Studying the Birch and Swinnerton-Dyer Conjecture for Modular Abelian Varieties Using Magma

23 Pages·2012·0.2 MB·English

Studying the Birch and Swinnerton-Dyer
Conjecture for Modular Abelian Varieties
Using Magma
William Stein
Harvard University, Cambridge, Massachusetts, USA
1 Introduction
In this paper we describe the Birch and Swinnerton-Dyer conjecture in the
caseofmodularabelianvarietiesandhowtouseMagmatodocomputations
with some of the quantities that appear in the conjecture. We assume the
reader has some experience with algebraic varieties and number theory, but
donotassumethereaderhasproflciencyworkingwithellipticcurves,abelian
varieties, modular forms, or modular symbols.
In Section 2 we quickly survey abelian varieties, modular forms, Hecke
algebras,modularcurves,andmodularJacobians,thendiscussShimura’scon-
struction of abelian varieties attached to modular forms. In Section 3 we sur-
vey many quantities associated to an abelian variety, including the Mordell-
Weilgroup,torsionsubgroup,regulator,Tamagawanumbers,realvolume,and
Shafarevich-Tategroup,andusethesetostatethefullBirchandSwinnerton-
Dyer conjecture for modular abelian varieties. Section 4 contains some com-
putational results from other papers about the Birch and Swinnerton-Dyer
conjecture.
The rest of the paper is about how to use the package that I wrote for
Magma to carry out an explicit computational study of modular abelian va-
rieties. Section 5 is about modular symbols and how to compute with them
in Magma. In Section 6 we state a theorem that allows us to use Magma
to compute subgroups of Shafarevich-Tate groups of abelian varieties. In Sec-
tion 7 we discuss computation of special values of L-functions. Section 8 is
about computing Tamagawa numbers, and in Section 9 we describe how to
computeadivisorandmultipleoftheorderofthetorsionsubgroup.Allthese
computationsarepulledtogetherinSection10toobtainaconjecturaldivisor
and multiple of the order of the Shafarevich-Tate group of a modular abelian
varietyofdimension20.WeflnishwithSection11,whichcontainsanexample
in which the level is composite and elements of the Shafarevich-Tate group
only becomes \visible" at higher level.
2 William Stein
Taken together, these computations give evidence for the Birch and
Swinnerton-Dyer conjecture and increase our explicit understanding of mod-
ular abelian varieties.
2 Modular Abelian Varieties
AnellipticcurveE overtherationalnumbersQisaone-dimensionalcommu-
tativecompactalgebraicgroup.Suchacurveisusuallygivenastheprojective
closureofana–necurvey2 =x3+ax+b,withaandbinQ.Thepointsover
the real numbers R of y2 =x3¡x+1 are illustrated in Figure 1. If P and Q
2
R P
(cid:1)
Q
1 2
PSfrag replacements R
¡R 2
(cid:0)
¡2
Fig. 1. AddingP =(0;1)toQ=(1;1)togetR=(¡1;¡1)ony2 =x3¡x+1
are two distinct points on E, we flnd their sum as follows: draw the unique
line through them and let (x;y) be the third point of intersection of this line
with E. Then the sum of P and Q is R =(x;¡y), as illustrated in Figure 1.
For more about elliptic curves, see [33, 34].
This paper is about abelian varieties, which are compact (commutative)
algebraicgroupsofdimensionpossiblygreaterthan1.Forexample,theCarte-
sian product of two elliptic curves is an abelian variety of dimension 2.
Explicit equations for abelian varieties are vastly more complicated than
for elliptic curves, so algorithms for computing with abelian varieties without
recourse to explicit algebraic equations are of great value. In this paper we
focusonsuchalgorithmsinthecasewhentheabelianvarietyisendowedwith
extra structure coming from modular forms.
