Manin symbols and modular forms

(first draft)

William A. Stein^{∗}

February 17, 1999

Contents

1 Introduction 1 1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Modular symbols and modular forms 2 2.1 Modular symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Manin symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 Cuspidal modular symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 Duality between modular symbols and modular forms . . . . . . . . . . . . . . . . 5

2.4.1 Weight 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4.2 Higher weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.5 Complex conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.6 Eichler-Shimura . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Linear maps 7

3.1 Linear operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.1.1 Action on modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.1.2 Action on modular symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.1.3 Hecke operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2 Action on Manin symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Generating H_{1}(Γ, Z) 9

1 Introduction

“The object of numerical computation is theoretical advance.”

– Atkin

∗University of California, Berkeley, U.S.A., was@math.berkeley.edu

The definition of the spaces S_{k}(Γ) of modular forms as functions on the upper half plane 𝔥 satisfying a certain equation is very abstract. The definition of the Hecke operators even more so.

We are fortunate that we now have methods available which allow us to transform the vector space of cusp forms of given weight ≥ 2 and level N into a concrete object, which can be explicitely computed. We have the work of Atkin-Lehner, Birch-Swinnerton-Dyer, Cremona, Manin, Mazur, Merel, and many others to thank for this.

The Eichler-Selberg trace formulas, as developed in [HJ] and [WA], can be used to compute characteristic polynomials of Hecke operators and hence gain some information about spaces of modular forms. It is also sometimes possible to write down explicit basis in terms of Θ-series and to compute the action of Hecke operators on their q-expansions. Other methods include computing the Hecke operators and q-expansions using Brandt matrices and quaternion algebras as in [P] or [K], or the module of supersingular points in “characteristic N” as exploited by Mestre and Oesterle in [M1].

Though the above methods are each beautiful and well suited to certain applications, we will not discuss them further here. Instead we focus on the modular symbols method, as it also has many advantages. We will only discusss the theory in this summary paper, leaving an explicit description of the objects involved for later. Nonetheless there is a definite gap between the theory on the one hand, and an efficient running machine implimentation on the other. To impliment the algorithms hinted at below requires making absolutely everything completely explnicit and then finding intelligent and efficient ways of performing the necessary manipulations. This is a nontrivial and tedious task, with room for error at every step.

Our exposition follows very closely that of [M].

1.1 Notation

Let Γ be a finite index subgroup of SL_{2}(Z) and k ≥ 2 an integer. If k is odd, assume −1 ∉ Γ, so that the modular forms theory is nonempty. Let P^{1}(Q) = Q ∪ {∞}.

2 Modular symbols and modular forms

2.1 Modular symbols

Let M be the Z-module generated by formal symbols {α, β}, α, β ∈ P^{1}(Q), subject to the relations

{α, β} + {β, γ} + {γ, α} = 0.

Thus {α, β} = −{β, α} and {α, α} = 0. There is a left action of g ∈ GL_{2}(Q) given by g.{α, β} = {gα, gβ}.

Let

V_{k} = Sym_{Z}^{k−2}(Z × Z) = Z_{k−2}[X, Y]

be the free Z-module of homogeneous polynomimals in two variables of degree k − 2. There is a left action of g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ M_{2}(Z) given by

g.P (X, Y) = P (det(g)g^{−1}(X, Y)) = P (dX − bY, −cX + aY).

The space

M_{k} := V_{k} ⊗_{Z} M.

is equipped with a left action of M_{2}(Z) given by

g.(P ⊗ x) = g.P ⊗ g.x.

Let

M_{k}(Γ) := H_{0}(Γ, M_{k})

be the zeroth homology group. Thus M_{k}(Γ) is the quotient of M_{k} by the relations g.x = x for all x ∈ M_{k} and g ∈ Γ. The elements of M_{k}(Γ) are called modular symbols of weight k for Γ. As we will see later, using Shapiro’s lemma and an explicit computation,

M_{k}(Γ) ⊗ C ≅ H^{1}(Γ, V_{k} ⊗ C).

