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# ALGEBRA MATH 100

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Date Created: 09/09/15

Some Abelian Varieties with Visible ShafareviCh Tate Groups William Stein May 28 1999 Abstract We give examples of abelian variety quotients of the modular jaco bian JON with nontrivial visible ShafareviChTate groups In computing these examples we developed formulas for invariants of higher dimensional modular abelian varieties Introduction In 4 Cremona and Mazur studied visibility of Shafarevich Tate groups of el liptic curves E C J0Ni This paper extends some of these computations to higher dimensional A C J0Ni For each N S 1001 and N 1028 1061 we compute the analytic rank 0 new optimal A in J0N having nontrivial odd visible ShafarevichTate group visible in the new part of J0Ni A total of 19 such A were found having 111 of order the squares of 3 5 7 321113151i Our computations say little about invisible elements of 111 however see The algorithms developed for computing invariants of modular abelian varieties are also of some interest Acknowledgment It is a pleasure to thank Barry Mazur for his brilliant lectures on visibility Ken Ribet and Robert Coleman for explaining monodromy and John Cremona for explaining modular symbols The author would also like to thank Richard Taylor and Loic Merel for comments and Amod Agashe for many useful discussions and proving Theorem 27 Not at ion Let 52 N 52F0N C be the space of cusp forms of weight 2 for the subgroup To N C SL2Zi Thus 52 N can be identi ed with the differentials on the modular curve X0 Let J0 N denote the jacobian of X0 Denote by H1 X0 N Z the integral homology of X0 The Hecke algebra T acts in a compatible manner on 52N H1X0N Z and J0Ni The letters f and g Amod Agashe will be ajoint author of this paper as soon as he has had a chance to agree with what it says are reserved for newforms7 iiei7 normalized eigenforms for T in the complement of the old subspace of 52 Let fl7 i i i fa denote the Galois conjugates of f7 and If Annf the annihilator of f in Ti 1 Optimal abelian varieties and newforms An abelian variety quotient of Jo N is called optimal if the kernel is connected We associate to f an optimal quotient Af of 0 0 A IfJ0N A J0N A Af A 0 Let H H1X0N7 Z7 5392 5392 N7 and recall that integration de nes a non degenerate pairing 5392 X H A C7 hence a map H A HomC 527 C Composing with restriction to SQIf de nes a map 39i39f H A HomSgIf7 Ci Theorem 11 Af is an abelian vaiiety of dimension d with canonical Lseries d LAfs H Loci 3 i1 The complex uniformization of the tori AfC and A C is described by the following diagram 0 0 meo l l HlIfl A PAH l Hom527 C If HomSg7 C A HomSg If C AVC A J0NC A AC l l l 0 0 0 in which the vertical columns are exact but the rows are not Proof 16 and section 17 of 1 11 The Birch and SWinnerton Dyer Conjecture Let AZ be the Neron model of A Afr The Tamagawa number cp is the number of Fprational components of the special ber Appi A basis hlp H hd for the Neron differentials de nes a measure n on AR and we let 9 MARi Let w H1Q7 A A Hv H1QUA and set LU Kerwi If LA7 1 f 0 it is known 9 that AQ and LHA are both nite One then has the following fundamental and still open Conjecture 12 Birch7 SWinnerton Dyer7 Tate 7 milHmqu A 9quot lAQl AWQM 12 The Manin Constant Let Sf C C 52 N7 C denote the subspace of cusp forms spanned by the con jugates of There are two lattices in SAC One is the lattice SfZ of cusp forms with integer Fourier expansion at in nity The other SfAZ is got by pulling back the Neron differentials de ned above The Manin constant is Cf WW I SKAZN We are aware of no examples of newforms f for which Cf 1 It is reasonable to expect that one can extend methods