On a conjecture of Gross and Zagier

0.27 MB·English

by  Dongho Byeon

ON A CONJECTURE OF GROSS AND ZAGIER
DONGHOBYEON,TAEKYUNGKIM,ANDDONGGEONYHEE
ABSTRACT. LetE beanellipticcurvedefinedoverQofconductorN,letMbetheManinconstantofE,andCbe
theproductoflocalTamagawanumbersofE atprimedivisorsofN. LetKbeanimaginaryquadraticfieldinwhich
eachprimedivisorofNsplits,PKbetheHeegnerpointinE(K),andX(E/K)betheTate–ShafarevichgroupofEover
K. Also,let2uK bethenumberofrootsofunitycontainedinK. In[11],GrossandZagierconjecturedthatifPKhas
infiniteorderinE(K),thentheintegeruK C M (#X(E/K))1/2isdivisibleby#E(Q)tors.Inthispaper,weshowthat
· · ·
thisconjectureistrue.
5
1
0
2

v CONTENTS
o
N 1. Introduction 2

2
Part1. E(Q)tors hasorderapowerof2 4

] 2. PreliminariesforPart1 4
T
2.1. Kramer’sformula 4
N
2.2. IsogenyinvarianceoftheGross–Zagierconjecture 6
.
h 3. E(Q) Z/2Z Z/4Z 7
tors
t ≃ ⊕
a 4. E(Q) Z/2Z Z/2Z 8
tors
m ≃ ⊕
5. E(Q) Z/4Z 10
tors
≃
[ 5.1. Tamagawanumbers 10

2 5.2. (#X(E/K))1/2 11
v 5.3. Exceptionalcase 13
6
6. E(Q) Z/2Z 14
9 tors≃
2 6.1. Tamagawanumbers 14
6 6.2. (#X(E/K))1/2 15
0
6.3. Exceptionalcase 17
.
1
0
Part2. E(Q) hasapointoforder3 18
5 tors
1 7. PreliminariesforPart2 18
: 7.1. Optimalcurves 18
v
i 7.2. Cassels’theorem 18
X
8. E(Q) Z/2Z Z/6Z 18
tors
r ≃ ⊕
a 9. E(Q)tors Z/3Z 20
≃
9.1. Tamagawanumbers 20
9.2. Maninconstants 20
9.3. (#X(E/K))1/2 20
References 21
Date:November3,2015.
2010MathematicsSubjectClassification. 11G05.
Key words and phrases. elliptic curves, isogeny, optimal curves, Gross–Zagier conjecture, Birch and Swinnerton-Dyer conjecture,
Mordell–WeilGroups,Tamagawanumbers,Maninconstant,descentonellipticcurves.
ThefirstauthorwassupportedbyBasicScienceResearchProgramthroughtheNationalResearchFoundationofKorea(NRF)funded
bytheMinistryofEducation(NRF-2013R1A1A2007694). ThesecondauthorwaspartiallysupportedbyGlobalPh.D.FellowshipProgram
throughtheNationalResearchFoundationofKorea(NRF)fundedbytheMinistryofEducation(grantnumber2011-0007588). Thethird
author was supported by Basic Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of
Education(2013053914).
1
1. INTRODUCTION
ThegoalofthispaperistoproveaconjecturemadebyGrossandZagierin[11]concerningcertaindivisibility
amongarithmeticinvariantsofellipticcurves.Thisgivesatheoreticalevidencetothe“strongform”ofBirchand
Swinnerton-Dyer conjecture, predicting that the leading coefficient of the Hasse–Weil L-function of an elliptic
curveencodessomeprecisearithmeticinvariantsofthecurve.
In[11],GrossandZagiergaveaformulaforthefirstderivativeats=1ofL-seriesofcertainmodularforms.
In particular, they transferred the formula to the realm of L-functions of elliptic curves. So let E be an ellip-
tic curve defined over Q with conductor N. For a negative square-free integer d, we consider the quadratic
twist E ofE which isin generalnot isomorphictoE overQ butbecomesisomorphicoverthe imaginaryqua-
d
dratic field K =Q(√d). We denote the discriminant of K over Q by disc(K) which is equal to d when d 1
≡
(mod 4)andto 4d otherwise. We also assumea close relationbetweenE andK in sucha way thateachprime
number dividing N splits completely in K. This is called the Heegner condition or Heegner hypothesis in the
literature, which we assume throughout this paper. The corresponding L-functions are also strongly related:
we have L(E/K,s)=L(E/Q,s) L(E /Q,s). By computing root numbers, the Heegner condition forces that
d
·
L(E/K,1)=0. Throughoutthispaper,weusethefollowingnotations.
N istheconductorofE.
•
w istheNérondifferentialofE overQand w 2:= w w¯ isthecomplexperiod.
• k k E(C)| ∧ |
hˆ istheNéron–TateheightattachedtoE.
R
•
M is the Manin constant of E, i.e., if f is the newformattached to E and p :X (N) E is a modular
0
• →
parametrisation, then M is the ratio satisfying p w =M 2p if(t )dt . We have M Q and a famous
∗ ×
· ∈
conjectureofY.ManinisthatM=1forallstrongWeilcurvesE. Forgeneraldiscussionsontheconstant
andcurrentstatusabouttheconjecture,see[1].
P E(K)istheHeegnerpointoverK.Thisdependsontheellipticcurveanditsmodularparametrisation
K
• ∈
chosen.