A cuspidal modular form of weight 2 for
Studying the Birch and Swinnerton-Dyer Conjecture 3
a b
¡ (N)= 2SL (Z):N jc
0 ‰(cid:181)c d¶ 2 (cid:190)
is a holomorphic function f(z) on the upper half plane such that for all (cid:176) =
a b
2¡ (N) we have
(cid:181)c d¶ 0
az+b
f =(cz+d)2f(z);
(cid:181)cz+d¶
and which satisfles certain vanishing conditions at the cusps (see [13, pg. 42]
foraprecisedeflnition).Wedenotetheflnitedimensionalcomplexvectorspace
of all cuspidal modular forms of weight 2 for ¡ (N) by S (¡ (N)). Because
0 2 0
11
2¡ (N), cuspidal modular forms have a Fourier series representation
(cid:181)01¶ 0
1 1
f(z)= a qn = a e2…inz:
n n
nX=1 nX=1
The Hecke algebra
T=Z[T ;T ;T ;:::]‰End(S (¡ (N)))
1 2 3 2 0
is a commutative ring that is free of rank equal to dim S (¡ (N)) (for the
C 2 0
deflnition and basic properties of the Hecke operators T , see [13, x3] and the
n
references therein).
A newform is a modular form
f =q+ a qn
n
nX‚2
that is a simultaneous eigenvector for every element of the Hecke algebra and
such that the coe–cients fap : p - Ng are not the prime-index coe–cients of
another eigenform of some level that strictly divides N.
The group ¡ (N) acts as a discrete group of linear fractional transfor-
0
mations on the upper half plane; the quotient of the upper half plane by
this action is a non-compact Riemann surface. Its compactiflcation has the
structure of algebraic curve over Q, i.e., the compactiflcation is the set of
complex points of an algebraic curve X (N) deflned by polynomial equations
0
with coe–cients in Q.
AdivisoronanalgebraiccurveX isanelementofthefreeabeliangroup
generated by the points of X. For example, if f is a rational function on X
then
(f)=(formal sum of poles of f) ¡ (formal sum of zeros of f)
is a divisor on X, where the sums are with multiplicity. Two divisors D and
1
D are linearly equivalent if there is a rational function f on X such that
2
4 William Stein
D ¡D =(f). The Jacobian J of an algebraic curve X is an abelian variety
1 2
of dimension equal to the genus (number of holes in the Riemann surface
X(C)) of X such that the underlying group of J is naturally isomorphic to
the group of divisor classes of degree 0 on X. Let J (N) denote the Jacobian
0
of X (N).
0
Similarly, let
a b
¡ (N)= 2SL (Z):N jc and a·1 (mod N) ;
1 ‰(cid:181)c d¶ 2 (cid:190)
deflne X (N) similarly, and let J (N) be the Jacobian of X (N).
1 1 1
A modular abelian variety is an abelian variety A for which there exists
a surjective morphism J (N) ! A. Modular abelian varieties are appealing
1
objects to study. For example, it is a deep theorem that every elliptic curve
over Q is modular (see [7, 40, 41]), and this implies Fermat’s Last Theorem
(see[25,Cor.1.2]).In[27],KenRibetconjecturedthatthesimpleabelianva-
rietiesoverQof\GL -type"areexactlythesimplemodularabelianvarieties.
2
A closely related conjecture of Serre (see [29, pg. 179] and [28]) asserts that
every odd irreducible Galois representation
‰:Gal(Q=Q)!GL (F )
2 p
is\modular";thisconjectureisequivalenttotheassertionthat‰canbereal-
ized (up to twist) as the action of Gal(Q=Q) on a subgroup of the points on
someJ (N)(see[28,x3.3.1]forapartialexplanation).ThoughSerre’sconjec-
1
ture is still far from proved, it implies Ribet’s conjecture (see [27, Thm. 4.4]).
We now return to considering ¡ (N), though we could consider ¡ (N)
0 1
for everything in the rest of this section. The Hecke algebra T, which we
introduced above as a ring of linear transformations on S (¡ (N)), also acts
2 0
via endomophisms on J (N).