known for elliptic curves eg7 11 to show that Cf is at least coprime to 2 Later in this paper we give a formula for LAf1Qf where 9f is computed using the lattice SfZ Thus our BSD special value is off by the Manin constant For the remainder of this paper we of cially assume Conjecture 13 of 1 13 Connecting Mordell Weil and Shafarevich Tate Let g 6 52N be nonconjugate newforms and Af7 Ag the corresponding optimal quotients of Jo Let m C T be a maximal ideal such that A m A m C J0N Let p be the residue characteristic of m Assume that p l 2N HP N 5pc7 where cg are the Tamagawa numbers of A Under hypothesis such as these we expect there to be a commutative diagram 0 A AfQmAfQ A X A LUAflml A 0 H 0 A AgQmAgQ A X A H1Aglml A 0 with exact rows Here X is an abelian group7 H1Spec Z7Af A precise statement will not be given here as the purpose of this paper is merely to present a few algorithms and computational results 2 Algorithms 21 Modular Symbols Modular symbols give a presentation of the homology of the modular curve X0 Here we brie y review modular symbols for F0N More information on how to actually compute with them can be found in 3 7 and 12 De ne the space of modular symbols MN7 Z to be the free abelian group generated by symbols 15 such that 15 E P1Q Q U subject to the relations 0 16 67 7 ma ay 9ag all g 6 FoN The space of boundary symbols N Z is the free abelian group generated by symbols 1 a E P1 Q modulo the relations a 9a all y 6 TOW The cuspidal symbols N Z are the kernel of the boundary map 8a 5 5 aZ o a ltN 2 a MN 2 3 gm 2 The Hecke algebra act on MN Z and there is an involution Ha 7a lntegration de nes a pairing 52N gtlt MN Z A C The Manin Drinfeld theorem asserts that the image ofMN Z in HomSg N C is a lattice There is a natural isomorphism between N Z and H1 X0 N Z 22 The Method of Graphs We brie y review the method of graphs see 13 and 15 for more details Let M be a positive integer p a prime not dividing M and put N pMi Let D be the nitely generated free abelian group on the superingular points of X0 F17 ie the enhanced elliptic curves E E C where E is a supersingular elliptic curve de ned over F17 and C is a cyclic subgroup of order M and enhanced curves are identi ed if they are isomorphic in the evident way Let wE W where AutE is the group of Fpautomorphisms of El We have 10 S 12 and if p 2 5 then 10 g 3 The monodromy pairing on D is 39 7 ltE7E gt WE TfE E 0 1fE7 E The Hecke operators act on D in a way compatible with this pairingi De ne XNm ZaEEIZaE0 It is known that XNp C is isomorphic as a Hecke module to the subspace of 52 N C generated by newforms and oldforms of level pd for d Mi David Kohel 8 has implimented an algorithm which computes the action of the Hecke operators on XNp using the arithmetic of quaternion algebrasi 23 Enumerating quotients of J0N It is necessary to list all newforms of a given level Ni This can be done by decomposing the new subspace of the modular symbols N Q using the Hecke operators The characteristic polynomial of T2 is computed and then T2 is used to break up the space The process is applied recursively with T3 T5 H until it terminatesi After computing the decomposition we order the newforms as suggested by Cremona First by dimension Within each dimension7 in binary by the signs of the Atkin Lehner involutions7 eg7 7 7 7 7 7 etc When two forms have the same involutions7 order by lTrapl with ties broken by taking the positive trace rst For historical reasons this does not always agree with the ordering in Cremona s tables page 5 of There is only one case in our table in which the two ordering schemes disagree7 our 446B is Cremona s 446D 24 The Modular Polarization A polarization of an abelian variety A is an isogeny between A and its dual