The theory of Eichler and Shimura embeds modular forms in H^{1}(Γ, V_{k} ⊗ C).

2.2 Manin symbols

Let e = {0, ∞} ∈ M.

Proposition 2.1 (Manin’s trick). The elements g.e for g ∈ SL_{2}(Z) generate M.

Proof. (From [CR1].) Writing {α, β} = {0, β} − {0, α}, it suffices to show that every symbol of the form {0, α} is in the group generated by the g.e. Let

\frac{p_{−2}}{q_{−2}} = \frac{0}{1}, \frac{p_{−1}}{q_{−1}} = \frac{1}{0}, \frac{p_{0}}{1} = \frac{p_{0}}{q_{0}} , \frac{p_{1}}{q_{1}} , \frac{p_{2}}{q_{2}} , …, \frac{p_{r}}{q_{r}} = α

denote the continued fraction convergents of the rational number α. Then

p_{j}q_{j−1} − p_{j−1}q_{j} = (−1)^{j−1} for − 1 ≤ j ≤ r.

Hence

\{0, α\} = ∑_{j=−1}^{r} \{\frac{p_{j−1}}{q_{j−1}} , \frac{p_{j}}{q_{j}} \} = ∑_{j=−1}^{r} g_{j}.e

where g_{j} = ( \begin{matrix}(−1)^{j−1}p_{j} & p_{j−1} \\ (−1)^{j−1}q_{j} & q_{j−1} \end{matrix} ).

To a pair g ∈ SL_{2}(Z) and P ∈ V_{k} define the Manin symbol

[P, g] = g.(P ⊗ e) ∈ M_{k}(Γ).

The matrices

σ = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) and τ = ( \begin{matrix}0 & −1 \\ 1 & −1 \end{matrix} )

satisfy

σ^{4} = 1, τ^{3} = 1

and generate SL_{2}(Z).

Proposition 2.2. Let g ∈ SL_{2}(Z) and P ∈ V_{k}. The symbol [P, g] depends only on P and the class Γg. When g runs through SL_{2}(Z) and P runs through V_{k}, the Manin symbols generate M_{k}(Γ) (maybe, at least this should be true after tensoring with Q). Furthermore, they satisfy

[P, g] + [σ^{−1}P, gσ] = 0,
[P, g] + [τ^{−1}P, gτ] + [τ^{−2}P, gτ^{2}] = 0,

Proof. The first assertion follows from the construction of M_{k}(Γ). The correct version of the second assertion (should somehow) follow from Lemma 2.1. For the third assertion note the following relations:

e + σ(e) = {0, ∞} + {σ(0), σ(∞)}
= {0, ∞} + {∞, 0} = 0,
e + τ(e) + τ^{2}(e) = {0, ∞} + {τ(0), τ(∞)} + {τ^{2}(0), τ^{2}(∞)}
= {0, ∞} + {1, 0} + {∞, 1} = 0.

Thus

[P, g] + [σ^{−1}P, gσ] = g(P ⊗ e) + gσ(σ^{−1}P ⊗ e)
= gP ⊗ e + gP ⊗ gσe
= gP ⊗ g(e + σ(e)) = 0,
[P, g] + [τ^{−1}P, gτ] + [τ^{−2}P, gτ^{2}] = g(P ⊗ e) + gτ(τ^{−1}P ⊗ e) + gτ^{2}(τ^{−2}P ⊗ e)
= gP ⊗ g(e + τ(e) + τ^{2}(e)) = 0.

Theorem 2.3. The above relations generate all relations satisfied by the Manin symbols. (Maybe one must tensor with Q.)

Remark 2.4. Not only do not know whether or not this is true before tensoring with Q, and I do not know how to prove this.

The Manin symbol [P, g] can be written as Z-linear combinations of Manin symbols

[X^{q}Y^{k−2−q}, g], with 0 ≤ q ≤ k − 2.