0
In order to construct Galois representations attached to modular forms,
Goro Shimura (see [31, x1] and [32, x7.14]) associated to each newform f =
a qn a simple abelian variety A deflned over Q. Let I be the ideal of
n f f
ePlements of T that annihilate f. Then
A =J (N)=I J (N):
f 0 f 0
The dimension of A equals the degree of the fleld generated over Q by the
f
coe–cients a of f. Note that A need not be simple over Q.
n f
We will frequently mention the dual A_ below. The dual can be consid-
f
ered as an abelian subvariety of J (N), by using that Jacobians are canoni-
0
cally self dual and the dual of the quotient map J (N) ! A is an inclusion
0 f
A_ ,! J (N). Note that A_ is the connected component of the intersection
f 0 f
of the kernels of all elements of I .
f
We say that a newform g is a Galois conjugate of f if there is (cid:190) in
Gal(Q=Q) such that g = (cid:190)(a )qn. If g is a Galois conjugate of f, then
n
P
Studying the Birch and Swinnerton-Dyer Conjecture 5
A = A ; if g is not a conjugate of f then the only homomorphism from A
f g f
to A is the zero map. (A nonzero homomorphism A ! A would induce
g f g
an isogeny of Tate modules, from which one could deduce that f and g are
Galois conjugate.)
Wewillconcernourselvesalmostentirelywiththesemodularabelianva-
rietiesattachedtonewforms,because,asmentionedabove,thereareanumber
of algorithms for computing with them that do not require explicit algebraic
equations (see [2, 3, 9, 10, 15, 19, 37, 35]). Also, it follows from standard re-
sults about constructing spaces of cusp forms from newforms, which can be
found in [4, 22], that every modular abelian variety is isogenous to a prod-
uct of abelian varieties of the form A . (An isogeny of abelian varieties is a
f
surjective homomorphism with flnite kernel.)
3 The Birch and Swinnerton-Dyer Conjecture
In the 1960s Bryan Birch and Peter Swinnerton-Dyer did computations with
elliptic curves at Cambridge University on the EDSAC computer (see, e.g.,
[5]). These computations led to earth-shattering conjectures about the arith-
metic of elliptic curves over Q. Tate [39] formulated their conjectures in a
more functorial way that generalized them to abelian varieties over global
flelds (such as the rational numbers). We now state their conjectures below
for modular abelian varieties over Q.
Let A be a modular abelian variety. Mordell and Weil proved that the
f
abelian group A (Q) of rational points on A is flnitely generated, so it is
f f
isomorphic to Zr£T where T is the flnite group A (Q) of all elements of
f tor
flnite order in A (Q). The exponent r is called the Mordell-Weil rank of A .
f f
If f is a newform, the L-function of f is deflned by the Dirichlet series
L(f;s)= a n¡s.HasseshowedthatL(f;s)hasananalyticcontinuation
n‚1 n
to a holomPorphic function on the whole complex plane. The Hasse-Weil L-
function of A is
f
L(A ;s)= L(g;s)
f
Y
where the product is over the Galois conjugates g of f. The analytic rank of
A is ord L(A ;s).
f s=1 f
We are now ready to state the flrst part of the conjecture.
Conjecture 3.1 (Birch and Swinnerton-Dyer) The analytic rank of A
f
is equal to the Mordell-Weil rank of A .
f
Remark 1.
1.It is an open problem to give, with proof, an example of an elliptic curve
with analytic rank at least 4. No examples with analytic rank at least 3
were known until the deep theorem of [16, Prop. 7.4].
6 William Stein
2.When A is an elliptic curve, Conjecture 3.1 is the Clay Mathematics
f
Institute Millennium Prize Problem from arithmetic geometry [17], so it
has received much publicity.
In order to explain the conjecture of Birch and Swinnerton-Dyer about
the leading coe–cient of L(A ;s) at s = 1, we introduce the regulator, real
f
volume,Tamagawanumbers,andShafarevich-TategroupofA .Mostofwhat
f
we say below is true for a general abelian variety over a global fleld; the
notable exceptions are that we do not know that the L-function is deflned on
the whole complex plane, and there are hardly any cases in general when the
Shafarevich-Tate group is known to be flnite.
Let Af(Q)=tor denote the quotient of Af(Q) by its torsion subgroup,
so Af(Q)=tor is isomorphic to Zr, where r is the Mordell-Weil rank of Af.