2u isthenumberofrootsofunitycontainedinthefieldK. u =1forallimaginaryquadraticfieldsK
K K
•
exceptwhenK=Q(√ 1)andK=Q(√ 3),inthesecaseswehaveu =2andu =3respectively.
K K
− −
CistheTamagawanumberofE overQwhichisdefinedbytheproductC=(cid:213) C ofalllocalTama-
pN p
• |
gawanumbers.
NowthemaintheoremofGrossandZagier([11],TheoremI.6.3)hasthefollowingconsequence.
Theorem1.1([11],TheoremV.2.1).
w 2 hˆ(P )
K
L′(E/K,1)= M2 ku2k d·isc(K)1/2. (1)
· K·| |
NowtheBirchandSwinnerton-Dyerconjecturecomesintothepicture.Weassumehereandthereafterthatthe
HeegnerpointP hasinfiniteorder,sothatL(E/K,1)=0.Formoredetailsforthefollowingconjecture,werefer
K ′
6
[27],appendixC.16.
Conjecture1.2(BirchandSwinnerton-Dyer). Iford L(E/K,s)=1,thentheTate–ShafarevichgroupX(E/K)
s=1
ofE overK isfinite,andL(E/K,s)=BSD ,where
′ E/K
w 2 C2 hˆ(P ) #X(E/K)
K
BSD = k k · · · . (2)
E/K disc(K)1/2 [E(K):ZP ]2
K
| | ·
Remark. In the literature, the factorC2 in the right hand side of the equation (2) is replaced by the Tamagawa
numberofE overtheextensionK. However,bytheHeegnerhypothesis,anyprime pdividingN splitsinK like
p=pp,andthusthenumberisequaltothesquareC2 oftheTamagawanumberofE overQ.
Remark. TheTate–ShafarevichgroupX(E/K)isinfactfiniteinthiscase(cf.Theorem5in[15]).
Equatingtheabovetwoformulae(1)and(2),GrossandZagierobtainedthefollowingconjecture.
Conjecture1.3([11],ConjectureV.2.2,StrongGross–ZagierConjecture). IfP hasinfiniteorderinE(K),then
K
ZP hasfiniteindexinE(K)andwehave
K
[E(K):ZP ]=u C M (#X(E/K))1/2. (3)
K K
· · ·
2
AstheorderoftherationaltorsionsubgroupE(Q) clearlydividestheindex[E(K):ZP ],theyalsoobtained
tors K
aweakerversionoftheconjecture,whichwecall“theGross–Zagierconjecture”throughoutthispaper.
Conjecture1.4([11], ConjectureV.2.3, Weak Gross–ZagierConjecture). IfE(K) hasanalyticrank1, thenthe
integeru C M (#X(E/K))1/2isdivisibleby#E(Q) .
K tors
· · ·
Rational torsion subgroupsof elliptic curves E over Q are completely classified by Mazur [21]: E(Q) is
tors
isomorphictooneofthefollowinggroups:
Z/nZ for1 n 10, n=12,
≤ ≤
(Z/2Z Z/nZ forn=2,4,6,8.
⊕
In[18],Lorenziniobtainedthefollowingtheorem.
Theorem1.5([18],Proposition1.1). LetE beanellipticcurvedefinedoverQwithaQ-rationalpointoforder
k. Thenthefollowingstatementsholdwithatmostfiveexplicitexceptionsforagivenk. Theexceptionsaregiven
bytheirlabelsinCremona’stable[7].
(a) Ifk=4,then2 C,exceptfor‘15a7’,‘15a8’,and‘17a4’.
|
(b) Ifk=5,6,or12,thenk C,exceptfor‘11a3’,‘14a4’,‘14a6’,and‘20a2’.
|
(c) Ifk=7,8,or9,thenk2 C,exceptfor‘15a4’,‘21a3’,‘26b1’,‘42a1’,‘48a6’,‘54b3’,and‘102b1’.
|
(d) Ifk=10,then50 C.
|
Withoutexception,k Cifk=7,8,9,10or12.
|
Fortheexceptionsofaboveproposition,wecancheckthat#E(Q) dividesC M, exceptfor‘15a7’,which
tors
·
is considered in §5. So the only remaining cases for the validity of the conjecture are those when E(Q) is
tors
isomorphic to the following 6 groups: Z/2Z, Z/3Z, Z/4Z, Z/2Z Z/2Z, Z/2Z Z/4Z, and Z/2Z Z/6Z.
⊕ ⊕ ⊕
Ourgoalhereistoprovetheseremainingcases,thustocompletetheproofoftheconjecture.
Main Theorem. Let E be an elliptic curve defined over Q such that the rational torsion subgroup E(Q) is
tors
isomorphictooneofthe6groups: Z/2Z,Z/3Z,Z/4Z,Z/2Z Z/2Z,Z/2Z Z/4Z,andZ/2Z Z/6Z. LetK
⊕ ⊕ ⊕
beanimaginaryquadraticfieldsuchthatE(K)isof(analytic)rank1andthatK satisfiestheHeegnercondition.
Thentheconjecture1.4istrue,i.e.,#E(Q) dividesC M u (#X(E/K))1/2.
tors k
· · ·
Fromnowon, E alwaysdenotesanelliptic curvedefinedoverQ havingtorsionsubgroupisomorphicto one
of the above6 groups, and K is always an imaginary quadraticfield such that ord L(E/K,s)=1 and that K
s=1
satisfiestheHeegnerhypothesis.