Since [SL_{2}(Z) : Γ] is finite, M_{k}(Γ) is a finitely generated abelian group. In particular, we can write down an explicit basis which is then readily amenable to machine computation. We write [X^{q}Y^{k−2−q}, g] = [q, g] to simplify notation. Let g_{1}, …, g_{n} be a set of coset representatives for Γ in SL_{2}(Z). Then M_{k}(Γ) is generated by

{[q, a_{i}] : 0 ≤ q ≤ k − 2, 1 ≤ i ≤ n}

subject to the relations given by Proposition 2.2.

Note that M_{k}(Γ) may contain nontrivial torsion, which is not well understood (by me!).

2.3 Cuspidal modular symbols

In this section we assume k is even. There is a similar definition when k is odd.

Let D_{0} = Div^{0}(P^{1}(Q)) be the group of divisors of degree zero supported on P^{1}(Q) and note that GL_{2}(Q) acts on D_{0} on the left by linear fractional transformations.

Let C(Γ) = H_{0}(Γ) = H_{0}(Γ, D_{0}) be the zeroth homology of Γ ⊂ SL_{2}(Z) acting on D_{0} = Div^{0}(P^{1}(Q)), so H_{0}(Γ) is the free abelian group generated by the set of orbits Γ\P^{1}(Q). It is the free abelian group on the cusps of the modular curve X(Γ). There is a map M_{k}(Γ) → C(Γ) which on Manin symbols is

[P, g] ↦ P (1, 0)[g(∞)] − P (0, 1)[g(0)].

Let S_{k}(Γ) be the kernel of this map.

2.4 Duality between modular symbols and modular forms

2.4.1 Weight 2

The first homology group H_{1}(X_{Γ}, Z) of the modular curve X_{Γ}, viewed as a real 2-manifold, is a free abelian group of rank 2g, where g is the genus of X_{Γ}. The global differentials Ω(X) = H^{0}(X, Ω) on X_{Γ}, now viewed as a Riemann surface, form a g dimensional complex vector space. It is equal to the complex vector space S_{2}(Γ) of cusp forms. There is a nondegenerate pairing

H_{1}(X_{Γ}, Z) ⊗ Ω(X) → C
⟨γ, ω⟩ ↦ ∫_{γ} ω.

Taking coefficients in R we have

H_{1}(X_{Γ}, R) = H_{1}(X_{Γ}, Z) ⊗ R.

Extending the above pairing gives a natural injection of H_{1}(X_{Γ}, R) into the dual space of Ω(X). Since the two spaces have the same real dimension, this injection must be an isomorphism.

Suppose now that k = 2. We can identify modular symbols {α, β} for Γ as elements of H_{1}(X_{Γ}, R), and we have the formula,

⟨{α, β}, ω⟩ = ∫_{α}^{β} ω.

In fact, modular symbols were first introduced in this way by Birch in [B] in his work with Swinnerton-Dyer on the special value at s = 1 of the L-function associated to a (modular) elliptic curve.

2.4.2 Higher weight

The duality generalizes to higher weight. Let f : 𝔥 → C be a map. For g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(Q) and z ∈ 𝔥, define

f|[g]_{k}(z) = (cz + d)^{−k}f(gz)(det g)^{k−1}
f|[\overline{g}]_{k}(z) = (c\overline{z} + d)^{−k}f(gz)(det g)^{k−1}

We denote by S_{k}(Γ) (resp. \overline{S_{k}(Γ)}) the complex vector space of holomorphic (resp. antiholomorphic) cusp forms of weight k for Γ. There is a canonical isomorphism of real vector spaces between S_{k}(Γ) and \overline{S_{k}(Γ)} which associates to f the antiholomorphic modular form z ↦ \overline{f(z)}.

