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On the torsion structure of elliptic curves over cubic number fields
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On the torsion structure of elliptic curves over cubic number fields
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Onthetorsionstructureofellipticcurvesovercubicnumberfields. by JianWang ————————————————————————– ADissertationPresentedtothe FACULTYOFTHEUSCGRADUATESCHOOL UNIVERSITYOFSOUTHERNCALIFORNIA InPartialFulfillmentofthe RequirementsfortheDegree DOCTOROFPHILOSOPHY (MATHEMATICS) August2015 Copyright2015 JianWang ON THE TORSION STRUCTURE OF ELLIPTIC CURVES OVER CUBIC NUMBER FIELDS JIAN WANG wang50@usc.edu April22,2015 CONTENTS DEDICATION III ACKNOWLEDGEMENTS IV ABSTRACT V CHAPTER 1 INTRODUCTION 1 1.1 EllipticcurvesoverC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 EllipticcurvesoverF q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 CHAPTER 2 MODULAR CURVES OVERC 6 2.1 Modularcurves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Heckeoperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Modularforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 TheJacobians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 CHAPTER 3 MODULAR CURVES OVERZ[1/N] 17 3.1 Ellipticcurvesoverarbitraryschemes . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Y 1 (N) Z[1/N] asafinemodulispace . . . . . . . . . . . . . . . . . . . . . . . . 18 3.3 Y 0 (N) Z[1/N] asacoarsemodulispace . . . . . . . . . . . . . . . . . . . . . . . 20 I 3.4 Generalizedellipticcurves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 CHAPTER 4 KAMIENNY’S CRITERION 23 4.1 Groupschemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2 N´ eronmodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.3 Formalimmersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.4 Kamienny’scriterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 CHAPTER 5 CUBIC POINTS ONX 1 (N): PRIME POWER CASES 30 5.1 caseX 1 (169) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.2 caseX 1 (121) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.3 caseX 1 (49) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.4 caseX 1 (25) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.5 caseX 1 (32) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.6 caseX 1 (27) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 CHAPTER 6 CUBIC POINTS ONX 1 (N): COMPOSITE CASES 38 6.1 caseX 1 (143) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 6.2 caseX 1 (91) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 6.3 caseX 1 (65) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6.4 caseX 1 (77) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6.5 caseX 1 (55) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 6.6 caseX 1 (40) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 6.7 caseX 1 (22) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 6.8 caseX 1 (24) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 APPENDIX 45 BIBLIOGRAPHY 53 II DEDICATION Tomyparents. III ACKNOWLEDGEMENTS I would like to express my deepest gratitude to my advisor, Professor Sheldon Kamienny, for hisexcellentinspirationandpatience.Prof.Kamiennytaughtmearithmeticalgebraicgeometry fromthebasicknowledgetotheadvancedtechniques.Heprovidedthebestencouragementand advicesformystudyandresearch. IsincerelythankProfessorRobertGuralnick,ProfessorEricFriedlander,ProfessorAravind AsokandProfessorLeanaGolubchik,whoworkonmyqualifyingand/ordissertationcommit- tee.SpecialthanksgoestoProfessorCharlesLanskiforhelpingwithmyteachingskills. I would like to thank my fellow students: Yin Tian, Tian Zhang, Burton Newman, Ozlem Ejder and many more. I would also like to thank the department of mathematics for providing mewithanexcellentatmospherefordoingresearch. I would like to thank Andrew Sutherland at MIT and Andreas Schweizer at KAIST for valuablecommentsandcorrectionsoferrorsinthedraftofthisthesis. IV ABSTRACT LetE beanellipticcurvedefinedoveranumberfieldK.ThenitsMordell-WeilgroupE(K)is finitely generated: E(K) ∼ = E(K) tor ×Z r . For a fixed E/K, the torsion component E(K) tor canbecalculated.However,forageneralclassofellipticcurves,itisusuallydifficulttolistall thepossiblestructuresofE(K) tor .Ford≥ 1,wecanconsiderthefollowingproblem:whatare thepossiblestructuresofE(K) tor with[K :Q] =d? Ford = 1,2, the problem has been solved by Mazur [34] and Kenku-Momose [24], Kami- enny[17].Ford = 3,theworkofParent[38],Jeon-Kim-Schweizer[16]andNajman[36]lead ustoconjecturethatthepossiblestructuresofthetorsionofanellipticcurveovercubicnumber fieldsare: Z/mZ, m = 1−16,18,20−21; Z/2Z×Z/2mZ, m = 1−7. In this thesis we studied the cyclic case, and obtained complete results in the prime power case:supposeE isanellipticcurveoveracubicnumberfieldK andN =p n isaprimepower, then the cyclic groupZ/NZ does not occur as a subgroup of E(K) tor for N =32, 27, 25, 49, 121,169. This is a complete result since Parent [38] showed that the prime divisors of E(K) tor are ≤ 13. In order to prove that a cyclic subgroup of E(K) tor is exactly one of those shown in Conjecture1.1.2,weneedtoeliminatethecaseN = 2·13,3·13,5·13,7·13,11·13;2·11,3· 11,5·11,7·11;4·7,5·7,6·7,9·7;8·5,6·5,9·5;8·3,4·9.Forthesecompositecase,wegot the following partial results: supposeE is an elliptic curve over a cubic number fieldK andN hasmorethanoneprimedivisors,thenthecyclicgroupZ/NZdoesnotoccurasasubgroupof E(K) tor forN =143,91,65,77,55,22,40,24. V CHAPTER1 INTRODUCTION 1.1 ELLIPTIC CURVES OVERC LetE beanellipticcurvedefinedoveranumberfieldK.ThenitsMordell-WeilgroupE(K)is finitelygenerated: E(K) ∼ = E(K) tor ×Z r For a fixed E/K, the torsion component E(K) tor can be calculated. Actually, for an elliptic curveE/K withWeierstrassequation y 2 =x 3 +Ax+B such that A,B ∈ O K . Let ∆ = 4A 2 + 27B 2 and let P ∈ E(K) tor be a point. Then y(P) 2 is essentially bounded by ∆ (See [43] Exercise 8.11). However, for a general class of elliptic curves, it is usually difficult to list all the possible structures of E(K) tor . We can consider the following: Problem1.1.1. For a natural numberd, what are the possible structures ofE(K) tor with[K : Q] =d? Ford = 1,i.e.K =Q,Mazur[34]provedthatthetorsiongroupE(Q) tor ofanellipticcurve E overtherationalnumberfieldmustbeisomorphictooneofthefollowing: Z/mZ, m = 1−10,12; Z/2Z×Z/2mZ, m = 1−4. 1 For d = 2, the possible structures of the torsion of an elliptic curve over quadratic number fieldsweredeterminedbyKenku-Momose[24]andKamienny[17]: Z/mZ, m = 1−16,18; Z/2Z×Z/2mZ, m = 1−6; Z/3Z×Z/3mZ, m = 1−2; Z/4Z×Z/4Z. Ford = 3,Jeon, Kim, Schweizer [16] found all the torsion structures that appears infinitely oftenwhenonerunsthroughallellipticcurveoverallcubicfields: Z/mZ, m = 1−16,18,20; Z/2Z×Z/2mZ, m = 1−7. Najman [36] found a sporadic point on X 1 (21), so thatZ/21Z can also serve as torsion of ellipticcurvesovercubicfields.Wecanconjecturethat: Conjecture1.1.2. The possible structures of the torsion of an elliptic curve over cubic number fields are: Z/mZ, m = 1−16,18,20−21; Z/2Z×Z/2mZ, m = 1−7. In this thesis, we will discuss the cyclic case. The main results are Theorem 1.1.3 and The- orem1.1.4. Theorem 1.1.3. Suppose E is an elliptic curve over an cubic field K. If N = p n is a prime power, then the cyclic groupZ/NZ does not occur as a subgroup of E(K) tor for N =32, 27, 25, 49, 121, 169. Theorem 1.1.3 is a complete result since Parent [38] showed that the prime divisors of E(K) tor are≤ 13. In order to prove that a cyclic subgroup ofE(K) tor is exactly one of those shown in Conjecture 1.1.2, we need to eliminate the case N = 2·13,3·13,5·13,7·13,11· 13;2·11,3·11,5·11,7·11;4·7,5·7,6·7,9·7;8·5,6·5,9·5;8·3,4·9.. Theorem 1.1.4 givesapartialresult. Theorem 1.1.4. Suppose E is an elliptic curve over an cubic field K. If N has more than one prime divisors, then the cyclic groupZ/NZ does not occur as a subgroup of E(K) tor for N =143, 91, 65, 77, 55, 22, 40, 24. As is done for d = 1 and 2, the key idea is to consider the modular curves X 1 (N) and the mapπ : X 1 (N)−→ X 0 (N). We want to prove the nonexistence of cubic points onX 1 (N) for certainN. 2 For this purpose, we will use the Igusa’s theorem [14] which states thatX 1 (N) andX 0 (N) havegoodreductionatp-N. LetJ 1 (N) (respectively,J 0 (N)) be the Jacobian ofX 1 (N) (respectively,X 0 (N)). Then we have a canonical embedding: X 1 (N) ,→ J 1 (N) (respectively, X 0 (N) ,→ J 0 (N)). Suppose A is a quotient ofJ 0 (N), andL(A,s) is theL series associated toA (See [42] for the definition). Kolyvagin and Logach¨ ev [28] proves a Birch-Swinnerton-Dyer conjecture for modular abelian varieties, which states thatA has Mordel-Weil rank 0 ifL(A,1)̸= 0. For abelian varieties that areaquotientofJ 1 (N) Q ,thesameresultwasprovedbyKato[20]. Withtheincreasingofdegreed,thebound(1+ √ p d ) 2 predictedbytheRiemannhypothesis increases drastically. If we do reduction forX 1 (N) modulop, the points on e X 1 (N) may repre- sents not only cusps even for the small N ′ s we are interested in. Therefore, the methods used ford = 1and2arenotenough.Wewillusethreemethods: Method A: If N is big enough such that elliptic curves overF p 3 for a suitable p can not admitZ/NZasasubgroup,andifthedecomposition J 1 (N)−→A 1 ×···×A n hasallA i finite,i.e.L(A i ,1)̸= 0,1≤ i≤ n,wecanshowthenonexistenceofcubicpointson X 1 (N)inarelativelyeasyway. Method B: If we can not show all the quotients of J 1 (N) are finite, we can consider the decomposition J 0 (N)−→B 1 ×···×B m and collect the B i such that L(B i ,1) ̸= 0. Then investigate Hecke correspondences T 1 ,T 2 ,T 3 onthefinitequotientandmakeuseofKamienny’scriterion. Method C: If none of the two methods above work, we have to make use of reduction of the defining equations of X 0 (N). By investigating the Atkin-Lehner involutions, we may find somecontradiction. 