LetusbrieflyexplainhowtoprovetheMainTheorem. Thepresentarticleisdividedintotwoparts. Thefirst
part(§2 §6)isdealingwiththecasethatE(Q) hasorderapowerof2.WhenE(Q) containsfull2-torsion
tors tors
∼
subgroupE[2],i.e.,whenE(Q) Z/2Z Z/2Z,orZ/2Z Z/4Z,thesituationsarealoteasierthantheother
tors
≃ ⊕ ⊕
cases, and we can prove the Main Theorem by computing Tamagawa numbers using Tate’s algorithm (§3 and
§4). Fortheothercases, i.e., whenE(Q) Z/2ZorZ/4Z, therearecurveshavingTamagawanumbersnot
tors
≃
divisibleby#E(Q) ,soweneedtocomputethesizeofthe2-torsionpartoftheTate–Shafarevichgroupsover
tors
K using Kramer’s formula. There are some ‘exceptionalfamilies’ for whichC (#X(E/K))1/2 does not have
·
enoughpowerof2. Forthesecases,weavoiddifficultiesbyconsideringisogenyinvarianceoftheGross–Zagier
conjecture. Kramer’sformulaandtheisogenyinvariancearelocatedattheheartoftechniquesintheproof,soin
thepreliminarysection§2wegivesufficientbackgroundtothesetechniques.
The second part (§7 §9) is devoted to the case in which E(Q) has a rational torsion point of order 3.
tors
∼
WhenE(Q) Z/2Z Z/6Z,wecanprovetheMainTheorembycomputingonlyTamagawanumbers(§8).
tors
≃ ⊕
But whenE(Q) Z/3Z, thereare also curveshavingTamagawanumbersnotdivisibleby #E(Q) , so we
tors tors
≃
needtocomputethelowerboundofthesizeofthe3-torsionpartofTate–ShafarevichgroupsoverKusingCassels’
formulaorneedtocomputetheManinconstantsusingthephenomenonthatoptimalcurvesdifferbya3-isogeny
(§9). SameasPart1,preliminariesaresummarisedin§7.
Allexpicitcomputationsin thispaperweredoneusingSageMathematicsSoftware[30]. Whenwe docom-
putations with Weierstrass equations, we frequently change the variables of an equation to obtain another. In
particular,whenweusetheclause“makeachangeofvariablesvia[u,r,s,t]”,itshouldbeunderstoodtotakethe
changeofvariablesformulagivenby
x=u2x +r and y=u3y +u2sx +t.
′ ′ ′
3
Forthedetails,werefer[27],§III.1.
Part1. E(Q) hasorderapowerof2
tors
2. PRELIMINARIES FORPART 1
2.1. Kramer’sformula. InthissubsectionweintroduceaformulaofKramer[16],anddiscusshowtomeasure
thesizeoftheTate–Shafarevichgroupofellipticcurveusingit. Ofcoursethepurposeofthissectionistoprovide
atooltoshowtheMainTheoremforthecasesE(Q) Z/2ZorZ/4Z. Thus,throughoutthissubsection,we
tors
≃
assumeE(Q) Z/2ZorZ/4Z,andconsequentlyE(Q)[2] Z/2Z.
tors
SincetheTate≃–ShafarevichgroupX(E/K)isfinite(Theor≃em5in[15]),its2-primarypartX(E/K)[2¥ ]has
perfectsquareorder. Soifwefindanon-trivialelementinX(E/K)[2],(orequivalentlydim X(E/K)[2] 1),
F2 ≥
wecanimmediatelyseethat2 (#X(E/K))1/2. Sointhissubsection,weareconcentratingonhowtofindsuch
|
anon-trivialelement.
Let pbeaprimenumber.Weusethefollowingnotations.
i =dim CokerN=dim E(Q )/NE(K ),whereN:E(K ) E(Q )isthenormmap. Thisquantity
• p F2 F2 p p p → p
iscalledlocalnormindexofE at p.
Let
•
F = x Sel2(E/Q):x N (cid:213) Sel2(E/K ) .
p p
( ∈ ∈ pp !)
|
Thisgroupiscalledtheeverywhere-localnormgroup.
NS istheimageofthenormmapSel2(E/K) Sel2(E/Q),whichwedonotneedinthispaper.
′
• →
Theorem2.1([16],Theorem1). ThedimensionofX(E/K)[2](overF )isequalto
2
(cid:229) i +dim F +dim NS rankE(K) 2dim E(Q)[2],
ℓ F2 F2 ′− − F2
wherethesumistakenoverallprimes(includinginfinity)ofQ.
Backtoourcase. BecauserankE(K)=1andE(Q)[2] Z/2Z,byTheorem2.1,dim X(E/K)[2] 1ifand
≃ F2 ≥
onlyifthequantity
(cid:229) i +dim F +dim NS
ℓ F2 F2 ′
isgreaterthanorequalto4.
2.1.1. Local norm indices. For generalintroductionand useful facts aboutthe numbersi , we refer §4 of [19]
p
and§2of[16]. Weonlyconcernthosenumbersrelevanttooursituation. Theproofofthefollowingproposition
canbefoundin§2of[16].
Proposition2.2. LetE beanelliptic curveoverQ with E(Q)[2] Z/2ZandletK =Q(√d)beanimaginary
≃
quadratic field satisfying the Heegner hypothesis. The local norm indices i for various primes ℓ are given as
ℓ
follows.
0 ifD <0,
min
(a) Onehasi¥ =i(C R)=
| (1 ifD min>0.