There is a pairing

(S_{k}(Γ) ⊕ \overline{S_{k}(Γ)}) × M_{k}(Γ) → C

given by the rule

⟨f_{1} + f_{2}, P ⊗ \{α, β\}⟩ = ∫_{α}^{β} f_{1}(z)P (z, 1)dz + ∫_{α}^{β} f_{2}(z)P (\overline{z}, 1)dz

where f_{1} ∈ S_{k}(Γ) and f_{2} ∈ \overline{S_{k}(Γ)}.

Theorem 2.5. The following pairing, obtained from the above one, is nondegenerate:

(S_{k}(Γ) ⊕ \overline{S_{k}(Γ)}) × S_{k}(Γ) → C.

2.5 Complex conjugation

Let η = ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) and η̃ = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ). Assume in this section that η^{−1}Γη = Γ.

Proposition 2.6. The map ι which associates to f ∈ S_{k}(Γ) ⊕ \overline{S_{k}(Γ)} the function z ↦ f(−\overline{z}) is a complex linear involution of S_{k}(Γ) ⊕ \overline{S_{k}(Γ)} which exchanges S_{k}(Γ) and \overline{S_{k}(Γ)}.

Define an involution ι^{×} on M_{k}(Γ) by

ι^{×}(P ⊗ x) = −η̃P ⊗ ηx.

This involution is adjoint to ι with respect to the pairing of Theorem 2.5. Moreover ι^{×} acts as follows on Manin symbols

ι^{×}([P, g]) = −[η̃P, ηgη^{−1}].

Let S_{k}(Γ)^{+} denote the subspace of elements of S_{k}(Γ) fixed by ι^{×}.

Proposition 2.7. The bilinear pairing induced by the pairing ⟨., .⟩

S_{k}(Γ) × S_{k}(Γ)^{+} → C

is nondegenerate.

2.6 Eichler-Shimura

Eichler and Shimura found a way to embed modular forms into a cohomology group. There is also a way to embed modular symbols into the same cohomology group.

The complex vector space V_{k} ⊗_{Γ} SL_{2}(Z) is endowed with a right action of SL_{2}(Z) given by the formula

(P ⊗ g).γ = (γ^{−1}P) ⊗ (gγ).

Proposition 2.8. We have an isomorphism of complex vector spaces

H^{1}(SL_{2}(Z), V_{k} ⊗_{Γ} SL_{2}(Z) ⊗ C) ≅ M_{k}(Γ) ⊗ C.

Proof. This is Proposition 9 of [M]. The proof involves explicit computations with cocycles using the fact that SL_{2}(Z) is generated by σ and τ.

Remark 2.9. It might be possible to replace tensoring with C by something less severe.

Lemma 2.10 (Shapiro). Let H be a subgroup of a group G and let A be a Z[H]-module. Then

H^{q}(G, Hom_{H}(Z[G], A)) = H^{q}(H, A) for all q ≥ 0.

Corollary 2.11. There is an isomorphism

M_{k}(Γ) ⊗ C ≅ H^{1}(Γ, V_{k} ⊗ C).

Proof. Since Γ has finite index in SL_{2}(Z) there is an isomorphism

Hom_{Γ}(Z[SL_{2}(Z)], V_{k} ⊗ C) ≅ V_{k} ⊗_{Γ} SL_{2}(Z) ⊗ C.

Now apply Shapiro’s lemma.

Define the parabolic cohomology group H_{P}^{1} by the exactness of the following sequence

0 → H_{P}^{1} (Γ, V_{k} ⊗ C) → H^{1}(Γ, V_{k} ⊗ C) → ⨁_{cusps α} H^{1}(Γ_{α}, V_{k} ⊗ C)

where Γ_{α} is the stabilizer in Γ of the cusp α of X_{Γ}.

For f ∈ M_{k}(Γ) define a class in H^{1}(Γ, V_{k} ⊗ C) by the cocycle

γ ↦ ∫_{z_{0}}^{γ(z_{0})} f(z)\binom{z}{1}^{k−2}dz.

Here z_{0} is a basepoint, v^{k−2} denotes the image of v ⊗ ⋯ ⊗ v in Sym^{k−2}(C × C) and the integral is that of a vector-valued differential. There is a similiar construction for holomorphic differentials.

