Parametrizations of elliptic curves by Shimura curves and by classical modular curves

Kenneth A. Ribet Mathematics Department UC Berkeley Berkeley, CA 94720-3840

15 March 1996

Modular Parametrizations

Suppose that f = ∑ a_{n}q^{n} is a weight-two newform on Γ_{0}(N) with integer coefficients. Attached to f is an elliptic curve A = A_{f}, which is simultaneously an abelian subvariety and a quotient of J_{0}(N). We obtain a covering of curves

π = π_{f}: X_{0}(N) → A

by composing the Albanese map

X_{0}(N) → J_{0}(N), x ↦ (x) − (∞)

with the quotient J_{0}(N) → A.

The degree of this map, deg π, is of considerable interest:

• A good bound on deg π will prove the ABC Conjecture;

• If p divides deg π, then f is congruent mod p to an eigenform other than f in S_{2}(Γ_{0}(N)); the converse is close to being true.

It is not difficult to compute deg π for a given form f.

Bertolini and Darmon have suggested a relationship between deg π and the analogue of deg π in which X_{0}(N) is replaced by a Shimura curve:

Suppose that N = D · M, where D is the discriminant of an indefinite quaternion division algebra A over Q and M is prime to D. Let Γ_{0}^{D}(M) be the analogue of Γ_{0}(M) in which SL(2, Z) is replaced by the group of Norm-1 elements in a maximal order of A. According to Shimizu and Jacquet-Langlands, the space S_{2}(Γ_{0}^{D}(M)) is Hecke-isomorphic to the Dnew subspace of S_{2}(Γ_{0}(N)).

Let X_{0}^{D}(M) be the modular curve (Shimura curve) associated to Γ_{0}^{D}(M) and let J_{0}^{D}(M) be the Jacobian of this curve. Using the Eichler-Shimura relations, one can show that J_{0}^{D}(M) is Q-isogenous to an abelian subvariety of J_{0}(N): the D-new part of J_{0}(N).

In particular, A is isogenous to an elliptic curve A′ ⊆ J_{0}^{D}(M). We obtain a covering π′: X_{0}^{D}(M) → A′ by composing an Albanese map X_{0}^{D}(M) → J_{0}^{D}(M) with the dual of the inclusion A′ ↪ J_{0}^{D}(M).

Bertolini-Darmon posit a relation of the form

deg π′ =^{??} deg π ∏_{p|D} \frac{1}{c_{p}} (??)

where c_{p} = ord_{p} Δ_{A} is the number of Néron components of A_{f} at p. Intuitively, because of “level lowering,” the c_{p} control congruences between f and forms of level dividing N/p for some p|D, i.e., between f and D-old forms. Passing from Γ_{0}(N) to Γ_{0}^{D}(M) is to divest oneself of these latter forms.

Simple examples imply that (??) cannot be valid as it stands. For instance, take N = 14; then necessarily D = 14 and M = 1. The curves A and A′ are all of J_{0}(14) and J_{0}^{14}(1), respectively. Hence

deg π = deg π′ = 1.

However, c_{2} = 6 and c_{7} = 3. We have 1 ≠ \frac{1}{18}.

An obvious difficulty stems from interaction between the c’s and reducible Galois modules. (Level lowering works only for irreducible representations.) The curve A is [14C] in Antwerp IV; it has two rational subgroups of order 3 and a single rational subgroup of order 2. There are no D-new forms because there are no weight-two forms of level 1.

The reducibility is related to the fact that A′ is isogenous to A but not necessarily isomorphic to A. This suggests several strategies:

1. We can keep book on the component numbers c′_{p} for A′ as well as their analogues c_{p} for A.

2. We can define π′ using A instead of A′.

3. We can look only at cases when A = A′.

In the example with N = 14, Kurihara proved that A′ is [14D]. We have c′_{2} = 3 and c′_{7} = 6. Strategy #1 corresponds to looking at the formula

1 =^{?} \frac{1}{c_{2}c′_{7}} = \frac{1}{36}.

It is an improvement on the previous 1 = \frac{1}{18} because it is correct in Q^{∗}/(Q^{∗})^{2}!

Adopting strategy #2, we have deg π′ = 2. The formula

2 = 1/18

is again correct in Q^{∗}/(Q^{∗})^{2}.

A general result

To each modular elliptic curve A of conductor N, we can define a degree d(A) in Q^{∗}/(Q^{∗})^{2}: Choose π: X_{0}(N) → A and set

d(A) := deg π ∈ Q^{∗}/(Q^{∗})^{2}.

