Algorithmic Number Theory

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

Includes supplementary material: sn.pub/extras

Klappentext

. 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. . . . . . . . . . . .

Zusammenfassung

From the reviews:

"The book contains 39 articles about computational algebraic number theory, arithmetic geometry and cryptography. ... The articles in this book reflect the broad interest of the organizing committee and the participants. The emphasis lies on the mathematical theory as well as on computational results. We recommend the book to students and researchers who want to read about current research in number theory and arithmetic geometry and its applications." (R. Carls, Nieuw Archief voor Wiskunde, Vol. 6 (3), 2005)

Inhalt
Invited Talks.- Gauss Composition and Generalizations.- Elliptic Curves The Crossroads of Theory and Computation.- The Weil and Tate Pairings as Building Blocks for Public Key Cryptosystems.- Using Elliptic Curves of Rank One towards the Undecidability of Hilbert's Tenth Problem over Rings of Algebraic Integers.- On p-adic Point Counting Algorithms for Elliptic Curves over Finite Fields.- Number Theory.- On Arithmetically Equivalent Number Fields of Small Degree.- A Survey of Discriminant Counting.- A Higher-Rank Mersenne Problem.- An Application of Siegel Modular Functions to Kronecker's Limit Formula.- Computational Aspects of NUCOMP.- Efficient Computation of Class Numbers of Real Abelian Number Fields.- An Accelerated Buchmann Algorithm for Regulator Computation in Real Quadratic Fields.- Arithmetic Geometry.- Some Genus 3 Curves with Many Points.- Trinomials ax 7 + bx + c and ax 8 + bx + c with Galois Groups of Order 168 and 8 · 168.- Computations on Modular Jacobian Surfaces.- Integral Points on Punctured Abelian Surfaces.- Genus 2 Curves with (3, 3)-Split Jacobian and Large Automorphism Group.- Transportable Modular Symbols and the Intersection Pairing.- Elliptic Curves and CM.- Action of Modular Correspondences around CM Points.- Curves Dy 2 = x 3 x of Odd Analytic Rank.- Comparing Invariants for Class Fields …