The height pairing is a nondegenerate bilinear pairing h on Af(Q)=tor. The
regulator Reg of A is the absolute value of the determinant of a matrix
Af f
who entries are h(Pi;Pj), where P1;:::;Pr are a basis for Af(Q)=tor. When
A (Q) has rank zero, the regulator is 1.
f
We use a certain integral model of A to deflne the real volume and
f
Tamagawa numbers of A . The N¶eron model A of A , whose existence was
f f
established by N¶eron in [24] (see also [6, Ch. 1]), is a canonical object associ-
atedtoA thatisdeflnedoverZ.TheN¶eronmodelcanbereducedmodulop
f
for every prime p, and when base extended to Q, the N¶eron model is isomor-
phic to A . The N¶eron model A is determined, up to unique isomorphism, by
f
the following properties, which the reader unfamiliar with schemes can safely
ignore:AisasmoothcommutativegroupschemeoverZsuchthatwheneverS
is a smooth scheme over Z the restriction map
Hom(S;A)!Hom (S ;A)
Q Q
is a bijection.
The real volume › of A is the absolute value of the integral over
Af f
A (R) of h ^¢¢¢^h where h ;:::;h are a basis for the holomorphic 1-
f 1 d 1 d
forms on A. Using various identiflcations as in [1, x2.2.2] one sees that the
Z-span M of h ;:::;h can be viewed as a submodule of
1 d
W =S (¡ (N);Z)\(Cf '¢¢¢'Cf )
2 0 1 d
where f ;:::;f are the Gal(Q=Q) conjugates of f. We call the index of M
1 d
in W the Manin constant of A , and conjecture (see [1]) that the Manin
f
constant is 1. This conjecture would imply that a basis h ;:::;h can be
1 d
computed, since W can be computed.
The reduction modulo p of A is an algebraic group A over the flnite
Fp
fleldF withpelements.IfpdoesnotdivideN,thenthisgroupisconnected,
p
but when p divides N, the reduction A need not be connected. Let
Fp
' =A =A0
A;p Fp Fp
Studying the Birch and Swinnerton-Dyer Conjecture 7
betheflnitegroupofcomponents.TheTamagawanumberofA atp,denoted
f
c , is the number of F -rational components of the reduction of A modulo p,
p p
so c =#' (F ).
p A;p p
TheonlyobjectlefttodeflnebeforewestatethesecondpartoftheBirch
and Swinnerton-Dyer conjecture is the Shafarevich-Tate group of A . This is
f
a group that measures the failure of a certain local-to-global principle for A .
f
Togiveanexactdescription,weletH1(Q;A )betheflrstGaloiscohomology
f
group of A , which is a torsion group with inflnitely many elements of any
f
orderbiggerthan1(see[30]foraproofinthecasewhenA isanellipticcurve;
f
the top of page 278 of [8] also purports to contain a proof). More precisely,
H1(Q;A ) is the set of equivalence classes of maps c : Gal(Q=Q) ! A (Q),
f f
with flnite image, such that c((cid:190)¿) = c((cid:190))+(cid:190)c(¿), and two classes c and c
1 2
are equivalent if there exists P in A (Q) such that c ((cid:190))¡c ((cid:190))=(cid:190)(P)¡P
f 1 2
for all (cid:190) 2 Gal(Q=Q). For each prime p we deflne H1(Q ;A ) analogously,
p f
but with the rational numbers Q replaced by the p-adic numbers Q . Also,
p
we allow p=1, in which case Q =R. Then
p
X(Af)=ker0H1(Q;Af)¡! H1(Qp;Af)1:
primMesp•1
@ A
WearenowreadytostatethefullBirchandSwinnerton-Dyerconjecture
for modular abelian varieties A .
