The following code creates a file that contains a table which lists,
for each integer N in some range, the abstract group structure of
the subgroup of J
In this example, we investigate a 2-dimensional abelian variety B,
which is a quotient of J
0(43). The purpose of this example is to
show how numerical computation with modular symbols suggests
interesting arithmetic questions about familiar abelian varieties. In
the following example, we find that the conjecture of Birch and
Swinnerton-Dyer (plus the Manin c=1 conjecture) implies that the
first nontrivial Shafarevich-Tate group of an (optimal) modular
abelian variety has order
two. Thus the surprising existence of an
abelian varieties with non-square order could have been (but was not)
hinted at long ago by somebody playing around with a modular symbols
package (in fact, it was discovered by B. Poonen and M. Stoll
[PS99] while they were designing and implementing
algorithms for computing with Jacobians of genus-two curves).
> M43 := ModularSymbols(43,2); // Level 43, weight 2.
> H1 := CuspidalSubspace(M43); // H_1(X_0(43),Q)
> D := NewformDecomposition(H1); // factors corresponding to newforms
> A,B := Explode(D);
> A; // The homology of the elliptic curve "43A"
Modular symbols space of level 43, weight 2, and dimension 2
> B; // The homology of the 2-dimensional abelian variety "43B"
Modular symbols space of level 43, weight 2, and dimension 4
> LRatio(B,1); // L(B,1)/Omega_B
2/7
The Birch and Swinnerton-Dyer conjecture predicts that
the Shafarevich-Tate group of B has order
as given by the formula for
ShaAn in the
code below. To compute this value, it remains to
compute #B(Q) and the Tamagawa number c
43.
> T := TorsionBound(B,11); T; // #B(Q) divides this number
7
> // Compute the subgroup of B(Q) generated by (0)-(oo).
> C := SubgroupOfTorus(B,WindingElement(M43)); C;
Abelian Group isomorphic to Z/7
Defined on 1 generator
Relations:
7*C.1 = 0
> TamagawaNumber(B,43);
7
> ShaAn := LRatio(B,1)*TorsionBound(B,11)^2/TamagawaNumber(B,43);
ShaAn is the Birch and Swinnerton-Dyer conjectural
order of the Shafarevich-Tate group of B, under
the assumption that the Manin constant of B is 1.
> ShaAn;
2
One of the Galois conjugate newforms associated to B is given below.
> qEigenform(B,12);
q + a*q^2 - a*q^3 + (-a + 2)*q^5 - 2*q^6 + (a - 2)*q^7 - 2*a*q^8 - q^9
+ (2*a - 2)*q^10 + (2*a - 1)*q^11 + O(q^12)
> BaseRing(Parent(qEigenform(B,12)));
Univariate Quotient Polynomial Algebra in a over Rational Field
with modulus a^2 - 2
> qIntegralBasis(B,12);
[
q + 2*q^5 - 2*q^6 - 2*q^7 - q^9 - 2*q^10 - q^11 + O(q^12),
q^2 - q^3 - q^5 + q^7 - 2*q^8 + 2*q^10 + 2*q^11 + O(q^12)
]
By integrating homology against the differentials corresponding
to the two modular forms above, we obtain a lattice that
defines the complex torus A
B(C):
> Periods(B,97);
[
(-0.2259499583067642118739519224 -
1.766644676299599532273333140*i
0.5250281159132219433729491648 +
0.8066018577029307230283142371*i),
(0.5981563162241222986475767220 -
1.920085638612119493276485632*i
0.8241062742261960348649172082 -
0.1534409622571770568748354995*i),
(-0.8241062745308865105215286445 -
0.1534409623125199610031524920*i
-0.2990781583129740914919680434 -
0.9600428199601077799031497367*i),
(-0.5981563162241222986475767220 -
1.920085638612119493276485632*i
-0.8241062742261960348649172083 -
0.1534409622571770568748354995*i)
]
Finally, it is tempting to ask whether or not the (conjectural)
two-torsion element of the Shafarevich-Tate group of B suggested
above is "visible" in the sense that it is "explained by a jump in
the rank of the Mordell--Weil group of A"
(see [CM00]).