Theorem 2.12 (Eichler-Shimura). The map above gives rise to isomorphisms

M_{k}(Γ) ⊕ \overline{S_{k}(Γ)} → H^{1}(Γ, V_{k} ⊗ C)
S_{k}(Γ) ⊕ \overline{S_{k}(Γ)} → H_{P}^{1} (Γ, V_{k} ⊗ C).

3 Linear maps

3.1 Linear operators

Let Δ ⊂ M_{2}(Z) such that ΓΔΓ = Δ and such that Γ\Δ is finite. Note that Δ is a union of double cosets of Γ\M_{2}(Z)/Γ. Let R be a set of representatives of Γ\Δ.

3.1.1 Action on modular forms

Let M_{k}(Γ) be the space of modular forms of weight k for Γ. For f ∈ M_{k}(Γ), define an operator T_{Δ} by

T_{Δ}(f) = ∑_{α∈R} f|[α]_{k}

This is a well-defined linear action on M_{k}(Γ) which preserves the subspace S_{k}(Γ).

3.1.2 Action on modular symbols

Similiarly, define an operator T_{Δ} on the space M_{k}(Γ) of modular symbols by

T_{Δ}(x) = ∑_{α∈R} α.x.

3.1.3 Hecke operators

Suppose now that Γ = Γ_{1}(N). Let n ≥ 1 be an integer and set

Δ_{n} = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ M_{2}(Z) : det = n, N|c, N|(a − 1)\}.

Then the nth Hecke operator is T_{Δ_{n}}. If Γ = Γ_{0}(N) the condition that N|(a − 1) is relaxed to (N, a) = 1.

3.2 Action on Manin symbols

We now describe how to explicitely compute the action of the Hecke operators on M_{k}(Γ). Recall, we have an explicit “generators and relations” description of M_{k}(Γ) in terms of Manin symbols. The action of the Hecke operators (and other linear operators) described in the previous section is given in terms of modular symbols. We could describe the action of an operator on a Manin symbol by taking the Manin symbol, finding the corresponding modular symbol, acting by the operator, and then converting back to a sum of Manin symbols. This process is painfully inefficient as it involves repeated application of Proposition 2.1. This was how computations were originally done until Mazur and Merel described the action of the Hecke operators directly in terms of Manin symbols.

Let n > 0 be an integer. We denote by M_{2}(Z)_{n} the set of matrices of M_{2}(Z) of determinant n.

Definition 3.1 (Condition (M)). We say that an element ∑_{g} a_{g}g ∈ C[M_{2}(Z)_{n}] satisfies condition (M) if for all cosets C ∈ M_{2}(Z)_{n}/ SL_{2}(Z), we have in C[P^{1}(Q)],

∑_{g∈C} a_{g}([g(∞)] − [g(0)]) = [∞] − [0].

Note that the condition (M) depends neither on the level or the weight.

Suppose now that Γ = Γ_{1}(N) (or Γ_{0}(N)). There is a bijection between cosets Γ\ SL_{2}(Z) and pairs of integers (u, v) satisfying a certain equivalence. The bijection associates to a 2 × 2 matrix its bottom two entries. We may thus view the Manin symbols as pairs [P, (u, v)].

Theorem 3.2 (Merel). Let [P, (u, v)] be a Manin symbol. Suppose ∑_{g} a_{g}g ∈ C[M_{2}(Z)_{n}] satisfies condition (M). Then we have

T_{n}([P (X, Y), (u, v)]) = ∑_{g=( \begin{matrix}a & b \\ c & d \end{matrix} )∈M_{2}(Z)_{n}} a_{g}[P (aX + bY, cX + dY), (au + cv, bu + dv)]

where the sum is restricted to the matrices g such that gcd(au + cv, bu + dv) = 1 (if (n, N) = 1 this restriction is unnecessary).

Proof. See section 2 of [M].