f
Conjecture 3.2Let A=A be a modular abelian variety attached to a new-
f
form, and let r =ord L(A;s) be the analytic rank of A. Then
s=1
L(r)(A;1) c ¢› ¢Reg
= p A A ¢#X(A):
r! #AQ(Q) ¢#A_(Q)
tor tor
Recall that L(r)(A ;1) makes sense at s = 1 because A is attached to
f f
a modular form. Also Kato established in [18, Cor. 14.3] that if L(A ;1)6=0
f
then X(A ) is flnite, and Kolyvagin-Logachev ([20, Thm. 0.3]) proved that
f
if f is a modular form in S (¡ (N)) and ord L(f;s) • 1, then X(A ) is
2 0 s=1 f
flnite. When the theorems of Kato, Kolyvagin, and Logachev do not apply,
we do not know even one example of a modular abelian variety A for which
f
X(A ) is provably flnite. John Tate once remarked that Conjecture 3.2 (for
f
arbitraryabelianvarieties)relatesthevalueofafunctionwhereitisnotknown
to be deflned to the order of a group that is not known to be flnite.
The rest of this paper is about how to use Magma to gather computa-
tional evidence for Conjecture 3.2, a task well worth pursuing. Elliptic curves
are naturally surrounded by modular abelian varieties, so we want to under-
stand modular abelian varieties well in order to say something about Conjec-
tures3.1{3.2forellipticcurves.Doingexplicitcomputationsaboutthesecon-
jectures results in stimulating tables of data about modular abelian varieties,
which could never be obtained except by direct computation. Until [2, 15]
8 William Stein
there were very few nontrivial computational examples of Conjecture 3.2 for
abelianvarietiesintheliterature,soitisimportanttotesttheconjecturesince
wemightflndacounterexample.Tryingtocomputeinformationaboutacon-
jecture stimulates development of algorithms and theorems about that con-
jecture. Finally, our computations may lead to reflnements of Conjecture 3.2
in the special case of modular abelian varieties; for example, most objects in
Conjecture 3.2 are modules over the Hecke algebra so there should be more
precise module-theoretic versions of the conjecture.
4 Some Computational Results
In [2] we use Magma to compute some of the arithmetic invariants of the
19608abelianvarietyquotientsA ofJ (N)withN •2333.Overhalfofthese
f 0
A have analytic rank 0, and for these we compute a divisor and a multiple
f
of the order of X(A ) predicted by Conjecture 3.2. We flnd that there are
f
at least 168 abelian varieties A such that the Birch and Swinnerton-Dyer
f
Conjecture implies that #X(A ) is divisible by an odd prime, and we use
f
Magma to show that for 37 of these the odd part of the conjectural order
of X(A ) divides #X(A ) by constructing nontrivial elements of X(A )
f f f
using visibility theory. The challenge remains to show that the remaining 131
abelian varieties A have odd part of X(A ) divisible by the odd part of the
f f
conjectural order of X(A ) (we successfully take up this challenge for one
f
example of level 551 in Section 11 of the present paper).
In [9, x2 and x7] we investigate Conjecture 3.1{3.2 when A is a quotient
f
of J (p) with p prime. In particular, we compute some of the invariants of
1
every A for p•71.
f
It was once thought by some mathematicians that Shafarevich-Tate
groups of abelian varieties would have order a perfect square (or at least
twiceaperfectsquare).Thisisfalse,asweshowedinthepaper[36],wherewe
use Magma to prove that for every odd prime p<25000 there is an abelian
variety whose Shafarevich-Tate group has order pn2 with n an integer.
Much of the data mentioned above is of interest even if the full Birch
and Swinnerton-Dyer conjecture were known since this data could probably
never be discovered without considerable computation, even assuming the
conjectures were true.
The rest of this paper is about how to use Magma to do computations
with newform quotients A of J (N) as in [2]. These computations involve
f 0
modular symbols, which underly most algorithms for working with modular
abelianvarieties.(IhopetoaddfunctionalitytoafuturereleaseofMagmafor
computingdirectlywithmodularabelianvarieties,sothatnoexplicitmention
of modular symbols is required.)