arising from a very amply invertible sheaf see 14 J0 N is a Jacobian so it possesses a canonical polarization arising from the 9divisor and this induces the modular polarization 6f A 7gt Af A JoN Af By 14 7 Theorem 133 degt9f is a perfect square so we may de ne the modular degree 6f deg 9f The kernel of Of is the intersection of A with If J0 N so it measure intersections between A and other factors of Jo Proposition 21 With notation as in Theorem 11 Kert9f E CokerHIf 7gt fH Proof Delete the middle column of the diagram in Theorem 11 and apply the snake lemma D Using Lemma 257 to be proved later7 we see that the modular degree can be computed as follows Let gob 9020 be a basis for HomHZIf and a1 agd a basis for HUf Then 6f is the square root of the absolute value of the determinant of the matrix goiaj 25 Torsion We can obtain both upper and lower bounds on Af Q or and A Qmr For the examples considered in our table these bounds were sufficient to determine the odd parts of these groups Let XpX E ZX denote the characteristic polynomial of Tp acting on Af It is a polynomial having integer coef cients and degree equal to dim Af Proposition 22 Both lAfQmrl and lAJ Q orl divide gcdxpp 1 I P 2N 17 prime Proof Use the EichlerShimura relation and injectivity of rational torsion under reduction modulo an odd prime p since 1 e lt p 7 1 The difference of cusps 16 6 X0 N de ne a point 1 7 E J0N Proposition 23 The order of the image of 1 7 in AfC equals the order of the image of the modular symbol 15 in wow zgtgty gtf N7 Z Proof By the classical AbelJacobi theorem 107 ch lV7 Theorem 227 the modular symbol 15 maps7 via the period map7 to the point 1 7 E J0N Now use Theorem 11 1 ln particular7 the point 0 7 E J0N generates a cyclic subgroup of Af and this gives a lower bound on AfQmr 26 Component Groups Let p be a prime exactly dividing N and AJ the component group of A Af Thus we have the exact sequence 07A HAFP 7gt Am7gt0 with A the Neron model of A and AP connected Let XIf be the submodule of X XNp cut out by the annihilator of The monodromy pairing de nes a map X 7gt HomXIf7 Z Let 5f be the modular degree and wp the sign of the AtkinLehner involution Wp on The following will be proved in 18 Theorem 24 lCokerX 7gt HomXIf7 Zl2 W476 6f DiscXIfgtltXIfHZ 7 Azdfpll wq 1 WW Am p l wq71 27 Rational part of the special value Let MN7 Q and extend 7 to a map 7gt C Then 39i39f has a rational structure in the following sense Lemma 25 Let 01 gon be 1 Qbasis for HomMQ7 QIf and set 11 1XXsonIMQHQ Then n 2d and Ker 11 Ker f Proof This result is due to Shimura 17 but we sketch a proof To compute dimHomMQ QIf we may rst tensor with Cl Let g2 denote the weight 2 antiholomorphic cusp forms and E2 the weight 2 Eisenstein series for To Then MC is isomorphic as a Tmodule to S2 2 69 E2 propi 9 of 12 and the Eichler Shimura embeddingi Because of the Peterson inner product the dual HomMC C is also isomorphic as a Tmodule to 52 EB g2 69 E2 Since f is new by the Atkin Lehner multiplicity one theory 52 69 EB E2l1fl 52W 2l1fl has complex dimension 2d which gives the rst assertioni Next note that Ker f E C C Ker 11 E C because each map I gt gt lies in HomMQ CIf and Ker 11 C is the intersection of the kernels of all maps in HomMQ CIfi By Theorem 11 the image of 7 is a lattice so dimQ Ker f dimQ 7 2d Since 11 is the intersection of the kernels of n 2d independent linear functionals 01 i i son Ker 11 also has dimension dimMQ 7 2dr Since the dimensions are the same and there is an inclusion we have an equality Ker f C Ker 11 C which forces Ker f Ker 11i 1 Let V be a nite dimensional vector space over Ri A lattice L C V is a free abelian group of rank dimV such that RL Vi If LM C V are lattices the lattice index L M is the absolute value