1.2 ELLIPTIC CURVES OVERF q Letk =F q bethefinitefieldwithq =p n elements.LetE/k beanellipticcurveoverk.Wecan definethezetafunctionofE: Z E (x) = 1−a q (E)x+qx 2 (1−x)(1−qx) Herea q (E) =q+1−N q (E)andN q (E) = #E(k),thenumberofpointsofE overk. 3 The Riemann hypothesis for E is then the assertion that if Z E (q −s ) = 0, then Re(s) = 1 2 . WecanalsorephrasetheRiemannHypothesisasaboundona q .Infact,thebound |N q −q−1|≤ 2 √ q is equivalent to the Riemann hypothesis for E. Let a q (E) = t, E is called ordinary if (t,q) = 1, otherwise it is called supersingular. In the range proposed by Riemann Hypothesis, all the ordinarytappear,whilethesupersingulartonlyappearsinrestrictedcase.Explicitly,wehave: Theorem1.2.1 (Waterhouse [47]). The isogeny classes of elliptic curves overk are in one-to- one correspondence with the rational integers t having |t| ≤ 2 √ q and satisfying one of the following conditions: (1)(t,p) = 1; (2) Ifn is even:t =±2 √ q; (3) Ifn is even andp̸≡ 1 mod 3:t =± √ q; (4) Ifn is odd andp = 2 or3:t =±p n+1 2 ; (5) If either (i)n is odd or (ii)n is even andp̸≡ 1 mod 4:t = 0. The first of these are ordinary; the second are supersingular and have all their endomorphisms definedoverk;therestaresupersingularbutdonothavealltheirendomorphismsdefinedover k. TheFrobeniusendomorphismπ : (x,y)7−→ (x q ,y q )ofE satisfies π 2 −tπ+q = 0 The algebra A =Q⊗End k (E) is semisimple with center Φ =Q(π). Deuring [8] also clas- sified the endomorphism rings of elliptic curves over finite fields. Here we use the results of Waterhouse[47]instead. Theorem1.2.2(Waterhouse[47]). LetA =Q⊗End k (E)betheendomorphismalgebraofan isogenyclassofellipticcurves.TheordersinAwhichareendomorphismringsofcurvesinthe class are as follows: (1) If the curves are ordinary: all orders containingπ; (2) If the curves are supersingular with all endomorphisms defined: the maximal orders; (3) If the curves are supersingular with not all endomorphisms defined: the orders which containπ and are maximal atp, i.e. have conductors prime top. IfE isordinary,thenA =Q(π).SoEnd(E) ∼ =O(D)forsomeD|t 2 −4q,where O(D) =Z+ D+ √ D 2 Z 4 istheimaginaryquadraticorderofsomediscriminantD.Thereareh ′ (D)isomorphismclasses ofellipticcurvesdefinedoverF q withO(D)astheendomorphismring.Hereh ′ (D)istheclass numberoftheorderO(D). Kohel [27] gives the method to calculate End(E). We call an isogeny ϕ an ℓ-isogeny if |kerϕ| = ℓ. We restrict to prime ℓ. The classical modular polynomial ϕ ℓ ∈Z[X,Y] has the property ϕ ℓ (j(E 1 ),j(E 2 )) = 0⇐⇒j(E 1 )andj(E 2 )areℓisogenous Theℓ-isogenygraphG ℓ (F q )hasvertexset E(F q ) ={j(E/F q )} =F q and edges (j 1 ,j 2 ) for ϕ ℓ (j 1 ,j 2 ) = 0. The ordinary connected components of G ℓ (F q ) are ℓ- volcanoes. Anℓ-volcano of heighth has vertices in levelV 0 ,··· ,V h . Vertices inV 0 have endo- morphismringO(D 0 )withℓ-D 0 .VerticesinV k haveendomorphismringO(ℓ 2k D 0 ). Proposition1.2.3(Kohel[27]). Theℓ-volcano can be described as: (a) The subgraph onV 0 is a cycle. All other edges lie betweenV k andV k+1 for somek. (b) Fork > 0, each vertex inV k has one neighbor inV k−1 . (c) Fork <h, every vertex inV k has degreeℓ+1. 5 CHAPTER2 MODULARCURVESOVERC Modular curves are moduli spaces of elliptic curves with level structures. They are constructed asthequotientsofthe(extended)complexupperhalfplanebytheactionsofcongruentgroups. Inthischapter,weadapttheexpositorytreatmentsinStevens[45]andDiamond-Shurman[10]. 2.1 MODULAR CURVES LetH ={z ∈C|Im(z) > 0} be the upper half plane. We adjoin the cuspsQ∪{∞} toH and getthe extended upper half plane H ∗ =H∪Q∪{∞} WedefineatopologyonH ∗ byspecifyingabaseofneighborhoodsU ϵ (ϵ> 0)asfollows: U ϵ (z) ={w∈H| |w−z|<ϵ}, z∈H U ϵ (z) ={w∈H| |w−(z+iϵ)|<ϵ}∪{z}, z∈Q U ϵ (z) ={w∈H| Im(w)>ϵ}∪{∞}, z =∞ ThegroupGL + 2 (Q) ={α∈GL 2 (Q)| |α|> 0}actsonH by α = ( a b c d ) :z7−→αz = az+b cz+d ∈H ThisactionextendscontinuouslytoH ∗ andpreservesthecusps. 6 ForapositiveintegerN,wedefinethe principal congruence groupΓ(N)by Γ(N) = { α = ( a b c d ) ∈SL 2 (Z)|α≡± ( 1 0 0 1 ) mod N } Definition 2.1.1. The congruence groups are the subgroups, Γ ⊆ SL 2 (Z), which contain a principalcongruencegroup.ThesmallestintegerN > 0forwhich Γ(N)⊆ Γ iscalledthelevelofΓ. Inthisthesis,wearemainlyinterestedinthefollowingcongruencegroups: Γ 0 (N) = {( a b c d ) ∈SL 2 (Z)|c≡ 0 mod N } Γ 1 (N) = {( a b c d ) ∈ Γ 0 (N)|a≡d≡±1 mod N } IfN 1 ,N 2 arepositiveintegers,letN = lcm(N 1 ,N 2 )and Γ 0 (N 1 ,N 2 ) = {( a b c d ) ∈ Γ 0 (N 1 )|b≡ 0 mod N 2 } Γ 1 (N 1 ,N 2 ) = {( a b c d ) ∈ Γ 0 (N 1 ,N 2 )|a≡d≡±1 mod N } IfanintermediategroupΓsatisfies Γ 1 (N 1 ,N 2 )⊆ Γ⊆ Γ 0 (N 1 ,N 2 ), wesayΓisagroupoftype(N 1 ,N 2 ). ForacongruencegroupΓ,definethemodularcurves Y(Γ) = Γ\H X(Γ) = Γ\H ∗ Notation. Forsimplification,weusethefollowingnotation: X(N) =X(Γ(N)), Y(N) =Y(Γ(N)); X i (N) =X(Γ i (N)), Y i (N) =Y(Γ i (N)), i = 0,1; X i (N 1 ,N 2 ) =X(Γ i (N 1 ,N 2 )), Y i (N 1 ,N 2 ) =Y(Γ i (N 1 ,N 2 )), i = 0,1. Proposition2.1.2([10],§2.4). ThemodularcurveX(Γ)isHausdorff,connected,andcompact. Moreover, it can be given the structure of compact Riemann surface. 7 There are finitely many Γ-orbits inQ∪{∞} which represent the finitely many cusps of X(Γ).Denotethissetbycusps(Γ). LetΓbeacongruencegroupoftype(N 1 ,N 2 ).Thencusps(Γ)maybeidentifiedwiththeset {( x y ) ∈Z 2 |(x,y,N 1 ,N 2 ) = 1 } /∼ wheretheequivalencerelationisdefinedby ( x y ) ∼ ( x ′ y ′ ) ⇐⇒ ( x ′ y ′ ) = ( ax+by cx+dy ) forsome ( a b c d ) ∈ Γ Denotetheequivalenceclassof ( x y ) by[ x y ]. LetN = lcm(N 1 ,N 2 ),thenwehavethefollowing: (1) ( x y ) ≡± ( x ′ y ′ ) mod N =⇒ [ x y ] = [ x ′ y ′ ] (2) [ x+N 2 y y ] = [ x y ] = [ x y+N 1 x ] The modular curvesX(Γ) have the structure of nonsingular projective algebraic curves de- finedoverabelianextensionofQ.Infact, Proposition 2.1.3 ([42],§6.7). The curves X(Γ) for Γ of type (N 1 ,N 2 ) have models defined overQ such that the sequence of projections X 1 (N 1 ,N 2 )−→X(Γ)−→X 0 (N 1 ,N 2 ) and projections to lower levels are defined overQ. If Γ is an arbitrary congruence group of levelN, thenX(Γ) has a model defined overQ(ζ N ), whereζ N =e 2πi/N . Themodularcurvesaremodulispacesofellipticcurveswithlevelstructure. Theorem2.1.4([10],Theorem1.5.1). 1. There is a bijection{(E,C)} ∼ −−→ Y 0 (N) where (E,C)isanisomorphismclasswithE anellipticcurveandC acyclicsubgroupoforder N. 2. There is a bijection{(E,P)} ∼ −−→Y 1 (N) where(E,P) is an isomorphism class withE an elliptic curve andP a point of orderN. For each divisor N ′ of N such that (N ′ ,N/N ′ ) = 1, we have an Atkin-Lehner involution w N ′ onX 0 (N).Forx = (E,C)∈X 0 (N),itisdefinedby w N ′(E,C) := (E/C[N ′ ],(C +E[N ′ ])/C[N ′ ]) 8 Theysatisfythemultiplicationrule: w N 1 ◦w N 2 =w lcm(N 1 ,N 2 )/gcd(N 1 ,N 2 ) For each natural number ℓ, there is a Hecke correspondence T ℓ (See§2.2 for its definition) onX 0 (N).Forx = (E,C)∈X 0 (N),itisdefinedby T ℓ (x) = ∑ G<E,|G|=ℓ,G∩C={O} (E/G,(C +G)/G) AllT ℓ aremutuallycommutativeamongeachotherandsatisfies T m T n =T mn ,(m,n) = 1 2.2 HECKE OPERATORS Let N 1 ,N 2 be positive integers and Γ be a congruence group of type (N 1 ,N 2 ). For each g ∈ GL + 2 (Q),let Γ(g) = Γ∩g −1 Γg SinceΓ(g)⊆ Γ,thereisanaturalsurjection X(Γ(g)) π(g) −−→X(Γ) SincegΓ(g)g −1 = Γ(g −1 ),thereisamap X(Γ(g)) g −−→X(Γ(g −1 )) Nowconsiderthediagram X(Γ(g)) π(g) g // X(Γ(g −1 )) π(g −1 ) X(Γ) X(Γ) Thisdiagramdefinesaproperalgebraiccorrespondence,T(g),onX(Γ)×X(Γ)by T(g) :X(Γ(g)) π(g)×(π(g −1 )◦g) −−−−−−−−−→X(Γ)×X(Γ) WecallT(g)theHeckecorrespondenceassociatedtog. The correspondence T(g) may also be described in terms of the double coset ΓgΓ. Write ΓgΓasadisjointunionofrightcosets, ΓgΓ = k ⨿ i=1 g i Γ 9 andlet π Γ :H ∗ −→X(Γ) bethenaturalprojection.Then T(g)◦π Γ (z) = k ∑ i=1 π Γ (g i z) The Hecke correspondences define operators on the homology groups H ∗ (X(Γ);A) by T(g) 7−→ π(g −1 ) ∗ ◦ g ◦ π(g) ∗ , and on the cohomology groups H ∗ (X(Γ);A) by T(g) 7−→ π(g) ∗ ◦g −1 ◦π(g −1 ) ∗ ,whereAisanyabeliangroup. Foreachprimeℓ,letλ ℓ ∈GL + 2 (Q)bethematrix λ ℓ = ( 1 0 0 ℓ ) We then write T ℓ for the Hecke correspondence T(λ ℓ ). Let N = lcm(N 1 ,N 2 ). For each m ∈ (Z/NZ) ∗ ,letσ m ∈SL 2 (Z)suchthat σ m ≡ ( ∗ 0 0 m ) mod N andwrite⟨m⟩forT(σ m ).Forz∈H ∗ ,thesecorrespondencearegivenby T ℓ ◦π Γ (z) = ℓ−1 ∑ k=0 π Γ ( z+N 2 k ℓ ) +π Γ (σ ℓ (ℓz)), ifℓ-N ℓ−1 ∑ k=0 π Γ ( z+N 2 k ℓ ) , ifℓ|N and ⟨m⟩◦π Γ (z) =π Γ (σ m (z)) Theseoperatorsaremutuallycommutative.The Hecke algebraT Z forΓisthecommutative algebrageneratedbyHeckeoperatorsoverZ: T Z =Z[T ℓ ,⟨m⟩|ℓprime,m∈ (Z/NZ) ∗ ] WeextendthedefinitionsoftheHeckeoperatorstoT n and⟨n⟩foralln∈Z >0 by T n = Id, ifn = 1 T p T p r−1−⟨p⟩T p r−2, ifn =p r ∏ T p e i i , ifn = ∏ p e i i ⟨n⟩ = { ⟨n⟩, if(n,N) = 1 0, if(n,N)̸= 1 10 2.3 MODULAR FORMS For any matrixγ = ( a b c d ) ∈ SL 2 (Z) and any integerk, define the weight-k