The Kernel of the Modular Polarization of a Quotient of J_{0}(N)

William A. Stein^{∗} March 12, 1999

Contents

1 The Modular Polarization 1 2 The ∞-Component Group 3

1 The Modular Polarization

Suppose A is a quotient of J_{0}(N). Because J_{0}(N) is a Jacobian it possesses a canonical principal polarization. This induces a polarization on A. Definition 1.1. The modular polarization is the map δ : A^{∨} → A arising from autoduality of J_{0}(N).

[figure: A^{∨}; J_{0}(N); A; π^{∨}; δ; π]

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

Let f ∈ S_{2}(Γ_{0}(N)) be a newform, A = A_{f} the corresponding optimal quotient of J_{0}(N), and 𝔭_{f} = Ann_{T}(f) ⊂ T the annihilator of f in the Hecke algebra. Let H_{1} = H_{1}(X_{0}(N), Z) be the first integral homology of the modular curve X_{0}(N).

Theorem 1.2. Let Φ : H_{1} → A(C) be the period map. Then there is an exact sequence

0 → Φ(H_{1}[𝔭_{f}]) → Φ(H_{1}) → Ker(δ) → 0.

Proof. Step 1: Pass to lattices. Over the complex number we may write each of A and A^{∨} as complex tori T/Λ where T ≅ C^{d} and Λ is a lattice. The isogeny δ : A^{∨} → A induces maps T (A^{∨}) → T (A) and Λ(A^{∨}) → Λ(A). We thus obtain the following commuting diagram with exact rows and columns.

0  0
↓  ↓
0 ⟶  0  ⟶ Λ(A^{∨}) ⟶ Λ(A) ⟶ L ⟶ 0
↓  ↓  ↓  ↓
0 ⟶  0  ⟶ T (A^{∨}) ⟶ T (A) ⟶ 0 ⟶ 0
↓  ↓  ↓  ↓
0 ⟶ Ker(δ) ⟶ A^{∨}  ⟶^{δ} A ⟶ 0 ⟶ 0
↓  ↓
0  0

Applying the snake lemma we see that

Ker(δ) ≅ L = coker(Λ(A^{∨}) → Λ(A)).

Step 2: Identify lattices. Proposition 6 of [S] allows us to identify Λ(A) and Λ(A^{∨}) in terms of the integral homology H_{1} = H_{1}(X_{0}(N), Z).

First for J_{0}(N) we have

T (J) = Hom(S_{2}(Γ_{0}(N), C))

and an exact sequence

0 → H_{1} → T (J) → J(C) → 0.

As for A, we have

H_{1} ⟶^{Φ} T (A) → A(C) → 0

where

Φ(γ) = (∫_{γ} f_{1}, …, ∫_{γ} f_{d}).

(See below for the basis for T (A).) We thus have

T (A) = Hom(S[𝔭_{f}], C)
Λ(A) = Φ(H_{1})

In defining Φ we have chosen the basis f_{1}, …, f_{d} for S[𝔭_{f}], in order to obtain a basis for T (A). For A^{∨} we have

T (A^{∨}) = T (J)[𝔭_{f}]
Λ(A^{∨}) = H_{1}[𝔭_{f}]

Step 3: Compute L.

The map Λ(A^{∨}) → Λ(A) induced by δ is the restriction of Φ to H_{1}[𝔭_{f}]. Thus

L = \frac{Φ(H_{1})}{Φ(H_{1}[𝔭_{f}])},

which, combined with step 1, completes the proof.

2 The ∞-Component Group

Fix a newform f ∈ S_{2}(Γ_{0}(N)) as before and let A_{f} be the corresponding optimal quotient of J_{0}(N). Define groups

Y_{A} = H_{1}[𝔭_{f}]
Y_{A^{∨}} = Hom(H_{1}, Z)[𝔭_{f}]

Conjecture 2.1. There is an exact sequence of abelian groups

0 → Y_{A^{∨}} → Hom(Y_{A}, Z) → Ker(δ) → 0.

Evidence. It’s only a conjecture because I have not worked out all the details yet. The basic idea is that in computing Φ(H_{1})/Φ(H_{1}[𝔭_{f}]) we can replace Φ by any homomorphism Ψ eminating from H_{1} and satisfying

Ker(Ψ) = Ker(Φ).

Now let’s make a few assumptions related to the structure of the Tmodule H_{1} = H_{1}(X_{0}(N), C): Let

V_{f} = Hom(H_{1}, Z)[𝔭_{f}].

Assumption 1. dim_{Z} V_{f} = 2d, where d is the number of conjugates of f. Fix a basis φ_{1}, …, φ_{2d} for V_{f} and define

Ψ : H_{1} → Z,
Ψ(x) = (φ_{1}(x), …, φ_{2d}(x)).

Assumption 2. Ker(Ψ) = Ker(Φ)

Given these two assumptions, computing Ψ(H_{1})/Ψ(H_{1}[𝔭_{f}]) and computing the cokernel Y_{A^{∨}} → Hom(Y_{A}, Z) are the same thing.

I think both of these assumption can be shown by looking at characteristic polynomials of Hecke operators and using the Atkin-Lehner multiplicity one theory.

In analogy with the Grothendieck-Raynaud-Ribet description of the local component groups of Neron models we make the following definition.

Definition 2.2. The ∞-component group is

Φ_{A,∞} := Ker(δ).

References

[S] G. Shimura, On the factors of the jacobian variety of a modular function field, J. Math. Soc. Japan, 25, No. 3, 523–544 (1973).