(b) Let pbeanoddprime. If pisagoodprimeforE andisramifiedinK,thenonehasi =dim E[2](k),
p F2
wherekistheresiduefieldofQ . Otherwiseonehasi =0.
p p
2 if(D ,d) =+1,
(c) If 2 is a good prime for E and is ramified in K, then one has i = min Q2 where
2 (1 if(D min,d)Q2 =−1,
( , ) denotestheHilbertnorm-residuesymbol. Otherwise,onehasi =0.
− − Q2 2
2.1.2. Everywhere-local norm group. Now we provide a way to compute the everywhere-localnorm group F .
Thefollowingisthekey.
Proposition2.3([16],Proposition7). Theeverywhere-localnormgroupF istheintersectionofSel2(E/Q)and
Sel2(E /Q)insideH1(Q,E[2]) H1(Q,E [2]),whereE isthequadratictwistofE byd.
d d d
≃
4
LetE bethequadratictwistofE byd. Inparticular,supposeE isdefinedbytheWeierstrassequation
d
y2=x3+Ax2+Bx, (4)
whichhasdiscriminantD =24B2(A2 4B).ThenE hastheWeierstrassequationoftheform
d
−
y2=x3+Adx2+Bd2x. (5)
Thediscriminantoftheaboveequation(5)isgivenbyD =16d6B2(A2 4B).
d
−
Proposition 2.4. The 2-torsion subgroups E[2] and E [2] are canonically isomorphic as Gal(QQ)-modules.
d
|
Consequently, the Galois cohomology groups H (Q,E[2]) and H (Q,E [2]) are isomorphic. In particular, we
• • d
identifyH1(Q,E[2])=H1(Q,E [2])inthesequel.
d
Proof. TheGalois-equivariantisomorphismE[2] E [2]isgivenby(t,0) (dt,0). (cid:3)
d
→ 7→
Denote by P (resp. P ) the rational torsion point of order 2 in E (resp. E ) corresponding to (0,0) in the
d d
equation(4)(resp.(0,0)intheequation(5)). LetE (resp. E )betheellipticcurveE/ P (resp. E / P )andlet
′ d′ h i d h di
f (resp. f )bethecanonicalquotient2-isogenyE E (resp.E E ).
d → ′ d → d′
Proposition2.5. Therearecanonicalhomomorphisms
H1(Q,E[f ]) H1(Q,E[2]), and H1(Q,E [f ]) H1(Q,E [2]),
d d d
→ →
andtheyinduce
Self (E/Q) Sel2(E/Q), and Selfd(E /Q) Sel2(E /Q).
d d
→ →
Proof. Ifwedenotetheuniquedualrational2-isogenyoff byf ,thenwehaveacanonicalexactsequence
′
0 E[f ] E[2] E [f ] 0. (6)
′ ′
−→ −→ −→ −→
This defines a canonical map H1(Q,E[f ]) H1(Q,E[2]) on cohomology groups, and it restricts to the map
Self (E/Q) Sel2(E/Q)ofsubgroups.For→E andf theproofismutatismutandisthesame. (cid:3)
d d
→
Proposition 2.6. There are canoncial isomorphisms H1(Q,E[f ]) Q /Q 2 and H1(Q,E [f ]) Q /Q 2.
× × d d × ×
≃ ≃
Moreover,theisomorphismsarecompatibleinthesensethatthefollowingdiagramiscommutative:
H1(Q,E[f ]) // H1(Q,E[2])
5
❧❧❧❧❧❧∼❧❧❧❧❧❧❧❧5
Q×/Q×2≃H1(Q❘❘,❘m❘2❘)❘❘∼❘❘❘❘❘❘❘)) (cid:15)(cid:15) (cid:15)(cid:15) =
H1(Q,E [f ]) // H1(Q,E [2])
d d d
wheretheverticalmapinthemiddleisinducedbythecanonicalisomorphismintheProposition2.4.
Proof. Clearlytheisomorphismsm E[f ] and m E [f ]arecompatibleinthesensethelefttrianglecom-
2 2 d d
→ →
mutes. ByKummertheoryweknowH1(Q,m )=Q /Q 2,whencetheresultfollows. (cid:3)
2 × ×
Proposition2.7. LetG bethesubgroupofQ /Q 2 generatedbytheclassofA2 4B. ThenG isthekernelof
× ×
−
thehomomorphismsH1(Q,E[f ]) H1(Q,E[2])andH1(Q,E [f ]) H1(Q,E [2]). Thus,
d d d
→ →
Ker Self (E/Q) Sel2(E/Q) =G Self (E/Q) Self (E/Q).
→ ∩ ⊂
Similarly, (cid:0) (cid:1)
Ker Selfd(E /Q) Sel2(E /Q) =G Selfd(E /Q) Self (E /Q).
d d d d
→ ∩ ⊂
Proof. Weonlygiveapr(cid:0)oofforE andf . ForE andf(cid:1) ,everythingisthesameundermakingcertainnotational
d d
change.Fromtheshortexactsequence(6),wehavethelongexactsequenceofcohomologygroups:
h
0 E(Q)[f ] E(Q)[2] E (Q)[f ] H1(Q,E[f ]) H1(Q,E[2]) H1(Q,E [f ])
′ ′ ′ ′
→ → → −→ → → →···
BecauseweonlyconsiderthoseellipticcurveswithE(Q)[f ]=E(Q)[2],themapE(Q)[2] E (Q)[f ]isthezero
′ ′
→
map,andthisagainforcesusthath :E (Q)[f ] H1(Q,E[f ])isinjective.Theimageh (E (Q)[f ])isthekernel
′ ′ ′ ′
→
ofH1(Q,E[f ]) H1(Q,E[2]).