of the determinant of an automorphism of V taking L isomorphically onto Mi Extend the de nition to the case when M has rank strictly smaller than dim V by de ning L M 0 Lemma 26 Suppose Ti V A Wi i 12 are surjectz39ve linear maps such that Ker7391 Ker7392 Then 71L171Mll r2LIT2Mlt Proof Surjectivety and equality of kernels insures that there is a unique iso morphism L z 1 A 2 such that L71 72 et a be an automorphism of W1 such that 07391L MM Then earlma L071 L tn M 72MA Since conjugation doesn t change the determinant 72L 72M l detltwr1gtt ldetal ML mm B Let 52N Z be the space of cusp forms whose q expansion at in nity hass integer coef cients Let Q be the measure of the identity component of Af R with respect to an integral basis for SfZ 52N ZIfi Let e 000 6 MN Z denote the winding elementi Theorem 27 Let 1 be as in Lemma 25 Then i mam Z Te f Proof Let 39i39 39i39f be the period map de ned by a basis fl7 i H 7 d of conjugate newformsi The image of 39i39 which we identify with Cquot7 is an algebra with unit element 1 11 l i 7 1 equipped with an action of the Hecke operators Tp acts as a17 i H 191 where the components are the Galois conjugates of up Let Zd C Rd C Cd be the usual submodulesi Let Vol be the volume of the image of 1r N7 ZJr under Q Observe that Vol Zd and lLAf1l Zd eZdi Let W C Cd be the Zmodule spanned by the columns of a basis for SfZi Because 9 is computed with respect to a basis for SfZ7 Vol W T1 0 Because 52N7 Z is saturated7 Zd W 1 so Zd T1 W T1 The following calculation involves lattices in Rd l lt Te I Z01 I Zd Z Te 1 39 W 1 WM 1 W Z0I eZd QZd I Te 7 w e e 7 vol W W NT 1 lLAf71l W ltIgtezd ltIgteT1 w n The theorem now follows from lemmas 2 57 216 and the fact that f has real Fourier coef cients so LAf1 E R hence lLAf1l iLAf1i 1 Corollary 28 Let nf be the order of the image in Af 0f the point 0 7 00 E J0NQ Then L A 1 1 w E LZ f f Proof Let 1 denote the image of 0 7 E AfQ and set I Annz C Ti Since f is a hewform the Hecke operators Tp for plN act as 0 or i1 on AfQ end of section 6 of 1f pl N a standard calculation section 218 of combined with the AbelJacobi theorem shows that 1171 10 1zi Let C Zz denote the nite7 by Manin Drinfeld cyclic subgroup of AfQ generated by I so nf is the order of C There is an injection TI lt gt C sending Tp to Tp By the theorern7 we have iLAf1Q WE Me WE we We we WE we We we IIPelei T 1JeI 1Je W The nal inclusion follows from two observations By AbelJacobi7 I is exactly those elements of T which send e into l so I IJe E Z Second7 there is a surjective rnap T 11e TI a 1M6 sending t to t 11e7 so T 11e21 11e divides nf lCl lTIl D 28 Intersections Let g be nonconjugate newforrns and H H1X0N7 Z Proposition 29 AV AZp y 0 239 the mod p rank of HIf HIg is strictly less than rank HIf rankHUg Proof By 11 Af HIf resp7 Ag HUgl is the subrnodule of H which de nes Af resp7 Ag By reduction mod p we mean the map H A H Fp Suppose rankAf Ag rnodp lt rank Af rank Ag Since Af resp7 Ag is a kernel7 it is saturated7 so rank Af rnodp rank Af resp7 for Ag We conclude that the mod p linear dependence must involve vectors from both Af and Ag there is v E Af and w 6 Ag so that v10 Ornodp but 1 w E Ornodp Thus 51 E H is integral7 ie7 in J0NC we have v 7 710 0 But 1 g Af and w e Ag otherwise 1 and w would be Ornodp7 so v and 711710 are both nontrivial p torsion in AV7 AZ resp Conclusion 0 f 1v 7w E A N AZMp Conversely7 suppose 0 f z E A N AZHp Choose lifts rnodulo H to If E Af and 1g 6 Ag Thenpzf E Af resp7 pry E Ag7 but pzf pH resp7 pry pH because I f 0 Since zfizg 6 H7 pzfing p1f71gE Ornodp This is a nontrivial linear relation between Af and Ag D Corollary 210 fp gt 2 and the sign of some AtkinLehner involution for f is di erent than that for y then A N 1le 0 Proof