Component Groups of Optimal Quotients of J_{0}(N)

William A. Stein^{∗}

March 11, 1999

Abstract

1 Introduction

Let f ∈ S_{2}(Γ_{0}(N)) be a newform and A_{f} the corresponding optimal quotient of J_{0}(N). Let Φ_{A_{f} ,p} be the special fiber of the Neron model of A_{f} at p. The integer #Φ_{A_{f} ,p} is of great arithmetic interest. For example, it appears in the Birch and Swinnerton-Dyer conjecture (I lie: actually the cardinality of the rational component group appears there).

How can we compute #Φ_{A_{f} ,p}?

When A_{f} = E is an elliptic curve the Riemann-Roch theorem gives us a powerful tool: we have a simple explicit Weierstrass equation which defines E. Using this Tate provided an efficient algorithm for computing the order of Φ_{E,p}. I doubt that there is an analogue of the Weierstrass equation when dim A_{f} > 1!

Ribet, in a letter to Mestre, considered the case in which A_{f} = E is an elliptic curve which has multiplicative reduction at p, i.e., ord_{p}(N) = 1. Using Grothendieck’s monodromy pairing on character groups of certain tori he was able to provide a “Weierstrass-free” formula for Φ_{E,p}. In this paper we generalize Ribet’s method to A_{f} of arbitrary dimension and thus obtain an explicit formula for #Φ_{A_{f} ,p}.

The formula is:

#Φ_{A_{f} ,p} = \sqrt{deg(δ)} · \frac{# coker(X → Hom(X[𝔭], Z))^{2}}{disc(X[𝔭] × X[𝔭] → Z)}

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

where • δ : A_{f}^{∨} → A_{f} is the modular polarization.

• X is the character group of the torus associated to J_{0}(N)/F_{p}.

• 𝔭_{f} = Ann_{T}(f), where T is the Hecke algebra.

• X × X → Z is the Monodromy pairing. Each of these quantities can be explicitely computed. See [K], [M], and [S].

2 The Monodromy Pairing

[to be written – will contain the basic facts]

3 Ribet’s Formulas

[to be written – will contain Ribet’s formulas as motivation.]

4 General Formula

In this section we prove our main theorem.

Let f ∈ S_{2}(Γ_{0}(N)) be a newform and suppose p|N but p ∤ M = N/p. The Hecke algebra T acts on the fundamental group X_{J} of J_{0}(N). The prime ideal

𝔭_{f} = Ann_{T}(f) ⊂ T

cuts out a submodule

V_{f} = X_{J}[𝔭_{f}] ⊂ X_{J}

of X_{J}. Theorem 4.1.

#Φ_{A_{f} ,p} = \sqrt{deg(δ)} · \frac{# coker(X_{J} → Hom(V_{f}, Z))^{2}}{disc(V_{f}, V_{f})}

We now begin the proof.

Fundamental diagrams:

[figure: A^{∨}; J; A; π^{∨}; δ; δ̂; π; X_{A}; X_{J}; X_{A^{∨}}; π^{∗}; δ^{∗}; δ̂^{∗}; π_{∗}]

Monodromy pairings:

⟨ , ⟩_{J} : X_{J} × X_{J} → Z ⟨ , ⟩_{A} : X_{A}^{∨} × X_{A} → Z

Component groups:

0 → X_{J} → Hom(X_{J}, Z) → Φ_{J,p} → 0 0 → X_{A^{∨}} → Hom(X_{A}, Z) → Φ_{A,p} → 0

Step 1. Formulas: Lemma 4.2. Let x ∈ X_{A^{∨}}, y ∈ X_{A}. Then

⟨π^{∗}δ̂^{∗}x, π^{∗}y⟩_{J} = deg(δ)⟨x, y⟩_{A}.

Proof. This is a direct computation, using the functorial properties of the objects involved.

⟨π^{∗}δ̂^{∗}x, π^{∗}y⟩_{J} = ⟨π_{∗}π^{∗}δ̂^{∗}x, y⟩_{A}
= ⟨δ^{∗}δ̂^{∗}x, y⟩_{A}
= ⟨(δ̂ ∘ δ)^{∗}x, y⟩_{A}
= deg(δ)⟨x, y⟩_{A}

Proposition 4.3.

#Φ_{A,p} = \frac{disc(π^{∗}δ̂^{∗}X_{A^{∨}}, π^{∗}X_{A})}{deg(δ)^{d}}.

Proof. By the lemma we have

disc(π^{∗}δ̂^{∗}X_{A^{∨}}, π^{∗}X_{A}) = deg(δ)^{d} · disc(X_{A^{∨}}, X_{A}).

Now use that #Φ_{A,p} = # coker(X_{A^{∨}} → X_{A}) = disc(X_{A^{∨}}, X_{A}).

Proposition 4.4.

#Φ_{A,p} = \frac{[X_{A} : δ̂^{∗}X_{A^{∨}}] · disc(π^{∗}X_{A}, π^{∗}X_{A})}{deg(δ)^{d}}.