The element ∑_{g∈M_{2}(Z)_{n}} g ∈ C[M_{2}(Z)_{n}] satisfies condition (M). In Merel’s paper one can find other families of simpler (more sparse) elements satisfying condition (M).

4 Generating H_{1}(Γ, Z)

How can we generate H_{1}(Γ, Z) using modular symbols?

Theorem 4.1. Choose any α ∈ Q ∪ {∞}, it doesn’t matter which. Then the map

Γ → H_{1}(Γ, Z) : γ ↦ {α, γ(α)}

is a surjective group homomorphism.

So, knowing generators for Γ would be enough.

Now specialize to the case Γ = Γ_{0}(N). Here is one guess for what might be true.

Question 4.2. Do the Manin symbols (c, d) with (c, N) = (d, N) = 1 generate H_{1}(X_{0}(N), Z)?

When N is prime those Manin symbols lie in H_{1}(X_{0}(N), Z) because they correspond to paths from the non-∞ cusp to itself. Let (c, d) be such a Manin symbol and choose a, b so that M = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(Z). Then the Modular symbol corresponding to (c, d) is M.e = M.\{0, ∞\} = \{M(0), M(∞)\}. Since c and d are both coprime to N, the cusps [\frac{a}{c}] and [\frac{b}{d}] are the same (remember, we are assuming N is prime so that there are only two cusps). The only other way to force the cusps to be the same would be to force b and d to both be divisible by N, but then (c, d) would not be a Manin symbol.

I do not know if these Manin symbols are enough to generate all integral modular symbols. But, we can set up some computer computations to get an idea of whether or not we should expect this.

Computation 1. Let H = H_{1}(X_{0}(N), Z). Let V be the submodule of H generated by the modular symbols \{∞, γ(∞)\} where γ = ( \begin{matrix}a & b \\ N & d \end{matrix} ) and 0 < a < N. Let W be the submodule of H_{1}(X_{0}(N), Q) generated by the Manin symbols (c, d) for which both c and d are coprime to N. If things were as easy as imaginable then both V and W would be equal to H (and to each other). If W is properly contained in V then we learn that the answer to the question is NO. In the prime case, if V is properly contained in W we learn that V does not generate H, which is also interesting.

We compute V and W for 11 ≤ N ≤ 100, and the module index [V : W]. Whenever there is a - for the index, this means that that neither V nor W span H_{1}(X_{0}(N), Q) over Q. In these cases we didn’t compute the actual index.