Suppose wqf wqg and let G A N AZHp Observe that Wq acts as wqf rnodp on Avp and as wqg rnodp on A Hence Wq acts as both wqf rnodp and wqgg rnodp on G Since p gt 27 this is not possible when G a 0 El 3 Results This section contains tables computed using the above algorithms as impliment ed in the author s program HECKE a C program using LiDIA and NTL David Kohel7s Magma software and PARI Each factor Af of Jo N is denoted as follows N isogenyClass dimension The dimension frequently determines the factor so it is included in the notation We consider only the odd part of 111 so we only computed the odd parts of the arithmetic invariants of Afr Thus at this point we make the WARNING ONLY ODD PARTS OF INVARIANTS ARE GIVEN Tables 13 New Visible 111 Let nf be the largest odd square dividing the numerator of LAf 0in Table 1 lists those Af such that for plnf there exists a new factor Bg of J0N of positive analytic rank and such that A N BZHp 0 This is necessary and usually sufficient for the p torsion in the new visible part of 111 to be nonzero In many cases it could be seen that there were no other appropriate new factors by looking at the signs of the Atkin Lehner involutionsi Up to level 1001 our search was systematici The two examples after level 1001 were not found by systematic search ie there may be a gap In those cases for which 41N we put Cg a as we don7t know how to compute 02 exactly when the reduction is additivei Table 2 contains further arithmetic information about each explanatory factor The explanatory factors of level S 1028 are exactly the set of rank 2 elliptic curves of level S 10281 By 2 the explanatory factor at level 1061 is the rst surface of rank 4 and prime level Table 4 Component groups Table 4 gives the quantities involved in the formula for Tamagawa numbers for each of the Af from table 1 Table 5 Odd square numerator In order to nd the Af we rst enumerated those Af for which the numerator of LAf 1Qf is divisible by an odd square nfi For N lt 1000 these are given in table 5 Any odd visible 111 coprime to primes dividing torsion and up must show up as a divisor of the numerator and given BSD it must show up as a square divisor because the Mordell Weil rank of the explanatory factor is even It would be interesting to compute the conjectural order of 111 for each abelian variety in this table but not in table 1 and show when possible that the visible 111 is old Af nf wq up T TL1Qf 6A Bg 389E20 52 7 97 97 52 5 389A1 433D16 72 7 32 32 72 3 7 37 433A1 446F8 112 7 1 3 3 112 11 359353 446B1 563E31 132 7 281 281 132 13 563A1 571D2 32 7 1 1 32 32 127 571B1 655D13 34 7 11 1 34 32 19 515741 655A1 664F8 52 7 a1 1 52 5 664A1 681B1 32 7 11 1 32 3 53 68101 707015 132 7 11 1 132 13 800077 707A1 709030 112 7 59 59 112 11 709A1 718F7 72 7 11 1 72 7 151 35573 718B1 794014 112 7 31 3 112 3 7 11 47 35447 794A1 817E15 72 7 1 5 5 72 7 79 817A1 91609 112 7 a1 1 112 3g 11 17 239 91601 94406 72 7 a1 1 72 7 944E1 997H42 34 7 83 83 34 32 997B101 1001L7 72 7 111 1 72 7 19 472273 100101 1028E14 32112 7 a1 3 34 112 313 11 1028A1 1061D46 1512 7 553 553 1512 61 151 179 1061B1 Table 1 New Visible LU Bg rank wq up T 63 Comments 389A1 2 7 1 1 5 rst curve of rank 2 433A1 2 7 1 1 7 446B1 2 7 11 1 11 this is 446D in 3 563A1 2 7 1 1 13 571B1 2 7 1 1 3 655A1 2 7 11 1 32 664A1 2 7 11 1 5 68101 2 7 11 1 3 707A1 2 7 11 1 13 709A1 2 7 1 1 11 718B1 2 7 11 1 7 794A1 2 7 11 1 11 817A1 2 7 11 1 7 91601 2 7 31 1 3 11 944E1 2 7 11 1 7 997B1 2 7 1 1 3 99701 2 7 1 1 3 100101 2 77 131 1 327 1028A1 2 7 31 1 3 11 intersects 1028B mod 11 1061B2 4 7 1 1 151 rst surface of rank 4 2 Table 2 Explanatory factors 4462223 6555131 6642 83 6813227 7077101 7182359 7942397 8171943 91622229 9442459 100171113 102822257 Table 3 Factorizations