Let M be a space of weight 2 cuspidal modular symbols with trivial
character that corresponds to a Galois-conjugacy class of newforms,
and let AM(C) be the cokernel of the period map. G. Shimura
proved that AM(C) is the set of complex points of an abelian
variety AM defined over Q. Let N be the level of M and let
J0(N) be the Jacobian of the modular curve X0(N). Shimura
constructed AM as a quotient of J0(N) by an abelian subvariety.
More precisely, if I is the annihilator of M in the Hecke
algebra, then AM = J0(N)/I J0(N).
Magma
contains functions for computing the modular degree,
congruence modulus, upper and lower bounds on the order of the
torsion subgroup, and the order of the component group of the closed
fiber of the Néron model of AM at primes that exactly divide the
level of M.
We compute the first example of an optimal elliptic curve over Q
such that the congruence modulus does not equal the modular
degree. (See
[FM99]
for further discussion of this problem. We warn the reader that the
divisibility r | deg(φ) | rN
i cited there is
incorrect, as our
54B example shows.)
> E := ModularSymbols("54B");
> ModularDegree(E);
2
> CongruenceModulus(E);
6
We next verify directly that the congruence modulus is divisible by 3.
> A := ModularSymbols("27A"); A; // 27=54/2.
Modular symbols space of level 27, weight 2, and dimension 2
> A54 := ModularSymbols(A,54); A54; // all images of A at level 54.
Modular symbols space of level 54, weight 2, and dimension 4
> qE := qIntegralBasis(E,17);
> qA54 := qIntegralBasis(A54,17);
> &+qA54 - &+qE;
-3*q^4 + 3*q^5 - 3*q^8 + 3*q^10 - 3*q^11 + 9*q^13 + 3*q^16 + O(q^17)
> IntersectionGroup(E,A54); // however, the intersection is trivial.
Abelian Group of order 1
Ken Ribet proved that if E is an optimal elliptic curve quotient of
J0(N), with N prime, and if fE is the corresponding newform,
then the congruence modulus of fE equals the modular degree of E.
The author is aware of no counterexamples to the following more
general statement: "If E is an optimal elliptic curve of
square-free conductor, then the congruence modulus of the newform
fE attached to E equals the modular degree of E." An
analogous statement for abelian varieties is false, even at prime
level. The first counterexample is ModularSymbols("431F"),
which corresponds to an abelian variety of dimension 24. In this
case, the modular degree is 211.6947, whereas the congruence
modulus is 210.6947.
The following code makes a table of congruence moduli and modular
degrees for the elliptic curves of conductor near 54. Notice the
counterexample at level 54.
> for N in [53..55] do
> C := CuspidalSubspace(ModularSymbols(N,2));
> newforms := NewSubspace(C);
> D := EllipticFactors(newforms,19);
> for E in D do
> printf "%o:\t%o,\t%o\n", N, ModularDegree(E), CongruenceModulus(E);
> end for;
> end for;
53: 2, 2
54: 2, 6
54: 6, 6
55: 2, 2
ModularKernel makes sense even for spaces of modular symbols of
weight greater than 2. As in the case of weight 2, this number gives
information about congruences between modular forms. The following example
illustrates how ModularKernel suggest a congruence between a form
of level 10 and weight 4 with a form of level 5.
> M := ModularSymbols(10,4);
> S := CuspidalSubspace(M);
> D := NewformDecomposition(S); D;
[
Modular symbols space of level 10, weight 4, and dimension 2,
Modular symbols space of level 10, weight 4, and dimension 4
]
> #ModularKernel(D[1]);
10
> f := qEigenform(D[1],8);
> g := qEigenform(D[2],8);
> g2 := Evaluate(g,Parent(g).1^2);
> f-(g+6*g2); // a congruence modulo 10!
-10*q^3 + 20*q^4 + 10*q^5 - 20*q^6 - 10*q^7 + O(q^8)
We provide several functions for computing the orders of component groups
of optimal quotients of J0(N) at primes p that exactly divide N.