This invariant is well defined because π is defined mod (End A ⊗ Q)^{∗} = Q^{∗}. (Since A has multiplicative reduction at primes dividing D, A has no complex multiplication.)

Let d′(A) be the analogue of d(A) with X_{0}(N) replaced by X_{0}^{D}(M).

Theorem. We have the equality d′(A) = d(A) · ∏_{p|D} c_{p}^{−1} in Q^{∗}/(Q^{∗})^{2}.

Theorem 1 was shown by Bertolini-Darmon to be a consequence of BSD. We shall obtain it (without assuming BSD) by using the same “fundamental exact sequence” which intervenes in the proof of level-lowering.

The proof is by induction: we pass between X_{0}^{Dpq}(M) and X_{0}^{D}(pqM). Here p and q are distinct primes which are prime to DM. In the initial step D = 1; at the final stage Dpq is the D in the theorem. Fix D and re-define d(A) relative to X_{0}^{D}(pqM) and d′(A) relative to X_{0}^{Dpq}(M). Then the desired formula is

d′(A) ∼ d(A)c_{p}^{−1}c_{q}^{−1},

where “∼” means “mod squares.”

The proof hinges on a comparison between the mod p reduction of X_{0}^{Dpq}(M) and the mod q reduction of X_{0}^{D}(pqM). For D = 1, this comparison was carried out in my level-lowering papers — it follows from results of Deligne-Rapoport, Cherednik, Drinfeld, Grothendieck, Raynaud,. . ..

Deligne and Rapoport predicted in the introduction to their article that the mod q reduction of X_{0}^{D}(qL) for D > 1 would follow the same pattern as the mod q reduction of X_{0}(qL). (For us, L = pM.) Kevin Buzzard’s 1995 thesis contains a chapter which confirms this prediction.

In the case D > 1, the fundamental exact sequence was studied by Jordan-Livné.

After these preliminary incantations, I will cheat by studying only the first step in the induction: the comparison between the mod q reduction of X_{0}(pqM) and the mod p reduction of X_{0}^{pq}(M).

Each reduction gives rise to a free finite-rank abelian group endowed with Hecke operators T_{n} (for n ≥ 1), an action of the relevant Galois group (e.g., Gal(\overline{F}_{p}/F_{p})), and a monodromy pairing.

To fix ideas, consider X_{0}(pqM) mod q. Form the toric part T of the mod q reduction of the Néron model of the Jacobian of X_{0}(pqM). The abelian group in question is

L_{q} := Hom_{\overline{F}_{q}}(T , G_{m}).

The group L_{q} is explicitly the group of degree-0 divisors on the set S of supersingular points of X_{0}(pM)_{\overline{F}_{q}}. Its monodromy pairing

⟨ , ⟩: L_{q} × L_{q} ⟶ Z

is the restriction to L_{q} × L_{q} of the diagonal pairing Z^{S} × Z^{S} → Z such that ⟨s, s⟩ is half the number of automorphisms of the object corresponding to s.

One may view ⟨ , ⟩ as a map

L_{q} ⟶ Hom(L_{q}, Z)

whose cokernel is the group of components of J_{0}(pqM)_{F_{q}}.

The fundamental exact sequence relates L_{q} and its analogue Y_{p} for the mod p reduction of the Shimura curve X_{0}^{pq}(M). The sequence reads

0 → Y_{p} →^{ι} L_{q} → X_{q} → 0,

where X_{q} is the character group of the toric part of J_{0}(qM) × J_{0}(qM) in characteristic q. In particular, X_{q} is torsion free and p-old for Hecke operators.

The map ι can be made entirely canonical by choosing \overline{F}_{p} and \overline{F}_{q} in a sneaky way. It intertwines the monodromy pairings on Y_{p} and L_{q}.

Recall that pqM is the conductor of A. Choose a covering π: X_{0}(pqM) → A over Q, consider the induced maps J_{0}(pqM) ⟶^{⟵} A and their reductions in characteristic q. The map J_{0}(pqM)_{F_{q}} → A_{F_{q}} induces an injection π^{∗}: X_{A} ↪ L_{q}, where X_{A} ≈ Z is the character group of the connected component of A_{F_{q}}.

Let L ≈ Z be the saturation of π^{∗}(X_{A}) in L_{q} and let v ∈ L be a generator. Thus π^{∗}(X_{A}) = t · L = Ztv for some t ≥ 1.