operator [γ] k onmeromorphicfunctionsf :H−→Cby (f[γ] k )(τ) = (cτ +d) −k f(γ(τ)), τ ∈H Wesayf is weight-k invariant underΓif f[γ] k =f forallγ∈ Γ ForacongruencegroupΓ,sinceΓ⊇ Γ(N)forsomeN,thenitcontainsatranslationmatrix oftheform ( 1 h 0 1 ) :τ 7−→τ +h forsomeminimalh∈Z + .Everymeromorphicfunctionf :H−→Cthatisweight-k invariant underΓ has a corresponding functiong : D ′ −→C, whereD ′ ={q∈C||q| < 1,q̸= 0} is the punctureddisk,suchthatf(τ) =g(q h )whereq h =e 2πiτ/h . Iff isalsoholomorphiconHthengisholomorphiconD ′ andsoithasaLaurentexpansion. We say f is holomorphic at∞ if g extends holomorphically to q = 0. In this case, f has a Fourierexpansion: f(τ) = ∞ ∑ n=0 a n q n h Definition 2.3.1 ([10], 1.2.3). Let Γ be a congruent group and let k be an integer. A function f : H −→ C is a modular form of weight k with respect to Γ if the following (1)-(3) are satisfied 1. f isholomorphic, 2. f isweight-k invariantunderΓ, 3. f[α] k isholomorphicat∞forallα∈SL 2 (Z), 4. a 0 = 0intheFourierexpansionoff[α] k forallα∈SL 2 (Z). Ifinaddition(4)isalsosatisfied,thenf isacuspformofweightk withrespecttoΓ.Thespace ofmodularformsisdenotedM k (Γ),thatofcuspformsS k (Γ). The hyperbolic measureontheupperhalfplaneH isgivenby dµ(τ) = dxdy y 2 , τ =x+iy∈H 11 The volumeofX(Γ)isgivenby V Γ = ∫ X(Γ) dµ(τ) Definition2.3.2([10],5.4.1). LetΓbeacongruencegroup.The Petersson inner product ⟨−,−⟩ Γ :S k (Γ)×S k (Γ)−→C isgivenby ⟨f,g⟩ Γ = 1 V Γ ∫ X(Γ) f(τ)g(τ)(Im(τ)) k dµ(τ) SupposeM|N andd|(N/M).Let α d = ( d 0 0 1 ) Thentheinjectivelinearmap[α d ] k takesS k (Γ 1 (M))toS k (Γ 1 (N))[10,Exercise1.2.11]. Definition2.3.3([10],5.6.1). ForeachdivisordofN,leti d bethemap i d : (S k (Γ 1 (N/d))) 2 −→S k (Γ 1 (N)) givenby (f,g)7−→f +g[α d ] k Thesubspaceof oldformsatlevelN is S k (Γ 1 (N)) old = ∑ p|N prime i p ((S k (Γ 1 (N/p))) 2 ) and the subspace of newforms at level N is the orthogonal complement with respect to the Peterssoninnerproduct, S k (Γ 1 (N)) new = (S k (Γ 1 (N)) old ) ⊥ Let us recall some knowledge from functional analysis. If V is an inner product space and T isalinearoperatoronV,thenthe adjoint operatorT ∗ isthelinearoperatoronV definedby ⟨Tv,w⟩ =⟨v,T ∗ w⟩, forallv,w∈V TheoperatorT iscalled normalifTT ∗ =T ∗ T. Theorem 2.3.4 ([10], Theorem 5.5.3). In the space S k (Γ 1 (N)) endowed with the Petersson inner product, the Hecke operators⟨p⟩ andT p forp-N have adjionts ⟨p⟩ ∗ =⟨p⟩ −1 , T ∗ p =⟨p⟩ −1 T p Thus the Hecke operators⟨n⟩ andT n for(n,N) = 1 are normal. 12 LetV beafinite-dimensionalinnerproductspace.Let{T α } α∈I beafamilyofnormalopera- torsonV suchthatT α T β =T β T α forallα,β∈I.ThenitisafactinlinearalgebrathatV hasan orthogonal basis of simultaneous eigenvectors for{T α } α∈I . Since we are interested in modular forms, we say eigenform instead of eigenvectors. We say the eigenformf(τ) = ∑ ∞ n=0 a n (f)q n is normalized ifa 1 (f) = 1. Theorem2.3.5([10],Theorem5.5.4,5.8.2). 1. ThespaceS k (Γ 1 (N))hasanorthogonalba- sis of simultaneous eigenforms for the Hecke operators{⟨n⟩,T n |(n,N) = 1}. 2. Letf ∈S k (Γ 1 (N)) new beanonzeroeigenformfortheHeckeoperators{⟨n⟩,T n |(n,N) = 1}. Thenf is an eigenform for all Hecke operators{⟨n⟩,T n |n∈Z + }. 3. The set of normalized eigenform inS k (Γ 1 (N)) new is an orthogonal basis of the space. each such form satisfiesT n f =a n (f)f for alln∈Z + . 2.4 THE JACOBIANS Let X be a compact Riemann surface of genus g ≥ 1. ViewingX as a sphere with g handles, letα 1 ,··· ,α g be the longitudinal loops, andβ 1 ,··· ,β g the latitudinal loops. LetH 1 (X,Z) be the first homology group ofX. Then by Riemann surface theory (See for example Farkas-Kra [13]),wecanidentifyH 1 (X,Z)withtheZ-spanoftheproductofintegrals: H 1 (X,Z) =Z ∫ α 1 ×···×Z ∫ αg ×Z ∫ β 1 ×···×Z ∫ βg LetΩ 1 hol (X)bethedegree1holomorphicdifferentialsonX.LetΩ 1 hol (X) ∧ = Hom C (Ω 1 hol (X),C), thevectorspaceofC-linearmapsfromΩ 1 hol (X)toC.ThenitisalsoaresultofRiemannsurface theory[13]that Ω 1 hol (X) ∧ =R ∫ α 1 ×···×R ∫ αg ×R ∫ β 1 ×···×R ∫ βg Definition2.4.1([10],6.1.1). The JacobianofX isthequotientgroup Jac(X) = Ω 1 hol (X) ∧ /H 1 (X,Z) ∼ =C g /Z 2g Remark 2.4.2. For modular curves X(Γ), there is an isomorphism between weight-2 cusp formsandholomorphicdifferentialsonX(Γ)[10,Exercise3.3.6]: ω :S 2 (Γ) ∼ −−→ Ω 1 hol (X(Γ)) Therefore,theJacobiansofmodularcurvescanalsobedefinedas Jac(X) =S 2 (Γ) ∧ /H 1 (X,Z) 13 The group of degree zero divisorsofX is Div 0 (X) = { ∑ x∈X n x x n x ∈Z,n x = 0forallbutfinitelyx, ∑ x∈X n x = 0 } The subgroup of principal divisorsis Div ℓ (X) ={D∈ Div 0 (X)|D =div(f)forsomemeromorphicfunctionf onX} Definition2.4.3. The Picard groupofX isthequotientgroup Pic 0 (X) = Div 0 (X)/Div ℓ (X) Fixabasepointx 0 ∈X,thenX embedsinitsPicardgroupviathemap X −→ Pic 0 (X), x7−→ [x−x 0 ] Themap Div 0 (X)−→Jac(X), ∑ x n x x7−→ ∑ x n x ∫ x x 0 iswelldefinedandwehave Theorem2.4.4(Abel-Jacobi[10]§6.1). The above map induces an isomorphism Pic 0 (X) ∼ −−→Jac(X) Suppose we have a nonconstant holomorphic map h : X −→ Y of compact Riemann surfaces. Then h induces the forward map h P and the reverse map h P on the corresponding Picardgroups[10,6.2.2,6.2.5]: h P : Pic 0 (X)−→ Pic 0 (Y), h P [ ∑ x n x x ] = [ ∑ x n x h(x) ] h P : Pic 0 (Y)−→ Pic 0 (X), h P [ ∑ y n y y ] = ∑ y n y ∑ x∈h −1 (y) e x x wheree x istheramificationdegreeatx. Definition 2.4.5. Let Γ be a congruence group of type (N 1 ,N 2 ). For each g ∈ GL + 2 (Q), the Hecke operator onJ(Γ)isdefinedbyT(g) =π(g −1 ) P ◦g P ◦π(g) P viathesequence Pic 0 (X(Γ)) π(g) P − −− → Pic 0 (X(Γ(g))) g P −−−→ Pic 0 (X(Γ(g −1 ))) π(g −1 ) P − −−−− → Pic 0 (X(Γ)) 14 Letf ∈S 2 (Γ 1 (M f )) new beanewformatlevelM f .ByTheorem2.3.5,f isaneigenformof theHeckealgebraT Z =Z[{T n ,⟨n⟩|n∈Z + }].Theeigenvaluemap λ f :T Z −→C, Tf =λ f (T)f haskernel I f = ker(λ f ) ={T ∈T Z |Tf = 0} Themapλ f inducesaZ-moduleisomorphism T Z /I f ∼ −−−→Z[{a n (f)}] Let I f J 1 (M f ) ={[ϕ◦T] | T ∈I f ,[ϕ]∈J 1 (M f )} Definition2.4.6([10],6.6.3). The Abelian variety associated tof isdefinedasthequotient A f =J 1 (M f )/I f J 1 (M f ) Wedefineanequivalencerelationonnewforms e f ∼f ⇐⇒ e f =f σ forsomeautomorphismσ :C−→C Thecardinalityoftheequivalenceclass[f]isthenumberofembeddings σ :K f =Q[{a n (f)}],−→C Theorem 2.4.7 ([10], Theorem 6.6.6). The Jacobian J 1 (N) is isogenous to a direct sum of Abelian varieties associated to equivalence classes of newforms, J 1 (N)−→ ⊕ f A m f f Here the sum is taken over a set of representatives f ∈ S 2 (Γ 1 (M f )) at levels M f dividing N, and eachm f is the number of divisors ofN/M f . Similarly,forf ∈S 2 (Γ 0 (M f )) new ,wehave Definition2.4.8([10],§6.6). The Abelian variety associated tof isdefinedasthequotient B f =J 0 (M f )/I f J 0 (M f ) Theorem2.4.9 ([10],§6.6). The JacobianJ 0 (N) is isogenous to a direct sum of Abelian vari- eties associated to equivalence classes of newforms, J 0 (N)−→ ⊕ f B m f f Here the sum is taken over a set of representatives f ∈ S 2 (Γ 0 (M f )) at levels M f dividing N, and eachm f is the number of divisors ofN/M f . 15 Definition2.4.10. TheL-functionofA f isdefinedby L(A f ,s) = ∏ σ:K f ,→C L(f σ ,s) = ∏ σ:K f ,→C ∞ ∑ n=1 a n (f σ )n −s Theorem 2.4.11 (Kolyvagin-Logach¨ ev [28]). Let A/Q be an abelian variety, which is isoge- nous to an abelian subvariety ofJ 0 (N) such thatL(A;1)̸= 0, thenA(Q) has rank0. Theorem 2.4.12 (Kato [20]). Suppose ∆ is an intermediate group Γ 1 (N)⊆ ∆⊆ Γ 0 (N). Let A/Q be an abelian variety, which is isogenous to an abelian subvariety of J ∆ (N) such that L(A;1)̸= 0, thenA(Q) has rank0. 16 CHAPTER3 MODULARCURVESOVERZ[1/N] ThecomplexalgebraicX 0 (N) C ,associatedtotheRiemannsurfacesX 0 (N),hasamodelX 0 (N) Q definedoverQ.Furthermore,wehaveamodelX 0 (N) Z overZ. The model X 0 (N) Q is good enough to study most of questions concerning rationality of points on and morphisms between the X 0 (N) Q . However, in order to consider the reduction of X 0 (N) Z at primes not dividing N, Deligne-Rapoport [7] and Katz-Mazur [21] developed a moreconvenientapproachbyinterpretingX 0 (N) Z overZ[1/N]asamodulispace.Hereweuse theexpositorytreatmentsinBruin[5],Edixhoven[12]andDiamond-Im[9]. 3.1 ELLIPTIC CURVES OVER ARBITRARY SCHEMES LetSch bethecategoryofschemes.ForaschemeS,letSch/S bethecategoryofS-schemes. Wehavethefollowingdefinitionofellipticcurvesoverschemes. Definition 3.1.1 ([5]). LetS be a scheme. An elliptic curveE overS is then a proper smooth morphism of schemes p : E −→ S whose fibres are geometrically connected curves of genus one,togetherwithasectionO inE(S). Let (p : E −→ S,O) be an elliptic curve, and S ′ −→ S be a morphism of schemes. Then theschemeE S ′ =E× S S ′ ,togetherwiththesectionO S ′ :S ′ −→E S ′,isanellipticcurveover S ′ . Proposition3.1.2. E/S is an abelian group scheme. 17 Proof. See Deligne and Rapoport [7], II, proposition 2.7, or Katz and Mazur [21], Theorem 2.1.2. Definition 3.1.3 ([5]). The category of elliptic curves, denoted by Ell, is defined as follows. The objects of Ell are elliptic curves (p : E −→ S,O) over variable base schemes S. The morphisms from (p : E −→ S,O) to (p : E −→ S,O) are the pairs (f : S −→ S,g : E −→ E)suchthatthediagram E ′ g // p ′ E p S ′ f // S iscommutativeandthesectionO S ′ :S ′ −→E ′ inducedbyOequalsO ′ .Forsimplicity,wewill leavetherequirementonO andO ′ implicit. 