Our algorithm involves Grothendieck's monodromy pairing on the character
group of the toric part of the closed fiber at p of the Néron model
of J0(N); the theory behind this algorithm is described in
[Ste01] (or [Ste00]);
see [KS00] for a computationally-oriented introduction
to the algorithm.
When N is prime, we use the Mestre and Oesterlé method to
construct the character group of the torus, as described in
[Mes86].
In general, the ideal theory of quaternion algebras is used.
Note: In the appendix to [Maz77], Mazur and
Rapoport give an explicit formula for the order of the component group
of J0(N) at primes p≥5 that exactly divide N. Their
formula is not currently used by the ComponentGroupOrder
function.
The RealTamagawaNumber function computes the order of the
"component group at infinity".
The order of the component group at p. This is the
order of the group of /line(F)p-points of the component
group of the reduction modulo p of the Néron model of the abelian
variety attached to the space of modular symbols M.
At present, it is necessary that p exactly divides the level.
If Sign(M) is not equal to 0, then only the odd part of
the order is returned.
The order of the group of Fp-rational points of the
component group of the space of modular symbols M. We require M to be associated to a single
Galois-conjugacy class of newforms.
The number of connected components of AM(R).
The number of connected components of the subgroup AM(C) - of
AM(C) on which complex conjugation acts as -1
We compute the orders of the component groups
of some abelian varieties.
> X11 := ModularSymbols("11A"); // corresponds to X_0(11).
> ComponentGroupOrder(X11,11);
5
> TamagawaNumber(X11,11);
5
> RealTamagawaNumber(X11);
1
> MinusTamagawaNumber(X11);
1
> J37 := ModularSymbols("37"); J37;
Modular symbols space of level 37, weight 2, and dimension 4
> ComponentGroupOrder(J37,37);
3
> A, B := Explode(NewformDecomposition(J37));
> ComponentGroupOrder(A,37);
3
> ComponentGroupOrder(B,37);
1
We can also compute component groups of optimal quotients whose
dimension is greater than 1. The abelian varieties B and C below correspond to the Jacobians labeled 65B and 65A in
[FLS+01], respectively.
> J65 := ModularSymbols("65");
> A,B,C := Explode(SortDecomposition(NewformDecomposition(J65)));
> B;
Modular symbols space of level 65, weight 2, and dimension 4
> C;
Modular symbols space of level 65, weight 2, and dimension 4
> ComponentGroupOrder(B,5); // not the Tamagawa number
3
> ComponentGroupOrder(B,13);
3
> ComponentGroupOrder(C,5);
7
> ComponentGroupOrder(C,13);
1
> HeckeEigenvalueField(C);
Number Field with defining polynomial x^2 + 2*x - 1 over the
Rational Field
Mapping from: Univariate Quotient Polynomial Algebra in a over
Rational Field
with modulus a^2 + 2*a - 1 to Number Field with defining
polynomial x^2 + 2*x - 1 over the Rational Field given by a rule
[no inverse]
> ComponentGroupOrder(J65,5);
42
When the Atkin-Lehner involution Wp acts as +1 on a modular abelian
variety A, the order of the component group can be larger than the
Tamagawa number cp=[A(Qp):A0(Qp)] that appears in the conjecture
of Birch and Swinnerton-Dyer.
> AtkinLehner(B,5);
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
> ComponentGroupOrder(B,5);
3
> TamagawaNumber(B,5);
1
The real and minus Tamagawa numbers are defined for
spaces of modular symbols of any weight over the rationals.
> Del := ModularSymbols("1k12A");
> Del;
Modular symbols space of level 1, weight 12, and dimension 2
Next we see that the period lattice associated
to Δ is rectangular.
> RealTamagawaNumber(Del);
2
> MinusTamagawaNumber(Del);
2
> Periods(Del,40);
[
(-0.0004853381649299516049241304429*i),
(0.001140737449583079336044545337)
]
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