Formula: c_{q} · deg π = ⟨tv, tv⟩ = t^{2}⟨v, v⟩.

Corollary. We have d(A) ∼ c_{q}⟨v, v⟩.

The formula is proved by comparing the monodromy pairing ⟨ , ⟩_{A} on X_{A} with the pairing ⟨ , ⟩ on L_{q}. We have

⟨π^{∗}x, y⟩ = ⟨x, π_{∗}y⟩_{A}

for x ∈ X_{A} and y ∈ L_{q}. Let g be the generator of X_{A} such that π^{∗}g = tv. Then

t^{2}⟨v, v⟩ = ⟨π^{∗}g, π^{∗}g⟩ = ⟨g, π_{∗}π^{∗}g⟩_{A}.

Since π_{∗}^{∘}π^{∗} is multiplication by deg π and since ⟨g, g⟩_{A} = c_{q}, we obtain the announced formula c_{q} deg π = t^{2}⟨v, v⟩.

Analogous reasoning yields c_{p} deg π′ = u^{2}⟨v′, v′⟩ for some u ∈ Z. Here, ⟨ , ⟩ denotes the monodromy pairing on Y_{p}, but this is the just the restriction to Y_{p} of the pairing we have been using on L_{q}. Further, v′ denotes a generator of the saturation L′ of (π′)^{∗}(X′_{A}) in Y_{p}, where X′_{A} is the character group associated with A mod p.

The key point is that L′ = L, so that v′ = ±v. Indeed, L could have been defined as the subgroup of elements ζ of L_{q} satisfying T_{n}ζ = a_{n}ζ for all n ≥ 1. Here, a_{n} is the nth coefficient of the modular form giving rise to the isogeny class of A.

Thus \frac{c_{p} deg π′}{u^{2}} = \frac{c_{q} deg π}{t^{2}} , and we get the Theorem.

Now compare:

\frac{c_{p} deg π′}{u^{2}} = \frac{c_{q} deg π}{t^{2}} ⟷ deg π′ =^{??} \frac{deg π}{c_{p}c_{q}} .

The two are identical when u = 1 and t = c_{q}.

Assume that π_{∗}: J_{0}(pqM) → A is an optimal quotient. Then an application of the snake lemma shows that

t = # coker(Φ_{J_{0}(pqM)} →^{π_{∗}} Φ_{A}).

The Φ’s are component groups computed in characteristic q. The source group Φ_{J_{0}(pqM)} is Eisenstein and should be regarded as an error term. In fact, if A is isolated in its isogeny class, then A[ℓ] is irreducible for all prime numbers ℓ, so that π_{∗} = 0 and t = c_{q}.

If A is now an optimal quotient of the Shimura curve J_{0}^{pq}(M), the equality u = 1 amounts

to the statement that the characteristic p map Φ_{J_{0}^{pq}(M)} →^{π′_{∗}} Φ_{A} is surjective.

Theorem. Assume that M = 1 and that A is isolated in its isogeny class. Then u = 1, t = c_{q} and deg π′ = deg π · c_{p}^{−1}c_{q}^{−1}.

The hypotheses are satisfied, for example, if pq = 57 and A = [57E]. One has deg π = 4, c_{3} = 2 and c_{19} = 1. The Theorem asserts that deg π′ = 2.

Proof of the Theorem

As explained earlier, we have t = c_{q}. Further, we can assert that

deg π′ = u^{2} \frac{deg π}{c_{p}c_{q}}

where u is a divisor of c_{p}. By symmetry, u divides c_{p} as well. But c_{p} and c_{q} are relatively prime: no A[ℓ] can be finite over Z!

It would be of considerable interest to treat at least the more general case where A is isolated in its isogeny class and has conductor Mpq with M ≥ 1. The only potential problem occurs if there is a prime ℓ such that A[ℓ] is finite both at p and at q. Further, there is no difficulty unless when p ≡ q ≡ 1 mod ℓ.

In this situation, A[ℓ] lies in J_{0}^{pq}(M)[𝔪] for some maximal ideal 𝔪 in the Hecke ring T. It will probably be helpful to understand the dimension of J_{0}^{pq}(M)[𝔪], i.e., the multiplicity of the irreducible representation A[ℓ] in J_{0}^{pq}(M). We know that multiplicities can be > 1 in some situations involving Shimura curves, but there is currently no complete analysis.