This is the body of a letter that I (Ken Ribet) wrote to J.-F. Mestre in early November, 1987. I did not publish the theorems in this letter because I expected that they would be incorporated into an article by Mestre and J. Oesterlé on the “graph method” for calculating eigenvalues of Hecke operators on spaces of cusp forms. My understanding is that the formulas in this letter have not appeared in print and that they retain some interest. I am happy to make them available at this time (May, 2001). My formulas have been generalized to some extent by William A. Stein, who calculates component groups of optimal abelian variety quotients of Jacobians of modular curves.

I return, first, to the notations you had on the board: p is a prime number, M is prime to p, N is the product pM. We suppose given an elliptic curve E, plus a map π : X_{o}(N) → E, over Q. We assume that the induced map on Albanese varieties π_{∗} : J_{o}(N) → E has connected kernel. Finally, we assume that E has bad (hence multiplicative) reduction at p. This situation occurs, in particular, if E has conductor N and is given as a “strong (Taniyama-)Weil curve.”

Next, we have some notation relative to the Néron models of E and J = J_{o}(N) at p. I will let X be the character group of the torus attached to J_{F_{p}}, so that X is the group of degree-0 divisors on the set C of isomorphism classes of supersingular elliptic curves over \overline{F}_{p}, which are enhanced by Γ_{o}(M) structures. A typical element of X is a sum ∑ λ_{C}(C), where the λ_{C} are integers which sum to 0. Let X_{E} be the character group associated with E. We chose a generator of this cyclic group. The natural map π^{∗} : X_{E} → X then sends this generator to a certain element ∑ λ_{C}(C) of X. We can identify X_{E} with the group of multiples of this sum. There is a second natural map π_{∗} : X → X_{E}. The endomorphism of X_{E} gotten by following π^{∗} by π_{∗} is multiplication by n = deg(π).

There are natural symmetric Z-valued pairings ⟨ , ⟩ and ⟨ , ⟩_{E} on X and X_{E}. These are the monodromy pairings considered by Grothendieck in SGA7. The component groups Φ and Ψ attached to J and E are then the cokernels of the maps

X → Hom(X, Z)

and

X_{E} → Hom(X_{E}, Z)

deduced from these pairings. The map

π_{∗} : Φ → Ψ

that π induces on component groups can be seen in the commutative diagram

0 → X → Hom(X, Z) → Φ → 0
↓ π_{∗} ↓ β ↓ π_{∗}
0 → X_{E} → Hom(X_{E}, Z) → Ψ → 0,

(1)

in which β is Hom(π^{∗}, Z).

Lemma 1 The map π_{∗} : X → X_{E} is surjective.

Proof . The map (in characteristic 0) π^{∗} : E → J is injective, because its transpose π_{∗} has connected kernel, by assumption. In particular, the map E[ℓ] → J[ℓ] is injective, for all prime numbers ℓ. The map which π^{∗} induces on tori

E^{toric} → J^{toric}

is then injective (in the appropriate sense), as one sees by lifting these tori to Z_{p}. This injectivity corresponds to the surjectivity asserted by the lemma.

Formula 1 The cokernel of π_{∗} : Φ → Ψ has order gcd(…, λ_{C}, …).

Proof . The cokernel in question is

Hom(X_{E}, Z)/{X_{E} + β(Hom(X, Z)}.

By the lemma, we have X_{E} ⊆ β(Hom(X, Z), since Hom(X, Z) contains X. Therefore the cokernel is simply

Hom(X_{E}, Z)/β(Hom(X, Z).

After some fooling with Ext groups, we see that this quotient map be identified with the torsion subgroup of X/X_{E} ^{1}. However, the gcd in the formula is the order of the torsion subgroup of X/X_{E}.

1Actually, this claim is wrong; I think that you get instead the dual of the indicated torsion group

Formula 2 We have

card(Ψ) = gcd (λ_{C}w_{C} − λ_{C′}w_{C′}),
C,C′

where w_{C} is the integer Aut(C)/2, as in your notation.

Proof . Consider the map

γ : X → Hom(X_{E}, Z)  (2)

gotten by composing

π_{∗} : X → X_{E}

and the map

X_{E} → Hom(X_{E}, Z)

which is ⟨ , ⟩_{E}. By the lemma, its cokernel is Ψ. To obtain the formula, we notice that the image of γ is generated by the elements γ(D − D′) for D, D′ ∈ C. For D and D′ in C, γ(D − D′) maps the generator ∑ λ_{C}(C) of X_{E} to λ_{D}w_{D} − λ_{D′}w_{D′}, since ⟨ , ⟩ is deduced by restriction from the pairing C, C′ ↦ δ_{C,C′}w_{C} on Z^{C}. It should be remarked here that we are using the compatibility between ⟨ , ⟩_{E} and ⟨ , ⟩ which is summarized by the relation

⟨ξ, π_{∗}(η)⟩_{E} = ⟨π^{∗}(ξ), η⟩  (3)

for ξ ∈ X_{E} and η ∈ X.

Formula 3 We have

n card(Ψ) = ∑ λ_{C}^{2}w_{C}.

Proof . Let g be our chosen generator of X_{E}. Then

card(Ψ) = ⟨g, g⟩_{E},

so that

n card(Ψ) = ⟨π_{∗}π^{∗}g, g⟩_{E} = ⟨π^{∗}g, π^{∗}g⟩.

The quantity ⟨π^{∗}g, π^{∗}g⟩ visibly coincides with the sum ∑ λ_{C}^{2}w_{C}.

Ken RIBET