The following computation suggests, but does not prove,
that this is the case.
> G := MordellWeilGroup(EllipticCurve(A)); G;
Abelian Group isomorphic to Z
Defined on 1 generator (free)
> IntersectionGroup(A,B);
Abelian Group isomorphic to Z/2 + Z/2
Defined on 2 generators
Relations:
2*$.1 = 0
2*$.2 = 0
Let M be a space of modular symbols the corresponds to
a Galois-conjugacy class of newforms.
The period map attached to M is a linear map
(AmbientSpace(M)) -> Cd,
where d is the dimension of the space of modular forms
associated to M.
The cokernel of the period map is a complex torus AM(C).
The terminology "period mapping" comes from the
fact that there are (often?) mereomorphic functions on Cd
whose periods are the image of the integral cuspidal modular
symbols under the period mapping.
In the functions below, M must not be a +1 or -1 quotient
and must be cuspidal.
The period mapping attached to the space of modular symbols M, computed using n terms of
the q-expansions of modular forms associated to M.
Prec: RngIntElt Default:
The period mapping attached to the space of modular symbols M, computed
to precision Prec. If no precision is given, uses default precision.
The complex period lattice associated to the space of modular symbols M,
computed using prec terms of the q-expansions
of modular forms associated to M.
Prec: RngIntElt Default:
The period mapping attached to the space of modular symbols M, computed
to precision Prec. If no precision is given, uses default precision.
The value
rj(f) = int0i∞ f(z) zj dz.
Let M be a space of modular symbols over a field K. For many
purposes it is useful to have a surjective map
π: (AmbientSpace(M)) -> V,
where V is a vector space over K and
ker(π) is the same as the kernel of the period mapping.
A surjective linear map from the ambient space of the space of modular symbols M
to a vector space, such that the kernel of this map
is the same as the kernel of the period mapping.
A surjective linear map from the ambient space of the space of modular symbols M
to a vector space, such that the kernel of this map
is the same as the kernel of the period mapping.
This map is chosen in such a way that the image of
IntegralBasis(CuspidalSubspace(AmbientSpace(M)))
is the standard Z-lattice.
(Note that M must be defined over Q.)
> M := ModularSymbols(33); M;
Full Modular symbols space of level 33, weight 2, and dimension 9
> S := CuspidalSubspace(M);
> N := NewSubspace(S);
> phi := RationalMapping(N);
> [phi(x) : x in IntegralBasis(S)];
[
( -2 4/3),
( -4 2/3),
( -2 2/3),
(-2 0),
( -2 -2/3),
(-4 0)
]
Notice that the image of the basis
IntegralBasis(S)
for H
1(X
0(33), Z)
is not Z x Z. However,
IntegralMapping(N) is normalized so that
the image is Z x Z:
> int := IntegralMapping(N);
> [int(S.i) : i in [1..Dimension(S)]];
[
( 2 -1),
( 1 -2),
( 1 -1),
( 0 -2),
( 0 -1),
(-1 -1)
]
Consider a quotient A
f of J
0(N) attached to a newform f∈S
2(Γ
0(N)).
Using
IntegralMapping and the Abel-Jacobi theorem, we can see the image
in A
f(Q) of the point (0) - (∞) ∈J
0(N)(Q). In the
level 97 example below, this image has
order 8, which is the numerator of (97 - 1)/12.
> Af := ModularSymbols("97B"); Af;
Modular symbols space of level 97, weight 2, and dimension 8
> int := IntegralMapping(Af);
> // Let x be the modular symbol {0,oo}
> x := AmbientSpace(Af)!<1,[Cusps()|0,Infinity()]>;
> int(x);
(-5/8 1/4 -1/4 0 0 1/4 3/8 1/4)
> Numerator((97-1)/12);
8
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