Proof. Using the formula from the previous proposition we have

#Φ_{A,p} = \frac{disc(π^{∗}δ̂^{∗}X_{A^{∨}}, π^{∗}X_{A})}{deg(δ)^{d}}
= \frac{[π^{∗}X_{A} : π^{∗}δ̂^{∗}X_{A^{∨}}] · disc(π^{∗}X_{A}, π^{∗}X_{A})}{deg(δ)^{d}}

Note that π^{∗} is an injection so it does not change the index: [π^{∗}X_{A} : π^{∗}δ̂^{∗}X_{A^{∨}}] = [X_{A} : δ̂^{∗}X_{A^{∨}}].

Step 2. Identify Φ_{A,p} inside of X_{J}: Proposition 4.5.

Φ_{A,p} ≅ coker(X_{J} → Hom(π^{∗}X_{A}, Z))

Proof. Let X̃_{J} denote the image of X_{J} in Hom(π^{∗}X_{A}, Z). We have the following diagram with exact rows and columns:

0 ↓

0  ⟶ K
↓  ↓
0 ⟶ X̃_{J} ⟶ Hom(π^{∗}X_{A}, Z) ⟶ W ⟶ 0
↓≅ ↓≅ ↓
0 ⟶ X_{A^{∨}} ⟶ Hom(X_{A}, Z) ⟶ Φ_{A,p} ⟶ 0
↓  ↓  ↓
0 ⟶  0  ⟶ L

↓ 0

The snake lemma implies that the sequence

0 → K → 0 → 0 → L → 0

is exact, hence K = L = 0 and so W = Φ_{A,p}.

Step 3. Relate π^{∗}X_{A} to V_{f}: Let

V_{f} = X_{J}[𝔭_{f}] = ⋂_{t∈Ann_{T}(f)} ker(t) ⊂ X_{J}.

By multiplicity one and T-invariance of π^{∗},

π^{∗}X_{A} ⊂ V_{f},

and the index is finite. Let

m = [V_{f} : π^{∗}X_{A}].

The discriminant expression of Φ_{A,p} is “homogeneous of degree 2 in V_{f}”:

Proposition 4.6.

#Φ_{A,p} = \frac{[X_{A} : δ̂^{∗}X_{A^{∨}}] · m^{2} · disc(V_{f}, V_{f})}{deg(δ)^{d}}.

Proof. Note that

disc(π^{∗}X_{A}, π^{∗}X_{A}) = m^{2} disc(V_{f}, V_{f}).

Now use (4.4).

The cokernel expression for Φ_{A,p} is “homogeneous of degree 1 in V_{f}”: Proposition 4.7.

#Φ_{A,p} = m · # coker(X_{J} → Hom(V_{f}, Z)).

Proof. Use (4.5) and properties of cokernels. Step 4. Cancel m: Proposition 4.8.

#Φ_{A,p} = \frac{deg δ^{d}}{[X_{A} : δ̂^{∗}X_{A^{∨}}]} · \frac{# coker(X_{J} → Hom(V_{f}, Z))^{2}}{disc(V_{f}, V_{f})}

Proof. Let c = #Φ_{A,p}. Using (4.6) and (4.7) we obtain

c = \frac{c^{2}}{c} = \frac{m^{2} · # coker(X_{J} → Hom(V_{f}, Z))^{2}}{\frac{[X_{A}:δ̂^{∗}X_{A^{∨}}]·m^{2}·disc(V_{f} ,V_{f})}{deg(δ)^{d}}}

Cancelling the m^{2}’s gives the formula. Step 5. Analyze deg(δ): Proposition 4.9.

\frac{deg(δ)^{d}}{[X_{A} : δ̂^{∗}X_{A^{∨}}]} = [X_{A^{∨}} : δ^{∗}X_{A}]

Proof. We have

deg(δ)^{d} = [X_{A} : deg(δ)X_{A}]
= [X_{A} : δ̂^{∗}X_{A^{∨}}] · [δ̂^{∗}X_{A^{∨}} : δ̂^{∗}δ^{∗}X_{A^{∨}}]
= [X_{A} : δ̂^{∗}X_{A^{∨}}] · [X_{A^{∨}} : δ^{∗}X_{A^{∨}}]

The last step uses that δ̂^{∗} is injective. Now divide both sides by [X_{A} : δ̂^{∗}X_{A^{∨}}].

Putting everything together we obtain

#Φ_{A,p} = \frac{# coker(X_{A} → X_{A^{∨}}) · # coker(X_{J} → Hom(V, Z))^{2}}{disc(V_{f}, V_{f})} .

Thus to prove the theorem it remains only to show that

# coker(X_{A} → X_{A^{∨}}) = \sqrt{deg(δ)}.

Step 6. Analyze cokernel of δ^{∗}:

Theorem 4.10.