→
5
We claim thatthis kernelis equalto G. Write E(Q)[2]= O,P,Q,P+Q , where O is the identity of E and
{ }
P E(Q), and similarly write E (Q)[f ]= O,T , where O is the identity of E . Clearly T E (Q). Since
′ ′ ′ ′ ′ ′
∈ { } ∈
E(Q)[2] E (Q)[f ]issurjectivebutE(Q)[2] E (Q)[f ]isthezeromap,thepointQismappedontoT under
′ ′ ′ ′
→ →
E(Q)[2] E (Q)[f ]. Then,h (T) H1(Q,E[f ])isdefinedbythe1-cocyle
′ ′
→ ∈
P ifs (Q)=P+Q=Q,
s s (Q) Q= 6
7→ − (0 ifs (Q)=Q.
However, this 1-cocycle corresponds to the 1-cocycle s s (√b)/√b defining an element H1(Q,m ), where
2
7→
A √A2 4B
b=A2 4B, since in the Weierstrass equation (4), Q correspondsto the point − ± − ,0 and thus
− 2 !
s (√A2 4B)
s (Q)=Q if and only if s √A2 4B = √A2 4B. Clearly the 1-cocycle s − defining an
− − 7→ √A2 4B
elementH1(Q,m )correspon(cid:16)dstoA2 4(cid:17)BinQ /Q 2. − (cid:3)
2 × ×
−
Recall (Proposition 2.3) that the everywhere-local norm group F is the intersection of two Selmer groups
Sel2(E/Q)andSel2(E /Q)insideH1(Q,E[2])=H1(Q,E [2]). Inordertoidentifyelementsintheintersection,
d d
weneedtofindb Q /Q 2suchthatb Self (E/Q) Selfd(E/Q)bydescentarguments(cf. [27],chapterX).
× ×
∈ ∈ ∩
InordertoensurethisisnottheidentityelementinF ,weshouldcheckb G. Thiswillbedonewhenwedeal
6∈
withE(Q) Z/4ZorZ/2Z.
tors
≃
2.2. IsogenyinvarianceoftheGross–Zagierconjecture. LetEandE beisogenousellipticcurvesdefinedover
′
Q,andK beanimaginaryquadraticfieldsatisfyingtheHeegnerhypothesis. Weconsiderthosecurveswithfixed
modularparametrisationsp :X (N) E andp :X (N) E .
0 ′ 0 ′
→ →
Proposition2.8. Letq :E E bearationalisogeny.
′
→
(a) IfthestrongGross–Zagierconjecture(Conjecture1.3)istrueforE thenitisalsotrueforE .
′
(b) Supposethatq respectsmodularparametrisationsofE andE ,i.e.,p =q p .Thenwehave
′ ′
◦
M2 C2 #X(E/K) M2 C2 #X(E /K)
· · = ′ · ′ · ′ . (7)
[E(K):ZP ]2 [E (K):ZP ]2
K ′ K′
(c) Let pbeaprime. If
(i) ord #E(K) =ord #E(Q) ,and
p tors p tors
(ii) ord #E(Q) ord u C M (#X(E/K))1/2 ,
p tors p K
≤ · · ·
then (cid:16) (cid:17)
ord #E (Q) ord u C M #X(E /K) 1/2 .
p ′ tors p K ′ ′ ′
≤ · · ·
Inparticular, ifE(K) =E(Q) , andif(cid:16)the weakGro(cid:0)ss–Zagierc(cid:1)onje(cid:17)cture(Conjecture1.4) forE is
tors tors
true,thenitisalsotrueforE .
′
Proof. (a)IsogenouscurvesE andE havethesameL-functionsandthesameBSD formulae,i.e.,L(E/K,s)=
′
L(E /K,s) and BSD =BSD (cf. Conjecture 1.2). The latter is a theoremof Cassels [5]. As the strong
′ E/K E/K
′
Gross–Zagierconjectureisobtainedbysimplyequatingtheseformulae,itisclearlyisogenyinvariant.
(b)LetP betheHeegnerpointforE definedbyP =q (P ). SinceL(E/K,s)=L(E /K,s),wehave
K′ ′ K′ K ′ ′ ′
w 2 hˆ(P ) M2
K
k k · = .
w 2 hˆ(P ) M2
k ′k · K′ ′
Similarly,fromBSD =BSD ,weget
E/K E/K
′
w 2 hˆ(P ) #X(E /K) C2 [E(K):ZP ]2
K ′ ′ K
k k · = · · .
kw ′k2·hˆ(PK′) #X(E/K)·C2·[E′(K):ZPK′]2
Equating,weobtaintheequation7.