Af p wp 1 Coker 1 DiscXIf Fp 389E20 389 7 597 97 433D16 433 7 33 7 37 32 446F8 223 7 3 3 11 359353 3 2 3 3 11 3 359353 563E31 563 7 281 13 281 281 571D2 571 7 1 32 127 1 655D13 131 7 1 32 19 515741 1 5 1 32 19 515741 664F8 83 1 5 1 681B1 227 7 1 3 53 1 3 1 3 52 5 707G15 101 7 1 13 800077 1 7 1 13 800077 709030 709 7 59 11 59 59 718F7 359 7 1 7 151 35573 1 2 1 7 151 35573 794G14 397 7 3 32 7 11 47 35447 3 2 3 311 3274735447 817E15 43 7 5 5779 5 19 1 7 79 916G9 229 1 3g 11 17 239 1 94406 59 7 1 7 1 997H42 997 7 83 32 83 83 1001L7 13 1 7 19 47 2273 1 11 7 1 719472273 1 7 1 7 19 47 2273 1028E14 257 1 313 11 1 1061D46 1061 7 553 55361 151 179 553 Table 4 Component groups 305D7 3 309D8 5 335E11 32 389E20 5 394A2 5 399G5 34 433D16 7 435G2 3 43604 3 446E7 3 446F8 11 455D4 3 473F9 3 50004 3 502E6 11 50614 5 524D4 3 530G4 7 538E7 3 551H18 3 553D13 3 555E2 3 55607 3 563E31 13 56403 3 571D2 3 579G13 3 5 597E14 19 602G3 3 60406 3 615F6 5 615G8 7 620D3 3 620E4 3 626F12 5 629G15 3 642D2 3 64405 3 644D5 3 655D13 32 660F2 3 662E10 43 664F8 5 668B5 3 67812 3 681B1 3 681110 3 68216 11 707G15 13 709030 11 718F7 7 721F14 32 72408 3 756G2 3 764A8 3 765M4 3 766B4 3 77209 3 790H6 3 794G12 11 794H14 52 79608 3 817E15 7 82004 3 825E2 3 844010 32 855M4 3 860D4 3 868E5 3 876E5 3 87802 3 884D6 3 885L9 32 894H2 3 90215 3 913G17 3 916G9 11 91802 5 918P2 3 925K7 3 932B13 32 933E14 19 934112 7 94406 7 946K7 3 949B2 3 951D19 3 959D24 3 964012 32 96611 3 97015 3 980F1 3 980J2 3 98617 5 989E22 5 993B3 32 996E4 3 997H42 32 998A2 3 998H9 3 999J 10 3 Table 5 Odd square numerator References 1 Al Agash On invisible elements of the TateShafarevich group Theorie des nombres 328 1999 3697374 2 Al Brumer The rank of J0N Ast risque 1995 no 228 3 41768 Columbia University Number Theory Seminar New York 1992 3 l E Cremona Algorithms for modular elliptic curves second ed Cam bridge University Press Cambridge 1997 4 1 El Cremona and B Mazur Visualizing elements in the Shafarevich Tate group Proceedings of the Arizona Winter School 1998 E Hl Darmon Fl Diamond and Rl Taylor Fermat s last theorem Curren t developments in mathematics 1995 Cambridge MA lnternatl Press Cambridge MA 1994 pp 17154 E F Diamond and 1 1m Modular forms and modular curves 1995 3 Cl Frey and Ml Miiller Arithmetic of modular curves and applications Al gorithmic algebra and number theory Heidelberg 1997 Springer Berlin 1999 pp 11748 E Dl Kohel Hecke module structure of quaternions 1998 9 Vl Al Kolyvagin and Dl Yul Logachev Finiteness of the ShafarevichTate group and the group of rational points for some modular abelian varieties Algebra i Analiz 1 1989 no 5 1717196 10 S Lang Introduction to modular forms SpringerVerlag Berlin 1995 With appendixes by Dl Zagier and Walter Feit Corrected reprint of the 1976 original 11 B Mazur Rational isogenies ofprime degree with an appendix by D Gold feld lnventl Math 44 1978 no 2 1297162 12 L Merel Universal Fourier expansions of modular forms On Artinls con jecture for odd 2dimensional representations Berlin Springer 1994 p p 59794 13 JlFl Mestre La m thode des graphes Exemples et applications Proceed ings of the international conference on class numbers and fundamental units of algebraic number elds Katata 1986 2177242 14 1S Milne Abelian varieties Arithmetic geometry Storrs Connl 1984 Springer New York 1986 pp 1037150 15 Kl Al Ribet On modular representations of gal q arising from modular forms lnventl Math 100 1990 no 2 4317476 16 Cl Shimura On the factors of the jacobian variety of a modular function eld 1 Math Soc Japan 25 1973 no 3 5237544 17 On the periods of modularfor39ms Math Ann 229 1977 2117221 18 Wl Stein Component groups of optimal factors of J0N Preprint 1999l

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