3.2 Y 1 (N) Z[1/N] AS A FINE MODULI SPACE Definition 3.2.1. LetC be a category, and let F : C −→ Set be a contravariant functor. We say that F is representable if there exists an object X ofC such that F is isomorphic to the contravariantfunctor h X :C−→Set T 7−→Hom C (T,X) By Yoneda’s lemma, the object X is unique up to unique isomorphism. If F is representable, wesaythatF isrepresentedbytheobjectX. Remark3.2.2. Suppose products exist in the categoryC. IfF :C −→ Set andG :C −→ Set arerepresentedbyobjectsX andY ,respectively,thenthefunctor F ×G :C−→Set T 7−→F(T)×G(T) isrepresentedbytheproductofX andY. Definition3.2.3([5]). A moduli problemforellipticcurvesisacontravariantfunctor P :Ell−→Set AnelementofP(E/S)iscalledaP-structureonE/S. Definition3.2.4([5]). WedefineEll P , the category of elliptic curves withP-structure,as: • Theobjectsarethepairs(E/S,α)withE→S anellipticcurveandα aP-structure, 18 • Themorphismsfrom(E ′ /S ′ ,α ′ )to(E/S,α)aretheelementsϕ∈Hom Ell (E ′ /S ′ ,E/S) suchthatP(ϕ)(α) =α ′ . Thereisaforgetfulfunctor F P :Ell P −→Ell sendingapair(E→S,α)totheellipticcurveE→S. FixanellipticcurveE→S,wehavetherepresentablemoduliproblem: h E/S =Hom Ell (−,E/S) Forsimplification,anh E/S -structureonE ′ /S ′ isdenotedasanE/S-structureonE ′ /S ′ ;the categoryEll h E=S isdenotedasEll E/S ;theforgetfulfunctorF h E=S isdenotedas F E/S :Ell E/S −→Ell Definition 3.2.5 ([5]). We say a moduli problem P ′ is relatively representable if for every ellipticcurveE/S thefunctor P ′ ◦F E/S :Ell E/S −→Set isrepresentable. Proposition3.2.6([5],5.1). LetP ′ be a moduli problem. The following are equivalent: (1)P ′ is relatively representable; (2) For every representable moduli problemP, the functor P ′ ◦F P :Ell P −→Set is representable; (3) For every representable moduli problemP, the simultaneous moduli problem P×P ′ :Ell−→Set E/S7−→P(E/S)×P ′ (E/S) is representable. Therefore, every representable moduli problem is relatively representable. Notation([5]). IfP isarepresentablemoduliproblem,wewriteE P −→M P fortherepresent- ingobjectinEll.IfP isarelativelyrepresentablemoduliproblemandE −→S isanyelliptic curve,wewrite E P,E/S // E M P,E/S // S fortherepresentingobjectinEll E/S . 19 Definition3.2.7 ([5]). LetP be a representable moduli problem. The schemeM P is called the fine moduli scheme associated toP, and the elliptic curve E P is called the universal elliptic curveoverM P . Definition3.2.8. LetP beamoduliproblem.Foreveryellipticcurvef :E−→S,thegroup Aut S (E) ={g :E ∼ −−→E|fg =f} acts on the setP(E/S) by functoriality ofP. We say thatP is rigid if for every elliptic curve E/S andeveryα∈P(E/S), {g∈ Aut S (E)|αg =α} ={Id E } The following theorem says rigidity is the key property for representability of relatively representablemoduliproblem. Theorem3.2.9([21],§4.7.). LetP be a relatively representable and affine moduli problem for elliptic curves overZ[1/N]-schemes. ThenP is representable if and only ifP is rigid. WeareinterestedinthefollowingmoduliproblemsonZ[1/N]-schemes: [Γ 1 (N)] :E/S7−→{embeddingsofgroupschemes(Z/NZ) S ,−→E[N]}; [Γ 0 (N)] :E/S7−→{subgroupschemesofE[N]isomorphic(Z/NZ) S }; Theorem3.2.10([5],Theorem7.1). For allN ≥ 1, the moduli problems[Γ 1 (N)],[Γ 0 (N)] are relatively representable. Moreover, [Γ 1 (N)] is representable for N≥ 4, and its moduli scheme is a smooth affine curve overSpecZ[1/N]. 3.3 Y 0 (N) Z[1/N] AS A COARSE MODULI SPACE Inthemoduliproblem[Γ 0 (N)],subgroupschemesofE[N]isomorphicto(Z/NZ) S havenon- trivialautomorphisms.Sothemoduliproblem[Γ 0 (N)]isnotrigidforanyN ≥ 1,thereforeby Theorem 3.2.9 it is not representable. So there does not exist fine moduli space X 0 (N) Z[1/N] . However,thereexistsacoarsemodulispaceX 0 (N) Z[1/N] . Definition3.3.1 ([12], 4.2.1). LetS be a scheme,F : Sch/S −→ Set a contravariant functor, andΦ : F −→ h X a morphism of functors. Then(X,Φ) is called a coarse moduli space forF if (a)ForeveryS-schemeSpec(k)withkanalgebraicallyclosedfield,Φ(k) :F(k)−→X(k) isbijective,and (b) For every S-scheme Y and every morphism Ψ : F −→ h Y , there exists a unique mor- phismf :X −→Y suchthatΨ =h(f)◦Φ. 20 Remark. IfΦisanisomorphism,thenX isafinemodulischemeforF. TheclassicalIgusa’stheoremcanbeformulatedasthefollowing[12,Theorem4.2.2]: Theorem3.3.2(Igusa[14]). LetN ≥ 1.ThenthereexistsacoarsemodulischemeY 0 (N) Z[1/N] for[Γ 0 (N)].TheZ[1/N]-schemeY 0 (N) Z[1/N] isanaffinesmoothcurve,withgeometricallyirre- ducible fibres. The natural bijection betweenY 0 (N) Z[1/N] (C) andH/Γ 0 (N) is an isomorphism of complex algebraic curves. Weusethefollowingexplicitversion[10,Theorem8.6.1]: Theorem 3.3.3 (Igusa [14]). Let N be a positive integer and let p be a prime with p - N. The modular curve X 0 (N) has good reduction at p. Moreover, reducing the modular curve is compatible with reducing the moduli space in that the following diagram commutes: S 1 (N) ′ gd ψ 1 // X 0 (N) e S 1 (N) ′ e ψ 1 // e X 0 (N) HereS 1 (N) ′ gd is the moduli space which is comprised of(E,C) withE having good reduction at p and ] j(E) ̸= 0,1728; e S 1 (N) ′ is the moduli space which is comprised of (E,C) /Fp with j(E)̸= 0,1728. 3.4 GENERALIZED ELLIPTIC CURVES InordertocompactifyY 1 (N) Z[1/N] andY 0 (N) Z[1/N] ,Deligne-Rapoport[7]interpretedthecusps ofX 1 (N) Z[1/N] andX 0 (N) Z[1/N] asgeneralizedellipticcurves. Let k be an algebraic closed field. For any N ≥ 1, the N´ eron N-gon (C N ,+) over k is definedasfollows(See[9]forthedetails). • ToobtaintheschemeC N overk ,weindexthecomponentsP 1 ’swithelementsofZ/NZ andrequireanormalizationr : ⨿ i∈Z/NZ P 1 −→ C N tosend(∞) i and(0) i+1 tothesame pointforeachi.Thusr restrictstoanisomorphism ⨿ i∈Z/NZ G m −→C reg N . • The+isthemorphismC reg N ×C N −→C N obtainedbythecommutativediagram ⨿ i∈Z/NZ G m × ⨿ i∈Z/NZ P 1 ((x) i ,(y) j )7−→(xy) i+j // r×r ⨿ i∈Z/NZ P 1 r C reg N ×C N + // C N 21 Definition3.4.1(Deligne-Rapoport). AgeneralizedellipticcurveoverS isapair(E,+)where E is a scheme of curves over S and + is an S-morphism E reg × S E −→ E such that: (a) + makesE reg acommutativegroupschemeoverS actingonE;(b)Thegeometricfibersof(E,+) areellipticcurvesorN´ eronpolygons. Remark3.4.2(Diamond-Im[9]). (a)AgeneralizedellipticcurveoverS issmoothifandonly if it is an elliptic curve; (b) A generalized elliptic curve over an algebraically closed field is eitheranellipticcurveoraN´ eronpolygon. 22 CHAPTER4 KAMIENNY’SCRITERION Inorder to introduce the Kamienny’scriterion, we needsome knowledgeof N´ eron models and formalimmersions.HerewefollowSilverman[44]andArnold[3]. 4.1 GROUP SCHEMES Definition 4.1.1 ([44], §IV.3). Let S be a scheme. A group scheme over S is an S-scheme π :G−→S andS-morphisms σ 0 :S−→G, i :G−→G, µ :G× S G−→G suchthatthefollowingdiagramscommute: 1. (identity element) G× S G µ G× S G µ S× S G p 2 // σ 0 ×1 88 r r r r r r r r r r G G× S S p 1 // 1×σ 0 88 r r r r r r r r r r G 2. (inverse) G× S G 1×i // G× S G µ G× S G i×1 // G× S G µ G δ G OO π // S σ 0 // G G δ G OO π // S σ 0 // G (HereG δ G − →G× S Gisthediagonalmap.) 23 3. (associativity) G× S G× S G 1×µ µ×1 // G× S G µ G× S G µ // G Example 4.1.2 ([44], IV.3.1.2). The additive group schemeG a overZ is the schemeG a = SpecZ[T].ThegrouplawonG a isgivenby G a × Z G a //G a SpecZ[T 1 ]× Z SpecZ[T 2 ] Spec(Z[T 1 ]× Z Z[T 2 ]) SpecZ[T 1 ,T 2 ] // SpecZ[T] wherethemorphismSpecZ[T 1 ,T 2 ]−→ SpecZ[T]isinducedbytheringhomomorphism Z[T]−→Z[T 1 ,T 2 ], T 7−→T 1 +T 2 For any ringR, we haveG a (R) = R with group law given by addition onR. The additive group schemeG a/S overan arbitrary schemeS is the group schemeG a × Z S obtained by base extension.Inparticular,G a/R = SpecR[T]. Example 4.1.3 ([44], IV.3.1.3). The multiplicative group schemeG m overZ is the scheme G m = SpecZ[T,T −1 ].ThegrouplawonG m isgivenby G m × Z G m //G m SpecZ[T 1 ,T −1 1 ]× Z SpecZ[T 2 ,T −1 2 ] Spec(Z[T 1 ,T −1 1 ]× Z Z[T 2 ,T −1 2 ]) SpecZ[T 1 ,T −1 1 ,T 2 ,T −1 2 ] // SpecZ[T,T −1 ] wherethemorphismSpecZ[T 1 ,T −1 1 ,T 2 ,T −1 2 ]−→ SpecZ[T,T −1 ]isinducedbytheringhomo- morphism Z[T,T −1 ]−→Z[T 1 ,T −1 1 ,T 2 ,T −1 2 ], T 7−→T 1 T 2 24 For any ring R, we haveG m (R) = R ∗ with group law given by multiplication on R ∗ . The multiplicative group schemeG m/S over an arbitrary scheme S is the group schemeG m × Z S obtainedbybaseextension.Inparticular,G m/R = SpecR[T,T −1 ]. Example4.1.4. Foranyn≥ 1,thereexistsashortexactsequence(theKummersequence): 0−→µ n −→G m x7→x n −−−→G m −→ 0 Thisdefinesagroupschemeµ n . Example4.1.5. Ifp = 0inS,wehaveashortexactsequence: 0−→α p −→G a F −−→G a −→ 0 whereF istheFrobeniusmap.Thisdefinesagroupschemeα p . Definition4.1.6. AgroupschemeGoveranoetherianschemeS is finite and flat ifandonlyif O G isalocallyfreeO S -moduleoffiniterank.TherankisalocallyconstantfunctiononS,and iscalledthe order ofG. Theorem4.1.7(Oort-Tate[46]). Afiniteflatgroupschemeoforderpoveranalgebraicfieldof characteristicp is isomorphic toZ/pZ,µ p orα p . Definition4.1.8. AfinitegroupschemeGoverk iscalled ´ etaleifk[G]isseparable,i.e. k[G]⊗ k k ∼ = k×···×k 4.2 N ´ ERON MODELS Definition4.2.1([44]IV§5). LetRbeaDedekinddomainwithfractionfieldK,andletE/K be an elliptic curve. A N´ eron model forE/K is a (smooth) group schemeE/R whose generic fiberisE/K andwhichsatisfiesthefollowinguniversalproperty: (N´ eron mapping property) LetX/R be a smooth R-scheme with generic fiber X/K, and let ϕ K : X /K −→ X /K be a rational map defined over K. Then thereexistsauniqueR-morphismϕ R :X /R −→E /R extendingϕ K . Theorem4.2.2(N´ eron). The N´ eron model of an elliptic curve exists. Proof. SeeBosch,L¨ utkebohmertandRaynaud[4].SeealsoSilverman[44]IV§6. 