(c) Let P (resp. P) be a generator of the group E(K)/E(K) (resp. E (K)/E (K) ), and let P =n P
′ tors ′ ′ tors K
(resp. P =n P). AsP =q (P )=nq (P),theindexn isdivisiblebyn . Theassumption(i)ord #E(K) =
K′ ′ ′ K′ K ′ p tors
6
ord #E(Q) impliesthatord [E(K) :E(Q) ]=0. Bytheequation7,wehave
p tors p tors tors
u2 M2 C2 #X(E/K) u2 M2 C2 #X(E /K)
K· · · = K· ′ · ′ · ′ ,
(#E(Q)tors)2·[E(K)tors:E(Q)tors]2 nn′ 2·(#E′(Q)tors)2·[E′(K)tors:E′(Q)tors]2
(cid:16) (cid:17)
andbytheassumption(ii)thelefthandsideoftheaboveequationisa p-adicinteger.Thus,
n
ord u C M #X(E /K) 1/2 ord ′ #E (Q) [E (K) :E (Q) ]
p K· ′· ′· ′ ≥ p n · ′ tors · ′ tors ′ tors
(cid:16) (cid:0) (cid:1) (cid:17) ord (cid:18)#E (Q(cid:0)) . (cid:1) (cid:19)
p ′ tors
≥
(cid:3)
Remark. By[9]Corollary4or[25],Theorem2,foragivenellipticcurveE definedoverQ,thereareatmost4
quadraticfieldsK suchthatE(K) =E(Q) .
tors tors
6
3. E(Q) Z/2Z Z/4Z
tors
≃ ⊕
Inthissection,weprovetheMainTheoremforthecaseswhenE(Q) isisomorphictoZ/2Z Z/4Z.
tors
⊕
Theorem3.1. SupposethatE(Q) isisomorphictoZ/2Z Z/4Z. Thenthe order8=#E(Q) dividesthe
tors tors
⊕
TamagawanumberCofE,exceptforthecurve‘15a3’,inwhichcaseC M=8.
·
From[17],table3,suchellipticcurvescanbeparametrisedbyoneparameterl Qby
∈
y2+xy l y=x3 l x2, (8)
− −
a 2 1 16a 2 b 2
wherel = = − ,withpositiveintegersa ,b havingnocommonprimedivisor,anda /b =
b −16 16b 2 6
(cid:18) (cid:19)
1/4. ThediscriminantoftheequationisD =l 4(1+16l )=0. Notethatsincewetakea andb relativelyprime,
6
therearenocommonprimedivisorof16a 2 b 2and16b 2except2.
−
Proposition3.2. Let pbeaprime.
(a) Ifm:=ord l >0, thenthe reductionofE modulo pis (split)multiplicativeoftypeI . Consequently
p 4m
theTamagawanumberat pofE isC =4m.
p
(b) Supposethat p=2. Ifm:=ord l <0,thenmisalwayseven,andtheminimalWeierstrassequationat
p
6
pisgivenby
y2+pzxy upzy=x3 ux2, (9)
− −
where u Z satisfying l =upm in Z , and where z is a positive integer. The reduction type of the
∈ ×p p
equationmodulo pisI withn=2z,whenceC =2z.
n p
Proof. (a)ThiscanbeshownbydirectlyapplyingTate’salgorithm(see[26],§IV.9)totheWeierstrassequation
(8).
(b) Since gcd 16a 2 b 2,16b 2 is a power of 2, if m = ord l =ord (16a 2 b 2)/16b 2 <0 then the
p p
− −
exponent m is always even. Changing Weierstrass equation (cf. [18], proof of Proposition 2.4), we get the
(cid:0) (cid:1) (cid:0) (cid:1)
equation(9). WeuseTate’salgorithmagainforthisequationtoobtaintheminimalityandreductiontype. (cid:3)
Let
S= pprimes:ord l >0 , T = pprimes:p=2,ord l <0 .
p p
6
Proposition 3.2 says that(cid:8)Theorem 3.1 is true w(cid:9)hen (i) #S (cid:8)2; or (ii) #S=1 and #T (cid:9)1. Thus the following
≥ ≥
propositionshowsTheorem3.1.
Proposition3.3. With possible finite numberofexceptions, we have #S 1 and moreover if T =0/, then #S
≥ ≥
2. The exceptions are exactly the following curves: ‘15a1’, ‘15a3’, ‘21a1’, ‘24a1’, ‘48a3’, ‘120a2’, ‘240a3’,
‘240d5’,and‘336e4’.Butinanycaseincludingtheseexceptions,wehave8 C M.
| ·
7
Proof. Write b =2nb with n 0 and b odd. The conditionT =0/ is equivalentto the conditionb =1. We
′ ′ ′
≥
dividetheproofaccordingtothevalueofn.
Suppose n=0. In this case 16a 2 b 2 is odd and so gcd 16a 2 b 2,16b 2 =1. Suppose that there is no
− −
primedividing16a 2 b 2. We thenhave16a 2 b 2= 1. Thisispossibleonlyif a =0, a contradiction. For
− − ± (cid:0) (cid:1)
thesecondstatement, assumeb =1. Inthiscase, we getl =(16a 2 1)/16. Ifthereisonlyoneoddprime p
−
dividing16a 2 1,thenwemusthave4a 1=1,acontradiction(a Z ).
>0
− 16a 2 4−b 2 4a 2 b 2 ∈
Supposen=1.Wehavel = − ′ = − ′ ,andgcd 4a 2 b 2,16b 2 =1. Iftherewerenoodd
16 4b 2 16b 2 − ′ ′
primedividing4a 2 b 2,wewouldh·ave′4a 2 b 2= ′ 1,whencea(cid:0)=0,acontradict(cid:1)ion.Ifb =2(equivalently
′ ′
− − ±
b =1), and if there were only one prime dividing 4a 2 b 2, then either one of the relations 2a 1=1 or
′ ′
− −
2a +1= 1wouldhold.Thuswemusthavea =1. Inthiscasewegetthecurve‘48a3’,havingC =C =4.