Remark 2.From a computational point of view, it is di–cult to give evidence
forConjecture3.1whenthedimensionisgreaterthan1incasesnotcoveredby
Studying the Birch and Swinnerton-Dyer Conjecture 9
thegeneraltheoremsofKato,Kolyvagin,andLogachev.Togivenewevidence
we would have to consider a modular abelian variety A with either f a
f
newform in S (¡ (N)) and ord L(f;s) > 1, or f a newform in S (¡ (N))
2 0 s=1 2 1
butnotinS (¡ (N))andord L(f;s)>0.WewouldthenshowthatA (Q)
2 0 s=1 f
isinflnite,andmorepreciselythatithastherankpredictedbyConjecture3.1.
In the above 2 cases the only known way to show that A (Q) is inflnite
f
is to exhibit a point of inflnite order in A (Q), and this seems to require
f
knowing equations for A . Also when L(A ;1) = 0, Conjecture 3.2 involves
f f
a regulator term, which we do not know how to compute without explicitly
flnding the points on a model for A . Thus we will focus on giving evidence
f
for Conjecture 3.2 in the case when L(f;1)6=0.
5 Modular Symbols
In this section we describe how modular symbols are related to homology
of modular curves, and illustrate how to compute with modular symbols in
Magma.Wealsodiscusscomputingdecompositionofmodularsymbolsspaces
and, for e–ciency reasons, computing in the +1 quotient.
Let N be a positive integer. The integral homology H (X (N);Z) of the
1 0
modular curve X (N) is a free abelian group of rank equal to the genus of
0
X (N). The Hecke algebra T = Z[T ;T ;T ;:::] acts on a H (X (N);Z) as
0 1 2 3 1 0
a ring of homomorphisms and makes H (X (N);Z) into a T-module. This
1 0
section is concerned with how to compute with this module using Magma.
Section 12 contains a complete log of all Magma computations given below.
Modularsymbolsprovideaflnitecomputablepresentationforthehomol-
ogyofX (N)alongwiththeactionoftheHeckealgebraTonthishomology.
0
TherelativerationalhomologyH (X (N);Q;cusps)istherationalhomology
1 0
of X (N) relative to the cusps; it is the flnitely generated free abelian group
0
of homology equivalence classes of geodesic paths from fi to fl, where fi and
fl lie in P1(Q)=Q[f1g. A flnite presentation for H (X (N);Q;cusps) can
1 0
be found in [23]. For simplicity, we typically compute H (X (N);Q;cusps)
1 0
flrst, then flnd H (X (N);Z) inside H (X (N);Q;cusps) if it is needed.
1 0 1 0
By deflnition, We now illustrate how Magma can compute a basis for
H (X (N);Q;cusps), and, given arbitrary fi and fl in P1(Q), flnd an equiva-
1 0
lent linear combination of basis elements.
M := ModularSymbols(389);
BASIS(M);
The output of BASIS(M) begins with the symbol f¡1/337,0g. Figure 2 on
page 10 illustrates how the expression f¡1/337,0g represents the relative
rational homology class determined by a geodesic path from ¡1=337 to 0 in
the upper half plane. The cusps determined by ¡1=337 and 0 are equivalent
by an element of ¡ (389), so the image of the geodesic path in the 32 holed
0
torus X (389)(C) is a closed loop.
0
10 William Stein
1
¡ 1 0
337
Quotient by ¡0(389)
...
[0]
31 more holes
PSfrag replacements [1]
Fig. 2. The Modular Symbol f¡1=337;0g
The following Magma code illustrates how to flnd the image in the rel-
ative homology of an arbitrary path between cusps. The extra < and > are
needed because we are considering modular symbols of weight k =2; in gen-
eral there is a coe–cient which is a homogenous polynomial of degree k¡2,
which is the flrst argument to the coercion. The CUSPS()j part of the expres-
sionisneededsothatthesequenceisasequenceofcusps(thisisnotrequired
if both cusps are rational numbers).
M ! <1, [CUSPS() j ¡1/337, INFINITY()] > ;
For more about computing with modular symbols, see [11, 12, 23, 37, 35].
Precise relationships between H (X (N);Q) and S (¡ (N)), along with
1 0 2 0
some linear algebra, make it possible for us to compute a basis of S (¡ (N))
2 0