25 Theorem 4.2.3 (Kodaira-N´ eron [25, 26, 37]). Let R be a Dedekind domain with field of frac- tionsK,letE beaN´ eronmodeloverRforanellipticcurveE/K,andletp⊂Rbeanynonzero prime ideal with residue fieldk p . Then 1. IfE has stable reduction at p, thenE(k p ) =E(k p ) 0 is an elliptic curve. 2. If E has semi-stable reduction at p, then there exists an extension k of k p of degree at most two so thatE(k) 0 =G m andE(k)/E(k) 0 ∼ =Z/nZ for some positive integern. 3. IfEhasunstablereductionatp,thenE(k p ) 0 =G a ,andE(k p )/E(k p ) 0 isafinitegroupof order at most four. Proof. SeeSilverman[44,§IV.9]. 4.3 FORMAL IMMERSION Definition4.3.1([3]). 1. Suppose Y,Z are locally Noetherian schemes and f : Y −→ Z is a locally finite type morphism. Then for a point y ∈ Y, we say that f is a formal immersionatyiftheinducedmap b O Z,f(y) −→ b O Y,y oncompletedlocalringsissurjective. 2. SupposeY,Z,S arelocallyNoetherianschemesandY andZ arefinitetypeandseparated overS and thatf : Y /S −→ Z /S is a locally finite typeS-morphism. Then for a section s∈ Y(S),we say thatf isa formal immersion alongs iff isa formal immersion for all y∈s(S). Proposition4.3.2([3],3.3). SupposeS is an integral Noetherian scheme andf :Y /S −→Z /S is a formal immersion along y 1 ∈ Y /S . Let y 2 ∈ Y /S such that (a) For some point t ∈ S, y 1 (t) =y 2 (t), and (b)f◦y 1 =f◦y 2 . Theny 1 =y 2 . 4.4 KAMIENNY’S CRITERION ForN prime, Kamienny [18] established an criterion for the nonexistence of rational points of modular curves over number fields. This criterion can be mutatis mutandis generalized to the cases whenN = prime power [19] or square free composite. In this section, we assumeN is a prime power or square free. The abelian varieties J 1 (N) /Q and J 0 (N) /Q are semistable and have good reduction at all primes ℓ- N. By Manin [32] and Drinfeld [11], the class of (0−∞) generates aQ-rational subgroupC ofJ 0 (N) /Q .LetJ /Q beaquotientofJ 0 (N) /Q withfiniteMordell-Weilgroup. 26 Let X (d) denote the d-th symmetric power of X 0 (N), i.e. the quotient space of the d-fold productX 0 (N)×X 0 (N)×···×X 0 (N)bytheactionofS d permutingthefactors.ThenX (d) isasmoothschemeoverS ′ = SpecZ[1/N].LetJ /S ′ betheN´ eronmodelofJ /Q . LetK be a number field of degreed overQ. Let (E,P)∈ X 1 (N)(K) andx = π(E,P)∈ X 0 (N)(K).Letx 1 ,··· ,x d betheimagesofxunderthedistinctembeddings φ i :K ,−→C, 1≤i≤d Wemayview(x 1 ,··· ,x d )asaQ-rationalpointofX (d) . A smooth projective curve X over an algebraically closed field k is called d-gonal if there exists a finite morphism f : X −→P over k of degree d. For d = 3 we say that the curve is trigonal. Also the smallest possible d is called the gonality of the curve. Considering the modularcurveX 0 (N)overC,wedenoteitsgonalityasGon(X 0 (N)). Lemma 4.4.1. Suppose Gon(X 1 (N)) > 3, J 1 (N)/ Q is finite, p > 2 is a prime not dividing N, and every point of degree 3 onX 1 (N) reduces modulop to the image of a cusp which is of degree≤ 3. ThenX 1 (N) has no noncuspidal point of degree3. Proof. IfpdoesnotdividetheorderofJ 1 (Q),thenthereductionmapisinjective: (4.1) J 1 (N)/ Q ,−→J 1 (N)/ Fp If p divides the order of J 1 (Q), i.e. there is a rational point of order p in J 1 (N), we look at its Zariski closure G in the N´ eron model of J 1 (N) over SpecZ p , whereZ p is the ring of p-adic integers. By Oort-Tate [46] the group scheme G must be the constant group scheme Z/pZ, and hence it is ´ etale. As such it must map isomorphically onto its reduction modulo p. Inotherwords,reductionmodulopisinjectiveonanypossiblerationalp-torsion.Thereforethe reductionmap(4.1)isalwaysinjective. Now suppose there is a noncuspidal point P of degree 3 on X 1 (N). Pick an element σ of order 3 in Gal(K/Q), where K is the normal closure of the defining field of P. Let Q be the cuspsuchthatP/ Fp =Q/ Fp .So ((P)+(P σ )+(P σ 2 )−(Q)−(Q σ )−(Q σ 2 ))/ Fp =O∈J 1 (N)/ Fp hencebytheinjectivityofthereductionmapabove (P)+(P σ )+(P σ 2 )−(Q)−(Q σ )−(Q σ 2 ) =O∈J 1 (N)/ Q ThenX 1 (N)hasadegree3function.SinceGon(X 1 (N))> 3,thisinimpossible. 27 AssumethatN islargeenoughsothatGon(X 0 (N))>d.Thenwemaydefineanembedding h :X (d) ,→J 0 (N)by h(y 1 ,··· ,y d ) = (y 1 +···+y d −d∞) WecomposethiswiththenaturalprojectionJ 0 (N)−→J toobtainamapf :X (d) −→J. Let f i = ∞ ∑ n=1 a n (f i )q n , 1≤i≤d beweight-twocuspformsonΓ 0 (N).Wewillsaythatf 1 ,··· ,f d satisfythelinearindependence conditionmodpifthevectors(a 1 (f 1 ),··· ,a d (f 1 )),··· ,(a 1 (f d ),··· ,a d (f d ))arelinearlyinde- pendentoverZ/pZ. Proposition4.4.2(Kamienny[18]). Supposep> 2 andp-N. The following are equivalent: 1. The map f : X (d) /S ′ −→ J /S ′ is a formal immersion along the section (∞,··· ,∞) in characteristicp. 2. Thereexistdweight-twocuspforms,associatedtoJ,thatsatisfythelinearindependence condition modp. 3. The firstd Hecke operatorsT 1 ,··· ,T d onH 1 (J,Z) are linearly independent modp. Lemma4.4.3. Letp-N beaprimenumberwithN > (1+ √ p d ) 2 andN-p 2d −1.Supposethat E isanellipticcurveoverK possessingaK-rationalpointP oforderN,andletx =π(E,P) be the corresponding point in X 0 (N)(K). Let ℘ be a prime ofO K above p and let k be the residue field of℘. Thenx 1/℘ =··· =x d/℘ =∞ /℘ . Proof. By the Kodaira-N´ eron Theroem [25, 26, 37, 44], we need to check that E has multi- plicativereductionat℘,butP doesnotspecializeto(E /k ) 0 . IfE hasgoodreductionat℘,thenE /k isanellipticcurvewithak-rationalpointP /k oforder N. By the Riemann hypothesis E(k) has order at most (1+ √ p d ) 2 . This is impossible under ourassumptionofN. IfE hasadditivereductionat℘,thentheindexof(E /k ) 0 inE /k is≤ 4.SoP mustspecialize to(E /k ) 0 ∼ =G a/k .ThenN dividethecardinalityp d ofk,whichisimpossiblesincep-N. If E has multiplicative reduction at ℘, and P specialize to (E /k ) 0 , then over an extension K /k ofdegreeatmost2,wehaveanisomorphism(E /K ) 0 ∼ =G m/K .ThenN mustdividethe cardinalityofK ∗ whichitselfmustdividep 2d −1.ThiscontradictsourassumptionofN. Theorem 4.4.4 (Kamienny’s creterion [18]). Let N be prime power or square free composite natural number such thatGon(X 0 (N))>d. Suppose there is a primep-N,p> 2 satisfying: 28 1. N-p 2d −1 andN > (1+ √ p d ) 2 2. The firstd Hecke operatorsT 1 ,··· ,T d onH 1 (J,Z) are linearly independent modp. Then there does not exist any elliptic curve with a point of order N rational over any field of degreed. 29 CHAPTER5 CUBICPOINTSONX 1 (N):PRIME POWERCASES 5.1 CASEX 1 (169) CalculationsinMagmashowsthedecomposition J 0 (169) =A 1 ×A 2 ×A 3 with dimA 1 = 2,dimA 2 = 3,dimA 3 = 3, and L(A i ,1) ̸= 0 for i = 1,3. So by Kolyvagin- Logach¨ ev[28],A 1 (Q)andA 3 (Q)areallfinite.TakeJ =A 1 ×A 3 .T 1 ,T 2 ,T 3 actsonH 1 (J,Z) ∼ = H 1 (A 1 ,Z)×H 1 (A 3 ,Z) =Z 4 ×Z 6 asfollowing: T 1 =I 10 T 2 =diag 2 1 0 −1 −1 −1 1 1 1 2 0 1 0 1 1 −1 , 1 3 −1 0 1 −2 0 4 −3 2 −1 −1 0 1 −1 1 −1 0 0 2 −1 1 0 −1 1 3 −2 0 1 −2 0 6 −4 2 −1 −2 T 3 =diag 2 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2 , 0 1 1 −1 0 −1 1 0 −1 −1 −1 0 1 0 −1 0 −1 0 0 1 0 0 0 −1 0 2 −2 0 −1 −1 1 3 −2 −2 −1 −2 30 Theyarelinearlyindependentmod3.Since169- 3 6 −1and169> (1+ √ 3 3 ) 2 ,thenbyTheorem 4.4.4,wearedone. 5.2 CASEX 1 (121) Wehavethedecomposition J 0 (121) =A 1 ×···×A 6 with dimA i = 1,1 ≤ i ≤ 6, and L(A i ,1) ̸= 0 for 2 ≤ i ≤ 6. Take J = A 2 ×···× A 6 . T 1 ,T 2 ,T 3 actsonH 1 (J,Z) ∼ = H 1 (A 2 ,Z)×···×H 1 (A 6 ,Z) =Z 2 ×···×Z 2 asfollowing: T 1 =I 10 T 2 =diag {( 1 0 0 1 ) , ( −1 0 0 −1 ) , ( 2 0 0 2 ) , ( −2 0 0 −2 ) , ( −2 0 0 −2 )} T 3 =diag {( 2 0 0 2 ) , ( 2 0 0 2 ) , ( −1 0 0 −1 ) , ( −1 0 0 −1 ) , ( −1 0 0 −1 )} Unfortunately, they are linearly dependent mod 3. So we need to reprove Lemma 4.4.3 for p = 5. We have 121- 5 6 −1, but 121 < (1+ √ 5 3 ) 2 ≈ 148.4. By Theorem 1.2.1, any elliptic curve E over k =F 125 can not have an k-rational points P of order 121. So, Lemma 4.4.3 is trueforp = 5,d = 3,N = 121.ThenbyTheorem4.4.4,wearedone. 5.3 CASEX 1 (49) J 1 (49)hasdecomposition: J 1 (49)−→A 1 ×···×A 5 whereA i hasdimensions:1,48,6,12,2andL(A i ,1)̸= 0foralli.SoJ 1 (49)(Q)isfinite. Let K be a cubic field. Let ℘ be a prime in O K lying above 3. Suppose x = (E,P) ∈ X 1 (49)(K). Since 49 > (1 + √ 3 3 ) 2 , then x ℘ is a cusp. We also know Gon(X 1 (49) > 3. ThereforebyLemma4.4.1,therearenononcuspidalcubicpointsonX 1 (49). 5.4 CASEX 1 (25) J 1 (25)hasdecomposition: J 1 (25)−→A 1 ×A 2 31 whereA i hasdimensions:8,4andL(A i ,1)̸= 0foralli.SoJ 1 (25)(Q)isfinite. We have 25 < (1+ √ 3 3 ) 2 ≈ 38.4. So we need to show any elliptic curveE overk =F 27 can not have an k-rational points P of order 25. Suppose E(k) has a point of order 25, then E(k) ∼ =Z/25Zsince25m> (1+ √ 3 3 ) 2 foranym> 1.ButbyTheorem1.2.1,E(k)̸∼ =Z/25Z. LetK beacubicfield.Let℘beaprimeinO K lyingabove3.Ifx = (E,P)∈X 1 (25)(K), then x ℘ is a cusp. We also know Gon(X 1 (25) > 3. Therefore by Lemma 4.4.1, there are no noncuspidalcubicpointsonX 1 (25). 