2 3
−
Supposen 5. InthiscasewecantakeanotherWeierstrassequation(cf. Equation(9))ofE ofthefollowing
≥
form:
y2+2nxy 2nuy=x3 ux2, (10)
− −
where u= a 2 22n 4. This equation has discriminant D = (22n+16u)22nu4 and c =22n+4u+16u2+24n,
− 4
−
so ord (D )=2n+4 and ord (c )=4. Moreover, by [27], Proposition VII.5.5, since its j-invariant has order
2 2 4
8 2n<0, E haspotentiallymultiplicativereductionmodulo2. Ifthisequation(10)isminimalattheprime2,
−
thenthecurvehasadditivereductionmodulo2(ord (c )>0). Tate’salgorithmsaysthatE hasreductionoftype
2 4
I for some k, with Tamagawa number 2 or 4. Suppose that the equation (10) is not minimal modulo 2. Then
∗k
wecantransform(10)intoaminimalmodelmodulo2,whichhasdiscriminantoforder2n+4 12=2n 8at
− −
2andc oforder0. Sincetheorderoftheminimaldiscriminantisevenand>0,andsinceE hasmultiplicative
4
reduction(ord c =0),we haveevenC byTate’salgorithm. AsC isevenand4 C forsomeodd p S,the
2 4 2 2 p
| ∈
proofofthiscaseiscompleted.
Remainingcases(n=2,3,and4)canbeshownsimilarly. (cid:3)
4. E(Q) Z/2Z Z/2Z
tors
≃ ⊕
Inthissection,weprovetheMainTheoremforthecaseswhenE(Q) isisomorphictoZ/2Z Z/2Z.
tors
⊕
Theorem4.1. SupposethatE(Q) isisomorphictoZ/2Z Z/2Z. Thenthe order4=#E(Q) dividesthe
tors tors
⊕
TamagawanumberCofE,exceptfortwocurves‘17a2’and‘32a2’.Forthesetwocaseswehave4=C M.
·
Following[17],wecantakeaWeierstrassmodeloftheform
y2=x(x+a)(x+b), (11)
wherea,b Zwitha=b=0=a. Notethataandbisingeneralnotrelativelyprime. Thediscriminantofthe
∈ 6 6 6
equation(11)isD =16(a b)2a2b2andc =16a2 16ab+16b2. Letc=a b=0.Ifthereisaprimepdividing
4
− − − 6
bothaandb,thenbychangingtheequationvia[p,0,0,0]ifnecessary,weassumemin(ord a,ord b)=1.
p p
WefirstinvestigatetheTamagawanumberC forprimes pdividingabc.
p
Proposition4.2. Let pbeaprime. Assumethateither(i) p aand p∤bc;or(ii) p band p∤ac. Thenwehave
| |
thefollowing.
(a) If pisodd,thenE hasreductionoftypeIordpD =I2ordp(a) modulo p,withevenTamagawanumberat p.
(b) Supposethatp=2.Ifm:=ord a=4andifb 1 (mod 4),thenEhasgoodreductionmodulo2,whence
2
≡
C =1. Otherwise,C iseven.
2 2
Proof. Weonlygivetheproofforthecase(i). Bythesymmetryoftherolesofaandbintheequation,thecase
(ii)followsimmediately.
(a)ThisisimmediatefromTate’salgorithm.
(b)Supposethat2 aand2∤bc. We doa case-by-casestudy. Inordertohelpreaderstore-constructproofs
|
oftheresultsinthefollowingtable,weremarkthatwemostlyapplyTate’salgorithmtotheWeierstrassequation
(11),whileforthecasem=4andb 1 (mod 4)andform 5,weapplythealgorithmtoanotherWeierstrass
a+b 1 ≡ab ≥
equationy2+xy=x3+ − x2+ x.
4 24
8
m b mod4 ReductionTypeofE at p=2 C
2
1 1or3 III 2
1 I 2or4
2 ∗n
3 I 2
∗0
1 III 2
3 ∗
3 I 2or4
∗n
1 I (good) 1
0
4
3 I 2or4
∗n
5 1or3 I even
2m 8
≥ −
(cid:3)
Proposition4.3. Let pbeaprimesuchthat p cand p∤ab.
|
(a) If pisodd,thenE hasreductionoftypeIordpD =I2ordp(c) modulo p,withevenTamagawanumberat p.
(b) Supposethat p=2. Ifm:=ord c=4andifa b 3 (mod 4),thenE hasgoodreductionmodulo2,
2
≡ ≡
whenceC =1. Otherwise,C iseven.
2 2
Proof. Wemakeachangeofvariablesvia[1, a,0,0],togetanotherequation
−
y2=x3+( 2a+b)x2+a(a b)x. (12)
− −
(a)ImmediatefromTate’salgorithmappliedtoequation(12).
(b)Let p=2. Similarasaboveproposition,theresultsfromTate’salgorithmappliedtotheequation(12)are
summarised as follows. In particular, when dealing with the cases a b 3 (mod 4) and m 4, we use the
2c b 1 ac ≡ ≡ ≥
equationy2+xy=x3+− − − x2+ xinstead.
4 16
m aandb mod4 ReductionTypeofE at p=2 C
2
1 any III 2
2 a 1 (mod 4)orb 1 (mod 4)(orboth) I forsomek 2or4
≥ ≡ ≡ ∗k
2or3 III 2
∗
4 a b 3 (mod 4) I (good) 1
0
≡ ≡
5 I even
2m 8
−
(cid:3)
Proposition4.4. Letpbeaprimedividingtwoofa,b,orc.Thenclearlyitdividesthethird.Bychangingvariables
intheequation(11)via[p,0,0,0]ifnecessary,weassumemin(ord a,ord b)=1. ThenE hasreductionoftype
p p
I forsomek,withevenTamagawanumber.
