlgorithmicNumberTheorySymposiabeganin1994atCornellUniversity inIthaca,NewYorktorecognizethegrowingimportanceofalgorithmicwork in the theory of numbers. The subject of the conference is broadly construed toencompassadiversebodyofmathematics,andtocoverboththetheoretical andpracticaladvancesinthe?eld. Theyhavebeenheldeverytwoyearssince: inBordeaux(Universit eBordeauxI)in1996,Portland(ReedCollege)in1998, Leiden(UniversiteitLeiden)in2000,andthepresentconferencehostedbythe MagmaComputationalAlgebraGroupattheUniversityofSydney. TheconferenceprogramincludedinvitedtalksbyManjulBhargava(Prin- ton),JohnCoates(Cambridge),AntoineJoux(DCSSICryptoLab),BjornP- nen(Berkeley),andTakakazuSatoh(Saitama),aswellas34contributedtalks invariousareasofnumbertheory. Inadditiontothemathematicalprogram,the conferenceincludedaspecialdinnertohonourAlfvanderPoortenofMacquarie University,ontheoccasionofhis60thbirthday. Eachpaperwasreviewedbyatleasttwoexpertsexternaltotheprogram committeeandtheselectionofpaperswasmadeonthebasisoftheserec- mendations. Weexpressourappreciationtothe66expertrefereeswhoprovided reportsonaverytightschedule. Refereeingofthesubmissionfromamemberof theMagmagroupwasorganizedbyJoeBuhler. Theprogramcommitteethanksthegenerousadvicefromorganizersofpre- ousANTSconferences,particularlyJoeBuhler,WiebBosma,HendrikLenstra, andBartdeSmit. TheconferencewasgenerouslysupportedbytheCollegeof ScienceandTechnology,theSchoolofMathematicsandStatistics(bothatthe UniversityofSydney),theAustralianDefenceScienceTechnologyOrganisation, andeSign. April2002 JohnCannon ClausFieker DavidKohel TableofContents InvitedTalks GaussCompositionandGeneralizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 ManjulBhargava EllipticCurves TheCrossroadsofTheoryandComputation. . . . . . . . . . 9 JohnCoates TheWeilandTatePairingsasBuildingBlocks forPublicKeyCryptosystems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 AntoineJoux UsingEllipticCurvesofRankOnetowardstheUndecidability ofHilbert sTenthProblemoverRingsofAlgebraicIntegers. . . . . . . . . . . . . 33 BjornPoonen Onp-adicPointCountingAlgorithmsforEllipticCurves overFiniteFields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 TakakazuSatoh NumberTheory OnArithmeticallyEquivalentNumberFieldsofSmallDegree . . . . . . . . . . . 67 WiebBosma,BartdeSmit ASurveyofDiscriminantCounting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 HenriCohen,FranciscoDiazyDiaz,MichelOlivier AHigher-RankMersenneProblem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 GrahamEverest,PeterRogers,ThomasWard AnApplicationofSiegelModularFunctions toKronecker sLimitFormula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 TakashiFukuda,KeiichiKomatsu ComputationalAspectsofNUCOMP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 MichaelJ. Jacobson,Jr. ,AlfredJ. vanderPoorten E?cientComputationofClassNumbersofRealAbelianNumberFields. . 134 St ephaneR. Louboutin AnAcceleratedBuchmannAlgorithmforRegulatorComputation inRealQuadraticFields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 UlrichVollmer VIII TableofContents ArithmeticGeometry SomeGenus3CurveswithManyPoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 RolandAuer,JaapTop 7 8 Trinomialsax +bx+candax +bx+c withGaloisGroupsofOrder168and8·168. . . . . . . . . . . . . . . . . . . . . . . . . . . 172 NilsBruin,NoamD. Elkies ComputationsonModularJacobianSurfaces. . . . . . . . . . . . . . . . . . . . . . . . . . 189 EnriqueGonz alez-Jim enez,JosepGonz alez,JordiGu`ardia IntegralPointsonPuncturedAbelianSurfaces. . . . . . . . . . . . . . . . . . . . . . . . . 198 AndrewKresch,YuriTschinkel Genus2Curveswith(3,3)-SplitJacobian andLargeAutomorphismGroup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 TonyShaska