5.5 CASEX 1 (32) The cyclic covering X 1 (32) −→ X 0 (32) can be factorised overQ into maps of degree 2 as follows: X 1 (32) α 1 − →X 2 α 2 − →X 3 α 3 − →X 4 =X 0 (32) whereX i =X 1 (32)/γ 2 4−i 5 . BytheworkofIgusa[14]weknowthatX 2 hasgoodreductionmodulop̸= 2.Theisomor- phismclassofeachellipticcurveE definedoverF q withapointP satisfying(π±1)P = 0or (π±17)P = 0contributes8pointstothereduction e X 2 ofX 2 overF q . Table5.1:EllipticcurvesoverF 27 t π N(π+1) N(π−1) N(π+17) N(π−17) = t+ √ t 2 −4q 2 =q+t+1 =q−t+1 =q+17t+289 =q−17t+289 0 3 √ −3 28 28 316 316 1 1+ √ −107 2 29 27 333 299 2 1+ √ −26 30 26 350 282 4 2+ √ −23 32 24 384 248 5 5+ √ −83 2 33 23 401 231 7 7+ √ −59 2 35 21 435 197 8 4+ √ −11 36 20 452 180 9 9+3 √ −3 2 37 19 469 163 10 5+ √ −2 38 18 486 146 Table5.1listedtheordinarycaseandsupersingularcaseofellipticcurvesoverF 27 .Wefind that noncuspidal points on e X 2 overF 27 comes from elliptic curves with endomorphism ring Z[ √ −23]orZ[ 1+ √ −23 2 ]. 32 Lemma5.5.1. Letxbeapointofdegree3onX 2 ,thenitdoesnotreducemodulo3toanypoint coresponding to an elliptic curveE with endomorphism ringZ[ √ −23] orZ[ 1+ √ −23 2 ]. Proof. Ishii[15]showedthatX 0 (32)hasadefiningequation: (5.1) Y 2 +2XY −14Y −X 3 +4X 2 −21X +54 = 0 withj-invariantgivenby j = (X 8 −16X 4 +16) 3 /X 4 (X 4 −16) Reductionmodulo3wehave Y 2 +2XY +Y −X 3 +X 2 = 0 j = (X 8 −X 4 +1) 3 /X 4 (X 4 −1) LetF 27 =F 3 [u]/(u 3 −u+1).Table5.2listthepointsonthereductionofX 0 (32)modulo3 whicharerationaloverF 27 .Foreachj,weseta 6 =−1/j andhavethefollowingellipticcurve withj(E) =j E :y 2 =x 3 +x 2 +a 6 Table5.2:PointsonthereductionofX 0 (32)modulo3whicharerationaloverF 27 X Y 1 Y 2 j ∞ ∞ ∞ − 0 −1 0 − 1 0 0 − u u 2 +u −u 2 −1 −u 2 −u−1 u+1 u 2 −1 −u 2 +u+1 −u 2 u−1 u 2 −u −u 2 −u+1 −u 2 +u−1 u 2 +1 u u 2 −u −u 2 −u+1 u 2 +u−1 u−1 u 2 −1 −u 2 +u+1 u 2 −u−1 u+1 u 2 +u −u 2 −1 −u 2 −u 2 +1 1 −u 2 +u−1 −u 2 −1 u 2 +u u 2 −u+1 −u 2 −u+1 −u 2 +u+1 u 2 −1 u 2 +u+1 −u 2 −1 −u 2 +u−1 −u 2 +u 1 −u 2 −u−1 −u 2 −u+1 u 2 −u u 2 −u 2 +u+1 −u 2 −u−1 −u 2 −u 1 −u 2 TocalculatetheendomorphismringEnd(E),weuse ϕ 2 (X,Y) =(X +Y) 3 −X 2 Y 2 +1485XY(X +Y)−162000(X +Y) 2 +41097375XY +8748000000(X +Y)−157464000000000 33 Reductionmodulo3,wehave ϕ 2 (X,Y) = (X +Y) 3 −X 2 Y 2 ThenwecancalculateE(F 27 ),π andEnd(E)aslistedinTable5.3. Table5.3:Endomorphismrings j a 6 E(F 27 ) π End(E) −u 2 −u−1(F) −u 2 +u+1 Z/24 2+ √ −23 Z[ √ −23] −u 2 (E) −u 2 −u+1 Z/24 2+ √ −23 Z[ √ −23] −u 2 +u−1(D) −u 2 −1 Z/24 2+ √ −23 Z[ √ −23] −u 2 −u+1(A) −u 2 Z/2×Z/12 2+ √ −23 Z[ 1+ √ −23 2 ] −u 2 +u+1(B) −u 2 −u−1 Z/2×Z/12 2+ √ −23 Z[ 1+ √ −23 2 ] −u 2 −1(C) −u 2 +u−1 Z/2×Z/12 2+ √ −23 Z[ 1+ √ −23 2 ] Consider the Atkin-Lehner involution ω on X 0 (32). Then ω(P) = P 0 ± P where P 0 = ω(P ∞ ). The reduction ofω modulo3 has the same modular interpretation as overQ. By using theadditivelawonX 0 (32),theω(P) iscalculatedaslistedintable5.4.Wecan seeifP corre- sponds to an elliptic curve with endomorphism ringZ[ √ −23], then ω(P) always corresponds to an elliptic curve with endomorphism ringZ[ 1+ √ −23 2 ]. If P corresponds to an elliptic curve with endomorphism ringZ[ 1+ √ −23 2 ], then ω(P) always corresponds to an elliptic curve with endomorphismringZ[ √ −23] On the other hand, if P were the projection of any of those points on X 2 rational inF 27 , ω(P)shouldcorrespondtoanellipticcurvewiththesameendomorphismringasP.Wehavea contradiction. Theorem5.5.2. There is no non-cusp point of degree 3 onX 1 (32) Proof. Kenku [22] shows that J(X 2 ) Q is finite of order dividing 2 9 × 5 2 . Let K be a cubic field. Let ℘ be a prime inO K lying above 3. If x ∈ X 2 (K), then x ℘ is a cusp. We also know Gon(X 2 )> 3.ThereforebyLemma4.4.1,thereisnononcuspidalcubicpointsonX 2 . Sinceγ 4 5 takesnon-cusppointofdegree3toanoncusppointofdegree3onX 1 (32),wecan concludethereisnonon-cusppointofdegree3onX 1 (32). 34 Table5.4:Pointsonthereductionof5.1modulo3whicharerationaloverF 27 P ω(P) P ∞ = (∞,∞) P 0 = (1,0) P 1 = (0,−1) P 2 = (0,0) P 3 = (u,u 2 +u)∗ P 24 = (−u 2 −u+1,u 2 ) P 4 = (u,−u 2 −1)∗ P 23 = (−u 2 −u+1,u 2 −u) P 5 = (u+1,u 2 −1)∗ P 18 = (−u 2 −1,u 2 −u+1) P 6 = (u+1,−u 2 +u+1)∗ P 17 = (−u 2 −1,u 2 +u) P 7 = (u−1,u 2 −u)∗ P 20 = (−u 2 +u+1,u 2 +u+1) P 8 = (u−1,−u 2 −u+1)∗ P 19 = (−u 2 +u+1,u 2 −1) P 9 = (u 2 +1,u) P 26 = (−u 2 −u−1,1)∗ P 10 = (u 2 +1,u 2 −u) P 25 = (−u 2 −u−1,−u 2 −u)∗ P 11 = (u 2 +u−1,u−1) P 22 = (−u 2 +u−1,1)∗ P 12 = (u 2 +u−1,u 2 −1) P 21 = (−u 2 +u−1,−u 2 +u)∗ P 13 = (u 2 −u−1,u+1) P 16 = (−u 2 ,1)∗ P 14 = (u 2 −u−1,u 2 +u) P 15 = (−u 2 ,−u 2 +1)∗ 5.6 CASEX 1 (27) ConsiderthecycliccoveringX 1 (27)−→X 0 (27).Theisomorphismclassofeachellipticcurve E definedoverF q withapointP satisfying(π±1)P = 0contributes9pointstothereduction ^ X 1 (27)overF q . Table5.5:EllipticcurvesoverF 125 t t 2 −4q N(π+1) =q+t+1 N(π−1) =q−t+1 9 −419 135 117 18 −176 144 108 We find that noncuspidal points on ^ X 1 (27) overF 125 comes from elliptic curves with endo- morphismringZ[ 1+ √ −419 2 ],Z[ √ −44],Z[ √ −11]orZ[ 1+ √ −11 2 ]. Lemma 5.6.1. Let x be a point of degree 3 on X 1 (27), then it does not reduce modulo 3 to any point corresponding to an elliptic curveE with endomorphism ringZ[ 1+ √ −419 2 ],Z[ √ −44], Z[ √ −11] orZ[ 1+ √ −11 2 ]. Proof. Ishii[15]showedthatX 0 (27)hasadefiningequation: (5.2) Y 2 +2XY +3Y −X 3 +X 2 +3X +9 = 0 35 withj-invariantgivenby j = (A+BY)/C where A =X 19 −12X 18 +27X 16 +441X 15 −1782X 13 −3159X 12 +9018X 10 −810X 9 −7290X 7 +22626X 6 −20412X 3 +729X +4374 B =X 18 +27X 15 −1782X 12 +9018X 9 −7290X 6 +729 C =X 6 −27X 3 Reductionmodulo5wehave A =X 19 −2X 18 +2X 16 +X 15 −2X 13 +X 12 +3X 10 +X 6 −2X 3 +4X +4 B =X 18 +2X 15 −2X 12 +3X 9 +4 C =X 6 −2X 3 Forj̸= 0,wesetA =− 1 j−3 andhavethefollowingellipticcurvewithj(E) =j E :y 2 +xy =x 3 +Ax+A Forj = 0,wehavethefollowingellipticcurvewithj(E 0 ) = 0 E 0 :y 2 +y =x 3 TocalculatetheendomorphismringEnd(E),weuse ϕ 2 (X,Y) =(X +Y) 3 −X 2 Y 2 +1485XY(X +Y)−162000(X +Y) 2 +41097375XY +8748000000(X +Y)−157464000000000 Reductionmodulo5,wehave ϕ 2 (X,Y) = (X +Y) 3 −X 2 Y 2 ThenwecancalculateE(F 125 )andEnd(E)aslistedinTable5.6. Consider the Atkin-Lehner involution ω on X 0 (27). Then ω(P) = P 0 + P where P 0 = ω(P ∞ ). The reduction ofω modulo5 has the same modular interpretation as overQ. By using the additive law on X 0 (27), We can see ω(P) corresponds to an elliptic curve with different endomorphismringasP.Thisisacontradiction. Theorem5.6.2. There is no non-cusp point of degree 3 onX 1 (27). Proof. WeknowX 1 (27)isnottrigonal.AndbyKenku[23],J 1 (27)(Q)isfiniteanditsorderis notdivisibleby5.SotheargumentinthecaseN = 32worksaswell. 36 Table5.6:Endomorphismrings j E(F 125 ) End(E) j(w(E)) End(w(E)) 2 Z/4×Z/36 Z[ 1+ √ −11 2 ] − a 2 +3a+2 Z/135 Z[ 1+ √ −419 2 ] 4a 2 +2a/3a 2 +2a+4 Z[ √ −44] a 2 +3a+4 Z/117 Z[ 1+ √ −419 2 ] 2a 2 +2a+1/3a 2 +4a+4 Z[ √ −44] a 2 +4a+2 Z/135 Z[ 1+ √ −419 2 ] 4a 2 +a/4a 2 +4a+1 Z[ √ −44] a 2 +4a+4 Z/117 Z[ 1+ √ −419 2 ] 4a 2 +2a/3a 2 +2a+4 Z[ √ −44] 2a 2 +2a Z/117 Z[ 1+ √ −419 2 ] 4a 2 +4a+4/3a 2 +4a+2 Z[ √ −11] 2a 2 +2a+1 Z/144 Z[ √ −44] a 2 +3a+4/3a 2 +3a+1 Z[ 1+ √ −419 2 ] 3a 2 +2a+2 Z/2×Z/72 Z[ √ −11] 4a 2 +a+4/4a 2 +2a+4 Z[ 1+ √ −419 2 ] 3a 2 +2a+4 Z/108 Z[ √ −44] a 2 +3a+2/a 2 +4a+4 Z[ 1+ √ −419 2 ] 3a 2 +3a+1 Z/135 Z[ 1+ √ −419 2 ] 3a 2 +4a+4/2a 2 +2a+1 Z[ √ −44] 3a 2 +3a+3 Z/117 Z[ 1+ √ −419 2 ] 4a 2 +4a+1/4a 2 +a Z[ √ −44] 3a 2 +4a+2 Z/2×Z/72 Z[ √ −11] 2a 2 +2a/4a 2 +2a+4 Z[ 1+ √ −419 2 ] 3a 2 +4a+4 Z/108 Z[ √ −44] a 2 +3a+4/3a 2 +3a+1 Z[ 1+ √ −419 2 ] 4a 2 +a Z/144 Z[ √ −44] a 2 +4a+2/3a 2 +3a+3 Z[ 1+ √ −419 2 ] 4a 2 +a+4 Z/117 Z[ 1+ √ −419 2 ] 4a 2 +4a+4/3a 2 +2a+2 Z[ √ −11] 4a 2 +2a Z/144 Z[ √ −44] a 2 +3a+2/a 2 +4a+4 Z[ 1+ √ −419 2 ] 4a 2 +2a+4 Z/117 Z[ 1+ √ −419 2 ] 3a 2 +2a+2/3a 2 +4a+2 Z[ √ −11] 4a 2 +4a+1 Z/108 Z[ √ −44] a 2 +4a+2/3a 2 +3a+3 Z[ 1+ √ −419 2 ] 4a 2 +4a+4 Z/2×Z/72 Z[ √ −11] 2a 2 +2a/4a 2 +a+4 Z[ 1+ √ −419 2 ] 37 CHAPTER6 CUBICPOINTSONX 1 (N):COMPOSITE CASES SincewehaveshownthereisnotorsionpointsoforderN = 32,25,27,49,121,169,itsuffices to prove there are no cubic points on X 1 (N) for N = 2·13,3·13,5· 13,7· 13,11· 13;2· 11,3·11,5·11,7·11;4·7,5·7,6·7,9·7;8·5,6·5,9·5;8·3,4·9. 6.1 CASEX 1 (143) Wehavethedecomposition J 0 (143) =A 1 ×···×A 5 38 with dimA 1 = 1,dimA 2 = 4,dimA 3 = 6,dimA 4 = 1,dimA 5 = 1, and L(A i ,1) ̸= 0 for i = 2,··· ,5.TakeJ =A 2 .T 1 ,T 2 ,T 3 actsonH 1 (J,Z) ∼ = H 1 (A 2 ,Z) =Z 8 asfollowing: T 1 =I 8 T 2 = 1 2 1 −2 −2 3 2 −1 0 0 0 0 1 1 −3 −2 −1 0 0 1 1 0 −1 −2 0 1 1 0 −1 1 2 −1 0 0 1 −1 1 2 −1 0 1 1 1 −2 −1 2 2 1 0 0 1 0 −1 1 2 1 0 0 0 −1 0 1 0 0 T 3 = −1 1 2 0 −2 3 0 0 2 0 0 −1 1 −3 2 0 1 −1 −2 3 2 −4 −1 3 0 0 2 −1 −1 2 0 −1 −1 1 2 −1 −1 2 2 −1 −1 0 2 −1 −1 3 0 −2 −1 1 1 −1 −1 2 1 −1 1 −1 0 0 1 −2 0 1 Theyarelinearlyindependentmod3.Since143- 3 6 −1and143> (1+ √ 3 3 ) 2 ,thenbyTheorem 4.4.4,wearedone. 6.2 CASEX 1 (91) Wehavethedecomposition J 0 (91) =A 1 ×···×A 4 with dimA 1 = 1,dimA 2 = 1,dimA 3 = 2,dimA 4 = 3, and L(A i ,1) ̸= 0 for i = 3,4. Take J =A 4 .T 1 ,T 2 ,T 3 actsonH 1 (J,Z) ∼ = H 1 (A 4 ,Z) =Z 6 asfollowing: T 1 =I 6 ,T 2 = 1 0 1 0 −1 0 0 0 0 0 −1 0 2 1 0 −1 2 −2 −1 −1 0 1 −2 1 −1 −1 0 −1 1 −1 0 −1 −1 −1 0 −1 ,T 3 = −1 −2 0 0 −1 1 −1 1 0 −1 0 −1 1 0 −2 0 1 −1 −1 −1 2 1 0 −1 0 −2 0 0 −2 0 1 −2 −2 −2 −1 −1 Theyarelinearlyindependentmod5.Since91- 5 6 −1and91< (1− √ 5 3 ) 2 < (1+ √ 5 3 ) 2 < 2·91,thenbyTheorem4.4.4,wearedone. 39 6.3 CASEX 1 (65) Wehavethedecomposition J 0 (65) =A 1 ×···×A 3 withdimA 1 = 1,dimA 2 = 2,dimA 3 = 2, andL(A i ,1)̸= 0 fori = 2,3. TakeJ = A 2 ×A 3 . T 1 ,T 2 ,T 3 actsonH 1 (J,Z) ∼ = H 1 (A 2 ×A 3 ,Z) =Z 4 ×Z 4 asfollowing: T 1 =I 8 T 2 =diag 2 0 0 −1 1 −2 1 −1 1 −1 2 −3 1 0 0 −2 , 1 −2 0 0 1 −3 0 0 1 −1 −2 1 0 −1 1 0 T 3 =diag −1 0 0 1 −1 3 −1 1 −1 1 −1 3 −1 0 0 3 , 2 −2 0 0 1 −2 0 0 1 −1 −1 1 0 −1 1 1 Theyarelinearlyindependentmod3.Since65- 3 6 −1and65> (1+ √ 3 3 ) 2 ,thenbyTheorem 4.4.4,wearedone. 