∗k
Proof. Ifm=n,thenwemayassumem>n=1withoutanylossofgenerality. ByTate’salgorithm,inthiscase
6
E hasreductionoftypeI withTamagawanumber2or4. Ifm=n=1,thenwecanwritea= pa andb= pb
∗k ′ ′
with(a,p)=(b,p)=1. Hence,
′ ′
ifa b (mod p),thenE hasreductionoftypeI modulo pwithTamagawanumber4;
• ′6≡ ′ ∗0
ifa b (mod p),thenE hasreductionoftypeI modulo pwithTamagawanumber2or4.
• ′≡ ′ ∗k
(cid:3)
RecallthatEisanellipticcurvedefinedbytheequationy2=x(x+a)(x+b)withdiscriminantD =16a2b2c2=
6
0wherea,b,c:=a b Z. Wealsohaveassumedthatmin(ord a,ord b) 1forallprimesp. Let
p p
− ∈ ≤
S:= pprimes:ord a>0,ord b>0 .
p p
If #S 2, then by Proposition 4.4, then(cid:8)the Tamagawa numberC of E i(cid:9)s divisible by 4. Thus the following
≥
propositionshowsTheorem4.1.
Proposition 4.5. Suppose that #S 1. Then 4 C with only two exceptions: ‘17a2’ and ‘32a2’. But in both
≤ |
exceptions,wehaveC=M=2.
Proof. ProofsaresimilartoProposition3.3. (cid:3)
9
5. E(Q) Z/4Z
tors
≃
Theorem5.1. IfE isanellipticcurvedefinedoverQ,havingrationaltorsionsubgroupE(Q) isomorphicto
tors
Z/4Z,thentheorder4=#E(Q) dividesu C M (#X(E/K))1/2.
tors K
· · ·
5.1. Tamagawanumbers. InordertoproveTheorem5.1,wefirstconsiderTamagawanumbersofE.
From[17],table3,suchellipticcurvescanbeparametrizedbyoneparameterl by
y2+xy l y=x3 l x2,
− −
where the discriminant of the equation l 4(1+16l )=0. This is the same as in section 3, but without further
6
restrictiononl . Letl =a /b ,witha ,b Zandgcd(a ,b )=1. ByProposition3.2(a),wemayassumea =1.
∈
SowebeginwiththefollowingWeierstrassequation
y2+b xy b 2y=x3 b x2, (13)
− −
withb Z. NotethatthiscurvehasdiscriminantD =(16+b )b 7andc =(16+16b +b 2)b 2. Ifb = 1,then
4
∈ ±
we haveeither‘15a8’or ‘17a4’,bothofwhichhaveM=4. So we mayassumethatthereisatleastoneprime
dividingb .
Let p be a prime dividing b , and let m:=ord b >0. Write b = pmu, for some u Z with gcd(u,p)=1.
p
∈
UsingTate’salgorithmappliedtoWeierstrassequationsy2+pz+1xy pz+2u−1y=x3 pu−1x2(whenm=2z+1
− −
isodd)ory2+pz+1xy pz+2u 1y=x3 pu 1x2(whenm=2ziseven),wecanfigureoutthereductiontypesand
− −
− −
Tamagawanumbersatprimes p b forE.
|
m p additionalconditions ReductionTypeofE at p C
p
m=2z+1forz Z any I 4
∈ ≥0 ∗1
p=2 I even
2z
6
m=2zforz Z u 3 (mod 4)andm=8 I (good) 1
∈ >0 p=2 ≡ 0
otherwise bad even
So,inthesequel,weassume
ord b isevenforallprime p;
p
•
thenumberofoddprimesdividingb is 1.
• ≤
Moreover,ifℓisanoddprimedividingb +16,thenE hasreductionoftypeIordℓ(b +16) atℓ.1 Wefurthermore
assumethroughoutthissection,that
ifℓisanoddprimedividingb +16,thenord (b +16)isodd.
ℓ
•
Supposethatthereisnooddprimepdividingb ,i.e.,b = 2mforsomepositiveintegerm.Aswecanseeinthe
±
abovetable,inordertoavoid4 C,wemayassumem=2ziseven. ApplyingTate’salgorithmtotheWeierstrass
|
equation(13),wehavethefollowingresults.
b Curve TamagawaNumber ManinConstant
22 ‘40a3’ C C =2 1 2
2 5
· ·
24 ‘32a4’ C =2 2
2
22zwithz 3 C =4
2
≥
22 ‘24a4’ C C =2 1 2
2 3
− · ·
24 singularcurve
−
26 ‘24a3’ C C =2 1 1
2 3
− · ·
28 ‘15a7’ C C =1 1 2
3 5
− · ·
22zwithz 5even C =2(z 4)
2
− ≥ −
22zwithz 5odd C =2(z 4)
2
− ≥ −
Sowhen b isapowerof2,thenweonlyneedtodealwiththecasesb = 22z with(i)z=4or(ii)z 3being
| | − ≥
odd.
1ThiscanbealsoshownbyTate’salgorithm,appliedtotheequationy2+b xy (b +16)2y=x3 (b +96)x2+192(b +16)x 128(b +
− − −
24)(b +16)forE.
10