# coker(X_{A} → X_{A^{∨}}) = \sqrt{deg(δ)}.

Proof. The construction of Neron models commutes with unramified base extension. Since ord_{p}(N) = 1 we can pass to an unramified extension of Q_{p} so that A has universal cover 𝔾_{m}^{d}. Applying the Raynaud-van der Put theorem we obtain an exact commutative diagram of rigid analytic spaces:

0  0
↓  ↓
0 ⟶ 0 ⟶ X_{A} ⟶ X_{A^{∨}} ⟶ L ⟶ 0
↓ ↓ ↓ ↓
0 ⟶ Λ ⟶ 𝔾_{m}^{d} ⟶ 𝔾_{m}^{d} ⟶ 0 ⟶ 0
↓ ↓ ↓ ↓
0 ⟶ K ⟶ A^{∨} ⟶ A ⟶ 0 ⟶ 0
↓  ↓
0  0

where all maps are induced by the map δ : A^{∨} → A, so K = Ker(δ) and L = coker(X_{A} → X_{A^{∨}}). Applying the snake lemma, with the connecting map going from K to L, we obtain the exact sequence

0 → Λ → K → L → 0.

Now compute the 1-motive dual, i.e., M′ = Hom(M, 𝔾_{m}), of the above exact diagram to obtain:

0 ⟵ 0 ⟵ 𝔾_{m}^{d} ⟵ 𝔾_{m}^{d} ⟵ L′ ⟵ 0
↑ ↑ ↑ ↑
0 ⟵ Λ′ ⟵ X_{A^{∨}} ⟵ X_{A} ⟵ 0 ⟵ 0
↑ ↑ ↑ ↑
K′ ⟵ 0 ⟵ 0 ⟵ 0

Here A′ = A = 0, (𝔾_{m}^{d})′ = X_{A} or X_{A^{∨}}. Note that dualizing is not in general exact, though it is exact on the X_{A} and 𝔾_{m}^{d} parts.

Adding in the cokernels at the top gives:

0  0
↑  ↑
0 ⟵ A ⟵ A^{∨} ⟵ K′ ⟵ 0
↑ ↑ ↑ ↑
0 ⟵ 0 ⟵ 𝔾_{m}^{d} ⟵ 𝔾_{m}^{d} ⟵ L′ ⟵ 0
↑ ↑ ↑ ↑
0 ⟵ Λ′ ⟵ X_{A^{∨}} ⟵ X_{A} ⟵ 0 ⟵ 0
↑ ↑ ↑
K′ ⟵ 0 ⟵ 0

All of the maps in this diagram are induced by

δ^{∨} : A^{∨} → A.

But δ is induced by the canonical polarization of the Jacobian J_{0}(N), so δ^{∨} = δ. This is the same diagram as before, but viewed from a different point of view! Thus:

K′ = K
L′ = Λ
Λ′ = L

If G is a finite group scheme then #G = #G′. Thus the exact sequence 0 → Λ → K → L → 0 together with the above equalities yields

#K = #L · #Λ = #L · #L′ = (#L)^{2}.

This completes the proof.

Question 4.11. In the proof we determined the structure of coker(X_{A} → X_{A^{∨}}) in terms of the abelian group Ker(δ). It is possible to compute Ker(δ) explicitely hence it also possible to compute coker(X_{A} → X_{A^{∨}}). Is it possible to somehow use this to obtain the structure of Φ_{A,p}, instead of just the order?

5 Examples

[a couple of explicit worked examples illustrating various principles]

6 Tables

[table of all c_{p} for level ≤ 100.] [table of supringsly LARGE c_{p}.] [table of analytic orders of � for all A_{f} of rank 0 and prime level ≤ 1500.]

The table will essentially look like:

f  odd(L(A_{f}, 1)/Ω(A_{f})) numer((N − 1)/12)  deg(δ)  c_{p}
389E20  5^{2}/97  97  5^{2} · 2^{?}  97
433D16  7^{2}/3^{2}  2^{2} · 3^{2}  3^{2} · 7^{2} · 37^{2} · 2^{?}
563E31  13^{2}/281  281  13^{2} · 2^{?}
571D2  3^{2}/1  5 · 19  2^{4} · 3^{4} · 127^{2}  1
709C30  11^{2}/59  59  11^{2} · 2^{?}
997H42  3^{4}/83  83
1061D46  151^{2}/(5 · 53)  5 · 53  61^{2} · 151^{2} · 179^{2} · 2^{?}
1091C62  7^{2}/(5 · 109)  5 · 109  2^{?}
1171D53  11^{2}/(3 · 5 · 13)  3 · 5 · 13
1283C62  5^{2}/641  641
1429B64  5^{2}/(7 · 17)  7 · 17
1481C71  13^{2}/(5 · 37)  2 · 5 · 37
1483D67  3^{2} · 5^{2}/(13 · 19)  13 · 19

References

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

[M] J.F. Mestre, La méthode des graphs. Exemples et applications, Taniguchi Symp., Proceedings of the international conference on class numbers and fundamental units of algebraic number fields (Katata, 1986), 217–242, Nagoya Univ., Nagoya, (1986).

[S] W.A. Stein, The Modular Polarization of a Quotient of J_{0}(N), in preparation, (1999).