6.4 CASEX 1 (77) Wehavethedecomposition J 0 (77) =A 1 ×···×A 6 withdimA 1 = dimA 2 = dimA 3 = dimA 5 = dimA 6 = 1,dimA 4 = 2, andL(A i ,1)̸= 0 for i = 2,··· ,6.TakeJ =A 4 ×A 5 .T 1 ,T 2 ,T 3 actsonH 1 (J,Z) ∼ = H 1 (A 4 ×A 5 ,Z) =Z 4 ×Z 2 as following: T 1 =I 6 T 2 =diag 0 2 −1 −2 2 −1 0 −2 1 −2 2 −2 −1 0 −1 −1 , ( −2 0 0 −2 ) T 3 =diag 1 −2 1 2 −2 2 0 2 −1 2 −1 2 1 0 1 2 , ( −1 0 0 −1 ) Theyarelinearlyindependentmod3.Since77- 3 6 −1and77> (1+ √ 3 3 ) 2 ,thenbyTheorem 4.4.4,wearedone. 40 6.5 CASEX 1 (55) Wehavethedecomposition J 0 (55) =A 1 ×···×A 4 with dimA 1 = dimA 3 = dimA 4 = 1,dimA 2 = 2, and L(A i ,1) ̸= 0 for i = 1,··· ,4. Take J =A 2 ×A 3 .T 1 ,T 2 ,T 3 actsonH 1 (J,Z) ∼ = H 1 (A 2 ×A 3 ,Z) =Z 4 ×Z 2 asfollowing: T 1 =I 6 T 2 =diag 1 0 −1 2 −2 −1 2 −2 −2 −2 3 0 0 −1 1 1 , ( −2 0 0 −2 ) T 3 =diag 0 0 2 −4 4 4 −4 4 4 4 −4 0 0 2 −2 0 , ( −1 0 0 −1 ) Theyarelinearlyindependentmod3.Since55- 3 6 −1and55> (1+ √ 3 3 ) 2 ,thenbyTheorem 4.4.4,wearedone. 6.6 CASEX 1 (40) J 1 (40)hasdecomposition: J 1 (40)−→A 1 ×···×A 9 whereA i hasdimensions:1,1,1,4,2,2,8,2,4andL(A i ,1)̸= 0foralli.SoJ 1 (40)(Q)isfinite. Let K be a cubic field. Let ℘ be a prime in O K lying above 3. Suppose x = (E,P) ∈ X 1 (40)(K). Since 40 > (1 + √ 3 3 ) 2 , then x ℘ is a cusp. We also know Gon(X 1 (40) > 3. ThereforebyLemma4.4.1,therearenononcuspidalcubicpointsonX 1 (40). 6.7 CASEX 1 (22) J 1 (22)hasdecomposition: J 1 (22)−→A 1 ×A 2 ×A 3 whereA i hasdimensions:1,1,4andL(A i ,1)̸= 0foralli.SoJ 1 (22)(Q)isfinite. 41 We have 22 < (1+ √ 3 3 ) 2 ≈ 38.4. So we need to show any elliptic curveE overk =F 27 can not have an k-rational points P of order 22. Suppose E(k) has a point of order 22, then E(k) ∼ =Z/22Zsince22m> (1+ √ 3 3 ) 2 foranym> 1.ButbyTheorem1.2.1,E(k)̸∼ =Z/22Z. LetK beacubicfield.Let℘beaprimeinO K lyingabove3.Ifx = (E,P)∈X 1 (22)(K), then x ℘ is a cusp. We also know Gon(X 1 (22)) > 3. Therefore by Lemma 4.4.1, there are no noncuspidalcubicpointsonX 1 (22). 6.8 CASEX 1 (24) Lemma6.8.1. LetE beanellipticcurve(overafieldk withbasepointO)andφaninvolution onE. Then either φ(P) = Q+P for a point Q∈ E of order 2, orφ(P) = Q−P for some pointQ∈E. Proof. By Silverman [43, III.4.7], Every morphism φ can be written as φ = τ Q ϕ, where ϕ is an isogeny and τ Q is translation by Q ∈ E. If φ is an involution with φ = τ Q ϕ, then it is an automorphism,i.e.φ∈Aut(E).Soϕ∈Aut(E,O).Wehave Id =φ 2 = (τ Q ϕ) 2 =τ Q (ϕτ Q )ϕ =τ Q (τ ϕ(Q) ϕ)ϕ =τ Q+ϕ(Q) ϕ 2 TakethevalueofO,wehave O =Id(O) =τ Q+ϕ(Q) ϕ 2 (O) =τ Q+ϕ(Q) (O) =Q+ϕ(Q) Thismeansτ Q+ϕ(Q) =Id,henceϕ 2 =Id. ConsiderthestructureofAut(E,O) Aut(E,O) ∼ = Z/2Z, j(E)̸= 0,1728 Z/4Z, j(E) = 0, char(k)̸= 2,3 Z/6Z, j(E) = 1728, char(k)̸= 2,3 SL 2 (F 3 ), j(E) = 0 = 1728, char(k) = 2 Z/4ZoZ/3Z, j(E) = 0 = 1728, char(k) = 3 weseethereisonlyoneelementinAut(E,O)oforder2.Therefore,ϕ =±Id. Ifϕ =Id,thenφ(P) =Q+P forapointQ∈E oforder2.Ifϕ =−Id,thenφ(P) =Q−P forsomepointQ∈E. Consider the cyclic covering X 1 (24) −→ X 0 (24). The isomorphism class of each elliptic curve E defined overF q with a point P satisfying (π± 1)P = 0 contributes 4 points to the reduction ^ X 1 (24)overF q . 42 Table6.1:EllipticcurvesoverF 7 3 t t 2 −4q N(π+1) =q+t+1 N(π−1) =q−t+1 8 −1308 352 24·14 16 −1116 24·15 328 32 −348 376 24·13 We find that noncuspidal points on ^ X 1 (24) overF 7 3 comes from elliptic curves with en- domorphism ringZ[ √ −1308],Z[ 1+ √ −327 2 ],Z[ √ −1116],Z[ 1+ √ −279 2 ],Z[ √ −124],Z[ 1+ √ −31 2 ], Z[ √ −348],Z[ 1+ √ −87 2 ]. Lemma6.8.2. Let x be a point of degree 3 onX 1 (24), then it does not reduce modulo 7 to any point corresponding to an elliptic curve E with endomorphism ringZ[ √ −1308],Z[ 1+ √ −327 2 ], Z[ √ −1116],Z[ 1+ √ −279 2 ],Z[ √ −124],Z[ 1+ √ −31 2 ],Z[ √ −348],Z[ 1+ √ −87 2 ]. Proof. Ishii[15]showedthatX 0 (24)hasadefiningequation: (6.1) Y 2 +4Y −X 3 −2X 2 +3X +4 = 0 withj-invariantgivenby j = (A+BY)/C where A =(X 2 −3) 3 (X 6 −9X 4 +3X 2 −3) 3 B =0 C =X 4 (X 2 −9)(X 2 −1) 3 Forj̸= 0,wesetA =− 1 j−3 andhavethefollowingellipticcurvewithj(E) =j E :y 2 +xy =x 3 +Ax+A Forj = 0,wehavethefollowingellipticcurvewithj(E 0 ) = 0 E 0 :y 2 +y =x 3 TocalculatetheendomorphismringEnd(E),weuse ϕ 2 (X,Y) =(X +Y) 3 −X 2 Y 2 +1485XY(X +Y)−162000(X +Y) 2 +41097375XY +8748000000(X +Y)−157464000000000 ThenwecancalculateE(F 7 3)andEnd(E). ConsidertheAtkin-Lehnerinvolutionω i ,i = 3,8,24onX 0 (24).ByLemma6.8.1,wehave ω i (P) =P i ±P whereP i =ω i (P ∞ ).Weclaimthatatleastoneofthemtakesthepositivesign. 43 Actually, suppose both ω 3 and ω 8 take the negative sign, i.e. ω 3 (P) = P 3 −P and ω 8 (P) = P 8 −P.Then ω 24 (P) =ω 3 ◦ω 8 (P) =ω 3 (P 8 −P) =P 3 −(P 8 −P) = (P 3 −P 8 )+P Denotethisinvolutionasω.Thenω(P) =Q+P forsomepointQoforder2.Wehave3points on ^ (X 0 (24))(F 7 3)oforder2.Theyare(0,−2),(1,−2)and(4,−2). The reduction of ω i modulo 7 has the same modular interpretation as overQ. By using the additive law on X 0 (24), We can see ω(P) corresponds to an elliptic curve with different endomorphismringfromthatofP.Thisisacontradiction. Table6.2:Atkin-LehnerinvolutiononX 0 (24)overF 7 3. P,(6points) (0,−2)+P (1,−2)+P (4,−2)+P Z/3Z×Z/120Z Z/364Z Z/370Z Z/359Z Z/3Z×Z/120Z Z/364Z Z/359Z Z/370Z Z/2Z×Z/156Z Z/322Z Z/342Z Z/366Z Z/2Z×Z/156Z Z/322Z Z/342Z Z/314Z Z/2Z×Z/156Z Z/3Z×Z/114Z Z/342Z Z/2Z×Z/160Z Z/2Z×Z/156Z Z/353Z Z/342Z Z/367Z Z/2Z×Z/156Z Z/357Z Z/342Z Z/314Z Z/2Z×Z/156Z Z/357Z Z/342Z Z/366Z Z/2Z×Z/156Z Z/353Z Z/342Z Z/2Z×Z/160Z Z/2Z×Z/156Z Z/3Z×Z/114Z Z/342Z Z/367Z Theorem6.8.3. There is no non-cusp point of degree 3 onX 1 (24) Proof. WeknowX 1 (24)isnottrigonal.AndJ 1 (24)(Q)hasdecomposition: J 1 (24)(Q)−→A 1 ×A 2 ×A 3 with A i of dimension 1, 2, 2. And L(A i ,1) ̸= 0 for all i which means J 1 (24)(Q) is finite. So theargumentinN = 32caseworksaswell. 44 APPENDIX Table6.3:PointsonthereductionofX 0 (27)modulo3whicharerationaloverF 5 3:I X Y j X Y j 2 4 2 ∞ ∞ − 3 1 − 3 0 − 4 3 2 4 1 0 a+2 a 2 +2a+4 4a 2 +2a a+2 4a 2 +a+4 3a 2 +3a+3 a+3 a+4 4a 2 +4a+4 a+3 2a+2 4a 2 +a a+4 a 2 +4 4a 2 +a+3 a+4 4a 2 +3a a 2 +a+1 2a+1 2a 2 +a 2a 2 +4a+1 2a+1 3a 2 2a 2 +2 2a+3 4a+3 3a 2 +4a+4 2a+3 2a+3 a 2 +4a+2 3a+3 4a 2 +3 4a 2 +4a+1 3a+3 a 2 +4a+3 4a 2 +2a+4 3a+4 3a 2 +2a+2 3a 2 +1 3a+4 2a 2 +2a+2 4a 2 +3a+4 4a 4a 2 2a 2 +4a+4 4a a 2 +2a+2 2a 2 +3a+3 a 2 2a 2 +2 4a 2 +4a+1 a 2 a 2 a 2 +3a+2 a 2 +3 a 2 +a+3a+2 a+2 a 2 +3 2a 2 +4a+3 3a 2 +a+4 a 2 +a a 2 +2a+2 3a+3 a 2 +a 2a 2 +a 4a 2 +3a+4 a 2 +a+3 2a 2 +a+1 4a 2 +2a+4 a 2 +a+3 a 2 +2a a 2 +3a+4 a 2 +a+4 a+4 2a 2 +2 a 2 +a+4 3a 2 +2a 3a+3 a 2 +2a 3a 2 +3a+2 4a 2 +a+4 a 2 +2a 3a 3a 2 +4a+4 a 2 +2a+1 4a 2 +4a+4 a a 2 +2a+1 4a 2 +2a+1 a 2 +3 a 2 +3a 4a 2 +3a+2 2a+2 a 2 +3a 4a 2 +a 4a 2 +3a+3 a 2 +3a+2 3a 2 +2a+3 a 2 a 2 +3a+2 2a 2a 2 +3a+2 a 2 +3a+3 a 2 +4a+3 4a+2 a 2 +3a+3 2a 2 +3 4a 2 +4a+2 a 2 +3a+4 4a 2 +a 4a 2 +3a a 2 +3a+4 4a 2 +3a+4 a 2 +a a 2 +4a a 2 +1 4a 2 +3a a 2 +4a 2a 2 +2a+1 a 2 +2 45 Table6.4:PointsonthereductionofX 0 (27)modulo3whicharerationaloverF 5 3:II X Y j X Y j a 2 +4a+2 4a 2 +3a+3 a 2 +a a 2 +4a+2 4a 2 +4a 2a+3 a 2 +4a+3 3a 2 +3a+2 3a 2 +a+3 a 2 +4a+3 4a+4 3a 2 +4a a 2 +4a+4 a 2 +3 2a 2 +a+2 a 2 +4a+4 2a 2 +2a+1 a 2 +4 2a 2 +4 a 2 +2a+4 a 2a 2 +4 3a 2a 2 +a+3 2a 2 +a 4a 2 +1 a 2 +4a+4 2a 2 +a 2a 2 +3a+1 4a 2 +a+4 2a 2 +a+1 2a 2 +4a+3 4a+3 2a 2 +a+1 4a 2 +4a+2 a 2 +2a 2a 2 +a+2 4a 2 +3 a 2 +2a 2a 2 +a+2 2a 2 +3a a 2 +3 2a 2 +2a+1 a 2 +a+3 3a 2 +4a+2 2a 2 +2a+1 2 a 2 +3a+2 2a 2 +2a+2 2a 2 +4a+1 4a 2 +2a+3 2a 2 +2a+2 4a 2 +2a+2 2a 2 +1 2a 2 +3a 3a+3 3a 2 +3a+3 2a 2 +3a a 2 +a+4 2a 2 +2a 2a 2 +3a+1 3a 2 3a 2 +a+4 2a 2 +3a+1 3a 2 +4a 4a 2 +1 2a 2 +3a+2 a 2 +a+3 2a+1 2a 2 +3a+2 3a 4a 2 +1 2a 2 +4a+1 4a 2 4a 2 +a 2a 2 +4a+1 2a 2 +2a a 2 +3a+4 2a 2 +4a+2 2a 2 +4a+3 3a 2 +4a+2 2a 2 +4a+2 4a 2 +3a 2a 2 +2a+1 2a 2 +4a+3 a 2 +4 2a 2 +a+3 2a 2 +4a+3 2a+2 2a 2 +2a+4 3a 2 4∗a 2 +3a 4a 2 +2a+3 3a 2 2a+2 2a 2 +3a+3 3a 2 +3 3a+2 2a 2 +2a+1 3a 2 +3 4a 2 +2a+4 a 2 +4a+4 3a 2 +4 3a 2 3a 2 +2a+2 3a 2 +4 a 2 +4 4a 2 +2a 3a 2 +a+1 4a 2 +3 a 2 +a+1 3a 2 +a+1 3a+2 3a 2 +1 3a 2 +2a 4a 2 2a 2 +3 3a 2 +2a a+2 3a 2 +a 3a 2 +2a+2 2a 2 +4a+2 2a 3a 2 +2a+2 2a 2 +2a+1 2a+3 3a 2 +2a+3 3a 2 +3a+3 3a 2 +4a 3a 2 +2a+3 a 2 +3a+3 2a 2 +4a+1 3a 2 +3a+1 3a 2 +4a+3 4a 2 +3a+1 3a 2 +3a+1 a 2 +2 2a 2 +a+2 3a 2 +3a+2 a 2 +3a+3 2a 2 +1 3a 2 +3a+2 3a 2 +a 3a 2 +2a 3a 2 +3a+3 4a 2 +2a 2a+2 3a 2 +3a+3 2a+1 2a 2 +3a+2 3a 2 +3a+4 4a 2 +2a 2a 3a 2 +3a+4 2a+4 a 2 +4 3a 2 +4a 3a 2 +3 4a+4 3a 2 +4a a 2 +2a+4 3a 2 +a 3a 2 +4a+2 a+3 a 2 +2 3a 2 +4a+2 4a 2 +a a 2 3a 2 +4a+3 3a 2 +3a+2 a+2 3a 2 +4a+3 a 2 +4a+4 4a 2 +4a+2 4a 2 +1 a 2 +2a+2 3a 2 +2a+4 4a 2 +1 a 2 +3a+3 2a 2 +2a 4a 2 +2 3a 2 +4a+2 2a 2 +4a+4 4a 2 +2 4a 2 +a+1 2a+1 4a 2 +a 2a 2 +3a 4a 2 +4a+4 4a 2 +a 2 3a 2 +3a+1 4a 2 +a+1 a+2 2a 2 +2a+4 4a 2 +a+1 2a 2 +2a+3 3a 2 +a+3 4a 2 +2a 2 a 2 +4a+2 4a 2 +2a 2a 2 +a 3a 2 +2a+2 4a 2 +2a+1 3a 2 +3 4a 2 +a+3 4a 2 +2a+1 4a 2 +a+2 4a+2 4a 2 +3a+1 3a 2 +a+4 3a 2 +2a+4 4a 2 +3a+1 4a 2 +3a+1 3a 2 +3a+1 4a 2 +3a+4 a+4 4a+3 4a 2 +3a+4 2a 2 +3a 3a 2 +3 4a 2 +4a a 2 +3a+4 3a 2 +2a 4a 2 +4a a 2 +4a+3 3a 2 +3 4a 2 +4a+2 3a+2 4a+4 4a 2 +4a+2 2a 2 +4a+1 2a 2 +3 4a 2 +4a+4 3a 2 +4 4a 2 +3a+3 4a 2 +4a+4 4a 2 +2a 4a 2 +3a+1 46 Table6.5:PointsonthereductionofX 0 (24)modulo7whicharerationaloverF 7 3:I X Y 1 Y 2 j ∞ ∞ ∞ − 0 −2 −2 − 1 −2 −2 − 3 4 −1 − 4 −2 −2 − 6 0 −4 − a 3a 2 +6a+6 4a 2 +a+4 5a 2 +5a+5 a+2 4a 2 +2a+6 3a 2 +5a+4 a 2 +3a+5 a+4 a 2 +4a 6a 2 +3a+3 6a+3 a+6 4a 2 +6a+3 3a 2 +a 4a 2a 6a 2 +6a+6 a 2 +a+4 6a 2 +a 2a+1 a 2 +4a+3 6a 2 +3a 4a 2 +4 2a+3 a 2 +a+1 6a 2 +6a+2 3a 2 +6a+6 2a+4 a 2 +5 6a 2 +5 2a 2 +6a+2 2a+5 2a 2 +2a+6 5a 2 +5a+4 3a 2 +3a+2 2a+6 3a 2 +2a+5 4a 2 +5a+5 6a 2 +6a 3a+1 4a+3 3a 3a 2 +a+5 3a+3 6a 2 +2 a 2 +1 6a 2 +6a 3a+4 6a 2 +4a+3 a 2 +3a 5a 2 +5a+4 3a+5 2a 2 +3a 5a 2 +4a+3 4a 2 +1 3a+6 6a 2 +4a+4 a 2 +3a+6 3a 2 +6a+2 4a+1 3a 2 +4a+1 4a 2 +3a+2 3a 2 +6a+2 4a+6 2a 2 +2 5a 2 +1 3a 2 +a+5 5a+6 4a+6 3a+4 4a 2 +4 6a 6a a+3 5a 2 +5a+5 6a+3 5a 2 +5a+3 2a 2 +2a 6a+3 6a+5 4a 2 +6a+5 3a 2 +a+5 a 2 +3a+5 a 2 +2 5a 2a+3 3a 2 +6a+2 a 2 +5 2a 2 +3a+5 5a 2 +4a+5 2a+6 a 2 +a+2 4a 2 +6a+5 3a 2 +a+5 6a 2 +5a a 2 +a+4 3a 2 +3a+6 4a 2 +4a+4 3a 2 +3a+2 a 2 +a+5 a 2 +5 6a 2 +5 5a 2 +4 a 2 +a+6 6a+5 a+5 4a 2 +5a a 2 +2a 4a 2 +4a 3a 2 +3a+3 6a 2 +6a a 2 +2a+1 2a 2 +5a+2 5a 2 +2a+1 5a 2 +5a+3 a 2 +2a+2 5a 2 +5a+2 2a 2 +2a+1 6a 2 +5 a 2 +2a+3 5a+3 2a 6a 2 +4a+2 a 2 +2a+5 2a 2 +2a+5 5a 2 +5a+5 4a 2 +2a a 2 +2a+6 5a 2 +a+1 2a 2 +6a+2 6a 2 +a 47 Table6.6:PointsonthereductionofX 0 (24)modulo7whicharerationaloverF 7 3:II X Y 1 Y 2 j a 2 +3a 6a 2 +2 a 2 +1 6a 2 +6a a 2 +3a+1 4a 2 +6a+2 3a 2 +a+1 4a 2 +3a a 2 +3a+2 5a+4 2a+6 4a 2 +3a+1 a 2 +3a+3 a 2 +a 6a 2 +6a+3 6a+3 a 2 +3a+6 2a 2 +5a+5 5a 2 +2a+5 3a 2 +3a+2 a 2 +4a+6 5a 2 +6a+6 2a 2 +a+4 4a 2 +3a+3 a 2 +5a 2a 2 +5a+5 5a 2 +2a+5 4a 2 +5a+5 a 2 +5a+1 2a 2 +3a+3 5a 2 +4a 5a a 2 +5a+3 3a 2 +6a+3 4a 2 +a 6a 2 +a+5 a 2 +5a+4 6a 2 +a+2 a 2 +6a+1 5a 2 +6a+3 a 2 +6a a 2 +4a 6a 2 +3a+3 6a+3 a 2 +6a+3 4a 2 +6a+1 3a 2 +a+2 6a 2 +3a+6 2a 2 3a+6 4a+4 6a 2a 2 +1 1 2 2a 2 +5a+4 2a 2 +5 2a 2 +a+1 5a 2 +6a+2 4a 2 +2a+6 2a 2 +6 4a+3 3a 5a 2 +5a+3 2a 2 +a+1 5a 2 +6a+3 2a 2 +a 6a 2 +2a+6 2a 2 +a+3 6a 2 +a+4 a 2 +6a+6 3a 2 +3a+2 2a 2 +a+4 4a 2 +5 3a 2 +5 6a 2 +6a+1 2a 2 +a+5 4a 2 +6a+3 3a 2 +a 4a 2 +3a 2a 2 +a+6 4a 2 +6 3a 2 +4 6a+3 2a 2 +2a+1 6a 2 +6a+4 a 2 +a+6 6a 2 +6a 2a 2 +2a+2 5a 2 +a+1 2a 2 +6a+2 4a 2 +5a+5 2a 2 +2a+3 3a 2 +6a+4 4a 2 +a+6 5a 2 +2a+3 2a 2 +2a+6 2a 2 +3a+2 5a 2 +4a+1 a 2 +4a+2 2a 2 +3a+1 6a+4 a+6 2a 2 +6a 2a 2 +3a+2 2a 2 +6a+4 5a 2 +a+6 2a 2 +4a 2a 2 +3a+5 3a 2 +2a+1 4a 2 +5a+2 5a 2 +4 2a 2 +3a+6 4a 2 +6a+6 3a 2 +a+4 a 2 +4a 2a 2 +4a+3 3a 2 +2a+1 4a 2 +5a+2 5a 2 +4a+1 2a 2 +4a+5 3a 2 4a 2 +3 4a 2 +2a+3 2a 2 +5a+1 5a 2 +3a+4 2a 2 +4a+6 3a 2 +6a+5 2a 2 +5a+2 5a 2 +6a+3 2a 2 +a 6a 2 +6a 2a 2 +5a+3 2a 2 +a+6 5a 2 +6a+4 4a 2 +2a 2a 2 +5a+4 1 2 3a 2 +2a 2a 2 +6a+1 5a 2 +3a+4 2a 2 +4a+6 2a 2 +4a+4 2a 2 +6a+2 2a 2 +3a+2 5a 2 +4a+1 4a 2 +2a+3 2a 2 +6a+4 6a 2 +a+4 a 2 +6a+6 4a 2 +2a+6 2a 2 +6a+5 5a 2a+3 2a 2 +5a+6 48 Table6.7:PointsonthereductionofX 0 (24)modulo7whicharerationaloverF 7 3:III X Y 1 Y 2 j 2a 2 +6a+6 5a+2 2a+1 a 2 +2a+1 3a 2 a 2 +4a+4 6a 2 +3a+6 6a 2 +4a 3a 2 +1 4a 2 +4a+5 3a 2 +3a+5 4a 2 +2a+3 3a 2 +4 3a 2 +2a+1 4a 2 +5a+2 5a 2 +5a+3 3a 2 +5 6a 2 +a+4 a 2 +6a+6 3a 2 +3a+2 3a 2 +6 a 2 +5a+1 6a 2 +2a+2 6a 2 +4a+6 3a 2 +a+1 a 2 +2a+4 6a 2 +5a+6 6a 2 +4a 3a 2 +a+2 5a 2 +6a+3 2a 2 +a 4a+3 3a 2 +a+5 4a 2 +3a+6 3a 2 +4a+4 6a 2 +5a+5 3a 2 +2a 1 2 2a 2 +1 3a 2 +2a+4 3a+1 4a+2 6a+6 3a 2 +2a+5 2a 2 +4a 5a 2 +3a+3 3a 2 +3a+2 3a 2 +2a+6 5a 2 +3a+2 2a 2 +4a+1 3a 2 +6a+6 3a 2 +3a 2a 2 +3a 5a 2 +4a+3 5a+2 3a 2 +3a+2 6a+1 a+2 6a 2 +6a+1 3a 2 +3a+3 5a 2 +3a+4 2a 2 +4a+6 5a 2 +6 3a 2 +4a 2a 2 +3a+6 5a 2 +4a+4 4a 2 +1 3a 2 +4a+1 6a 2 +5a+2 a 2 +2a+1 4a 2 +2 3a 2 +4a+2 2a 2 +2a+4 5a 2 +5a+6 3a 2 +6a+2 3a 2 +4a+5 2a 2 +2 5a 2 +1 6a 2 +6a 3a 2 +4a+6 2a 2 +6a+6 5a 2 +a+4 5a 2 +5a+3 3a 2 +5a 6a a+3 6a 2 +6a+2 3a 2 +5a+2 3a 2 +6a+5 4a 2 +a+5 3a 2 +3 3a 2 +5a+4 6a 2 +5a+5 a 2 +2a+5 4a 2 +2a+3 3a 2 +5a+6 a 2 +5a+2 6a 2 +2a+1 5a 2 +6a 3a 2 +6a+2 3a 2 +3a+2 4a 2 +4a+1 3a 2 +6a+2 3a 2 +6a+4 2a 2 +3a+2 5a 2 +4a+1 3a 2 +4a+4 3a 2 +6a+6 3a 2 +6a+6 4a 2 +a+4 3a 2 +3a+4 4a 2 +5 a 2 +6a+3 6a 2 +a 6a 2 +a+5 4a 2 +6 3a 2 +6a+6 4a 2 +a+4 4a 2 +2a+3 4a 2 +a 2a 2 +3a 5a 2 +4a+3 2a 2 +4a+4 4a 2 +a+1 4a 2 +2a 3a 2 +5a+3 3a 2 +3a+4 4a 2 +a+3 6a+6 a+4 3a 2 +4a+4 4a 2 +a+5 5a 2a+3 3a 2 +6a+2 4a 2 +2a 4a 2 +2a 3a 2 +5a+3 6a 2 +6a+2 4a 2 +2a+3 6a 2 +6a a 2 +a+3 4a 2 +2a+3 4a 2 +2a+5 a 2 +5a+4 6a 2 +2a+6 3a 2 +3 4a 2 +3a+4 5a 2 +a+1 2a 2 +6a+2 2a 2 +6a 4a 2 +3a+5 6a a+3 3a 2 +6a+2 49 Table6.8:PointsonthereductionofX 0 (24)modulo7whicharerationaloverF 7 3:IV X Y 1 Y 2 j 4a 2 +4a a 2 +5a+4 6a 2 +2a+6 5a+2 4a 2 +4a+2 5a 2 +5a 2a 2 +2a+3 5a 2 +5a+3 4a 2 +4a+3 a 2 +6a+3 6a 2 +a 5a 2 +2a+1 4a 2 +4a+4 4a 2 +2a+5 3a 2 +5a+5 5a 2 +6 4a 2 +5a a 2 +2 6a 2 +1 2a 2 +1 4a 2 +5a+5 3a 2 +a+6 4a 2 +6a+4 5a 2 +5a+3 4a 2 +5a+6 3a 2 +2a+2 4a 2 +5a+1 2a 2 +5a+6 4a 2 +6a+1 2a 2 +2 5a 2 +1 5a 2 +2a+1 4a 2 +6a+2 a 2 +5a+5 6a 2 +2a+5 6a 2 +5a+5 4a 2 +6a+4 4a 2 +2a+4 3a 2 +5a+6 6a 2 +3 4a 2 +6a+5 6a 2 a 2 +3 4a+3 5a 2 +4 6a 2 +2a a 2 +5a+3 6a 2 +6a 5a 2 +5 2a 2 +3 5a 2 3a 2 +2a+2 5a 2 +6 a 2 +6a 6a 2 +a+3 2a 2 +5a+4 5a 2 +a+1 a 2 +5a 6a 2 +2a+3 a 2 +2a+1 5a 2 +a+5 6a 2 +5a+5 a 2 +2a+5 4a 2 +2a+3 5a 2 +2a+1 5a 2 +4a+2 2a 2 +3a+1 3a 2 +3a+2 5a 2 +2a+2 2a 2 +5a+6 5a 2 +2a+4 2a 2 +3 5a 2 +2a+3 5a 2 +a+5 2a 2 +6a+5 3a 2 +2a 5a 2 +3a 4a 2 +3a+4 3a 2 +4a+6 3a 2 +3a+3 5a 2 +3a+1 a 2 +5a+2 6a 2 +2a+1 6a 2 +4a+6 5a 2 +3a+2 4a 2 +3a+5 3a 2 +4a+5 4a 2 +2a+3 5a 2 +3a+4 2a 2 +4a 5a 2 +3a+3 5a 2 +4a+1 5a 2 +3a+6 2a 2 +6a+6 5a 2 +a+4 5a 2 +5a+3 5a 2 +4a+1 4a 2 3a 2 +3 a 2 +4a 5a 2 +4a+4 6a 2 +3a+2 a 2 +4a+1 4a 2 +3a+1 5a 2 +4a+5 4a+3 3a 2a 2 +4a 5a 2 +5a+1 4a 2 +6a+3 3a 2 +a a 2 +4a+2 5a 2 +5a+3 4a 2 +2a+1 3a 2 +5a+2 3a 2 +3a+3 5a 2 +5a+4 3a+1 4a+2 5a 2 +2a+3 5a 2 +6a+1 a 2 +3a+2 6a 2 +4a+1 6a+3 5a 2 +6a+6 2a 2 +5a+5 5a 2 +2a+5 6a 2 +2a+6 6a 2 2a 2 +6a+6 5a 2 +a+4 3a 2 +6a+5 6a 2 +1 a 2 +4a 6a 2 +3a+3 3a 2 +2a+2 6a 2 +2 a 2 +4a+2 6a 2 +3a+1 2a+6 6a 2 +4 3a+1 4a+2 4a 6a 2 +5 4a 2 +6a+5 3a 2 +a+5 3a 2 +6a+2 50 Table6.9:PointsonthereductionofX 0 (24)modulo7whicharerationaloverF 7 3:V X Y 1 Y 2 j 6a 2 +a a 2 +5a+4 6a 2 +2a+6 6a+3 6a 2 +a+2 6a 2 +2 a 2 +1 6a+6 6a 2 +a+3 6a 2 +5a+5 a 2 +2a+5 2a 2 +3 6a 2 +a+4 6a 2 +5a a 2 +2a+3 6a 2 +3a+6 6a 2 +a+6 2a 2 +a+1 5a 2 +6a+2 5a 2 +6a 6a 2 +2a+2 2a+2 5a+1 4a 2 +2 6a 2 +2a+3 2a 2 +4a 5a 2 +3a+3 5a 2 +6a+3 6a 2 +2a+6 6a 2 +4a+5 a 2 +3a+5 5a 6a 2 +3a 3a 2 +3a+1 4a 2 +4a+2 2a 2 +6a+2 6a 2 +3a+1 6a 2 +5a+1 a 2 +2a+2 4a 2 +3a+3 6a 2 +3a+3 4a 2 +4a+6 3a 2 +3a+4 6a 6a 2 +3a+4 a 2 +6a+3 6a 2 +a 6a 2 +5 6a 2 +3a+5 a 2 +a+2 6a 2 +6a+1 5a 2 +5a+3 6a 2 +3a+6 2a 2 +6 5a 2 +4 3a 2 +3a+2 6a 2 +4a+3 a 2 +5 6a 2 +5 6a 2 +3 6a 2 +4a+4 4a 2 +2a 3a 2 +5a+3 6a+3 6a 2 +5a+4 2a 2 +4a+3 5a 2 +3a 6a 2 +4a+2 6a 2 +6a 3a 2 +6a 4a 2 +a+3 5a 2 +5a+4 6a 2 +6a+1 2a 2 +a+1 5a 2 +6a+2 4a 2 +5a 6a 2 +6a+5 a 2 +5a+2 6a 2 +2a+1 6a 2 +5a 51 Table6.10:StructuresofpointsonX 0 (24) F 125 j E(F 7 3) j E(F 7 3) 2a+6 Z/370Z 4a Z/328Z 4a+3 Z/3Z×Z/114Z 5a Z/359Z 5a+2 Z/363Z 6a Z/3Z×Z/120Z 6a+3 Z/2Z×Z/156Z 6a+6 Z/371Z a 2 +2a+1 Z/370Z a 2 +3a+5 Z/322Z a 2 +4a Z/359Z a 2 +4a+2 Z/363Z 2a 2 +1 Z/349Z 2a 2 +3 Z/2Z×Z/160Z 2a 2 +4a Z/3Z×Z/114Z 2a 2 +4a+4 Z/328Z 2a 2 +5a+4 Z/349Z 2a 2 +5a+6 Z/2Z×Z/160Z 2a 2 +6a Z/366Z 2a 2 +6a+2 Z/353Z 3a 2 +3 Z/322Z 3a 2 +a+5 Z/318Z 3a 2 +2a Z/349Z 3a 2 +2a+2 Z/2Z×Z/160Z 3a 2 +3a+2 Z/342Z 3a 2 +3a+3 Z/364Z 3a 2 +3a+4 Z/353Z 3a 2 +4a+4 Z/322Z 3a 2 +6a+2 Z/2Z×Z/156Z 3a 2 +6a+6 Z/3Z×Z/120Z 4a 2 +1 Z/328Z 4a 2 +2 Z/314Z 4a 2 +4 Z/357Z 4a 2 +2a+3 Z/2Z×Z/156Z 4a 2 +2a Z/3Z×Z/120Z 4a 2 +2a+6 Z/371Z 4a 2 +3a Z/328Z 4a 2 +3a+1 Z/314Z 4a 2 +3a+3 Z/357Z 4a 2 +5a Z/326Z 4a 2 +5a+5 Z/367Z 5a 2 +4 Z/367Z 5a 2 +6 Z/326Z 5a 2 +2a+1 Z/367Z 5a 2 +2a+3 Z/326Z 5a 2 +4a+1 Z/318Z 5a 2 +5a+3 Z/342Z 5a 2 +5a+4 Z/364Z 5a 2 +5a+5 Z/353Z 5a 2 +6a Z/328Z 5a 2 +6a+3 Z/3Z×Z/114Z 6a 2 +3 Z/366Z 6a 2 +5 Z/353Z 6a 2 +a Z/353Z 6a 2 +a+5 Z/366Z 6a 2 +2a+6 Z/318Z 6a 2 +3a+6 Z/370Z 6a 2 +4a Z/314Z 6a 2 +4a+2 Z/357Z 6a 2 +4a+6 Z/328Z 6a 2 +5a Z/363Z 6a 2 +5a+5 Z/359Z 6a 2 +6a Z/342Z 6a 2 +6a+1 Z/364Z 6a 2 +6a+2 Z/353Z 3a 2 +6a+5 Z/371Z 52 BIBLIOGRAPHY [1] Abhyankar,ShreeramS.Resolutionofsingularitiesofarithmeticalsurfaces.1965Arith- metical Algebraic Geometry (Proc. 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Abstract (if available)
Abstract
Let E be an elliptic curve defined over a number field K. Then its Mordell-Weil group E(K) is finitely generated: E(K)≅ E(K)tor × ℤʳ. In this thesis, I will discuss the cyclic torsion subgroup of elliptic curves over cubic number fields. I obtained complete results in the prime power case and partial results in the composite case.
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Creator
Wang, Jian
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Core Title
On the torsion structure of elliptic curves over cubic number fields
School
College of Letters, Arts and Sciences
Degree
Doctor of Philosophy
Degree Program
Mathematics
Publication Date
06/25/2015
Defense Date
04/22/2015
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University of Southern California
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cubic fields,elliptic curves,modular curves,OAI-PMH Harvest,torsion
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Kamienny, Sheldon (
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), Golubchik, Leana (
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), Guralnick, Robert M. (
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blandye@gmail.com,wang50@usc.edu
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Tags
cubic fields
elliptic curves
modular curves
torsion