\begin{matrix}N & [W : V] \\ 11 & 1 \\ 14 & 1 \\ 15 & 1 \\ 17 & 1 \\ 19 & 1 \\ 20 & 1 \\ 21 & 1 \\ 22 & 1 \\ 23 & 1 \\ 24 & 1 \\ 26 & 1 \\ 27 & 1 \\ 28 & 1 \\ 29 & 1 \\ 30 & - \\ 31 & 1 \\ 32 & 1 \\ 33 & 1 \\ 34 & 1 \\ 35 & 1 \\ 37 & 1 \\ 38 & 1 \\ 39 & 1 \\ 41 & 1 \\ 42 & - \\ 43 & 1 \\ 44 & 1 \\ 45 & 1 \\ 46 & 1 \\ 47 & 1 \\ 48 & - \\ 49 & 1 \\ 50 & 1 \\ 51 & 1 \\ 52 & 1 \\ 53 & 1 \\ 54 & - \\ 55 & 1 \\ 56 & - \\ 57 & 1 \\ 58 & 1 \\ 59 & 1 \\ 60 & - \\ 61 & 1 \\ 62 & 1 \end{matrix} \begin{matrix}N & [W : V] \\ 63 & 1 \\ 64 & 1 \\ 65 & 1 \\ 66 & - \\ 67 & 1 \\ 68 & 1 \\ 69 & 1 \\ 70 & - \\ 71 & 1 \\ 72 & - \\ 73 & 1 \\ 74 & 1 \\ 75 & 1 \\ 76 & 1 \\ 77 & 1 \\ 78 & - \\ 79 & 1 \\ 80 & - \\ 81 & 1 \\ 82 & 1 \\ 83 & 1 \\ 84 & - \\ 85 & 1 \\ 86 & 1 \\ 87 & 1 \\ 88 & - \\ 89 & 1 \\ 90 & - \\ 91 & 1 \\ 92 & 1 \\ 93 & 1 \\ 94 & 1 \\ 95 & 1 \\ 96 & - \\ 97 & 1 \\ 98 & 1 \\ 99 & 1 \\ 100 & - \\ 101 & 1 \\ 102 & - \\ 103 & 1 \\ 104 & - \\ 105 & - \\ 106 & 1 \\ 107 & 1 \end{matrix} \begin{matrix}N & [W : V] \\ 108 & - \\ 109 & 1 \\ 110 & - \\ 111 & 1 \\ 112 & - \\ 113 & 1 \\ 114 & - \\ 115 & 1 \\ 116 & 1 \\ 117 & 1 \\ 118 & 1 \\ 119 & 1 \\ 120 & - \\ 121 & 1 \\ 122 & 1 \\ 123 & 1 \\ 124 & 1 \\ 125 & 1 \\ 126 & - \\ 127 & 1 \\ 128 & 1 \\ 129 & 1 \\ 130 & - \\ 131 & 1 \\ 132 & - \\ 133 & 1 \\ 134 & 1 \\ 135 & - \\ 136 & - \\ 137 & 1 \\ 138 & - \\ 139 & 1 \\ 140 & - \\ 141 & 1 \\ 142 & 1 \\ 143 & 1 \\ 144 & - \\ 145 & 1 \\ 146 & 1 \\ 147 & 1 \\ 148 & 1 \\ 149 & 1 \\ 150 & - \\ 151 & 1 \\ 152 & - \end{matrix} \begin{matrix}N & [W : V] \\ 153 & 1 \\ 154 & - \\ 155 & 1 \\ 156 & - \\ 157 & 1 \\ 158 & 1 \\ 159 & 1 \\ 160 & - \\ 161 & 1 \\ 162 & - \\ 163 & 1 \\ 164 & 1 \\ 165 & - \\ 166 & 1 \\ 167 & 1 \\ 168 & - \\ 169 & 1 \\ 170 & - \\ 171 & 1 \\ 172 & 1 \\ 173 & 1 \\ 174 & - \\ 175 & 1 \\ 176 & - \\ 177 & 1 \\ 178 & 1 \\ 179 & 1 \\ 180 & - \\ 181 & 1 \\ 182 & - \\ 183 & 1 \\ 184 & - \\ 185 & 1 \\ 186 & - \\ 187 & 1 \\ 188 & 1 \\ 189 & - \\ 190 & - \\ 191 & 1 \\ 192 & - \\ 193 & 1 \\ 194 & 1 \\ 195 & - \\ 196 & - \\ 197 & 1 \end{matrix}

10

Conclusion: Neither obvious set of elements of H_{1}(X_{0}(N), Z) will, in general, generate. It might be necessary to look at

R. S. Kulkarni, An arithmetic-geometry method of the study
of the subgroups of the modular group, American Journal of mathematics
113, 1991, 1053-1133

and find explicit generators.

References

[B] B. Birch, Elliptic curves, a progress report, AMS conference on number theory, Stonybrook (1969), 396–400.

[CR1] J.E. Cremona, Algorithms for modular elliptic curves, 2nd edition, Cambridge University Press, (1997).

[CR2] J.E. Cremona, Modular symbols for Γ_{1}(N) and elliptic curves with everywhere good reduction, Math. Proc. Camb. Phil. Soc. 111 (1992), 199–218.

[HJ] H. Hijikata, Explicit formula of the traces of Hecke operators for Γ_{0}(N), J. Math. Soc. Japan. 26 (1974), 56–82.

[K] D. Kohel, Hecke module structure of quaternions, (1998), preprint.

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