The integer δ has been regarded with intense interest for the last decade. For one thing, primes dividing δ are “congruence primes for f”: if p divides δ, then there is a mod p congruence between f and a weight-two cusp form on Γ₀(N) which has integral coefficients and is orthogonal to f under the Petersson inner product. (See, e.g., Section 5 of ref. 2 for a precise statement.) For another, it is known that a sufficiently good upper bound for δ will imply the ABC Conjecture (3, 4). More precisely, as R. Murty explains in ref. 24, the ABC Conjecture follows from the conjectural bound

This note concerns relations between δ and analogues of δ in which J₀(N) is replaced by the Jacobian of a Shimura curve.

To define these analogues, it is helpful to give a characterization of δ in which π does not appear explicitly. For this, note that the map ξ: A J₀(N) which is dual to ξ may be viewed as a homomorphism A → J₀(N), since Jacobians of curves (and elliptic curves in particular) are canonically self-dual. The image of ξ is a copy of A which is embedded in J₀(N). The composite ξξ End A is necessarily multiplication by some integer; a moment’s reflection shows that this integer is δ. Let Γ₀^D(M) be the analogue of Γ₀(M) in which SL(2, Z) is replaced by the group of norm-1 units in a maximal order of the rational quaternion algebra of discriminant D. Let X₀^D(M) be the Shimura curve associated with Γ₀^D(M) and let J′ = J₀^D(M) be the Jacobian of X₀^D(M). The correspondence of Shimizu and Jacquet–Langlands (7) relates f to a weight-two newform f′ for the group Γ₀^D(M); the form f′ is well defined only up to multiplication by a nonzero constant. Associated to f′ is an elliptic curve A′ which appears as an optimal quotient ξ′ : J′ → A′ of J′. Using the techniques of Ribet (8) or the general theorem of Faltings (9), one proves that A and A′ are isogenous—i.e., that A′ belongs to . We define δ^D(M) Z as the composite ξ′(ξ′).

To include the case D = 1 in formulas below, we set δ¹(N) = δ.

Roberts (10) and Bertolini and Darmon (section 5 of ref. 11) have pointed out that the Gross–Zagier formula and the conjecture of Birch and Swinnerton-Dyer imply relations between δ and δ^D(M) in Q*/(Q*)². Bertolini and Darmon allude to the possibility that there may be a simple, precise formula for the ratio δ/δ^D(M). The relation which they envisage involves local factors for the elliptic curves A and A′ at the primes p|D.

While these factors may well be different for the two elliptic curves, we will ignore this subtlety momentarily and introduce only those factors which pertain to A. Suppose, then, that p is a prime dividing D, so that A has multiplicative reduction at p. Let c_p be the number of components in the fiber at p for the Néron model of A; i.e., c_p = ord_pΔ, where Δ is the minimal discriminant of A. As was mentioned above, δ controls congruences between f and newforms other than f in the space S of weight-two forms on Γ₀(N); analogously, δ^D(M) controls congruences between f and other forms in the D-new subspace of S. At the same time, level-lowering results such as those of Ribet (12) lead to the expectation that the c_p control congruences between f and D-old forms in S. This yields the heuristic formula:

1

To state our results, we need to be more precise about the numbers c_p and c_q which appear above. We set:

Theorem 1. One has

where the “error term”

In our proof of the theorem, we shall first prove a version of the displayed formula in which (D, p, q, M) is expressed in terms of maps between component groups in characteristics p and q. (See Theorem 2 below.) We then prove the second assertion of Theorem 1.

Before undertaking the proof, we illustrate Theorem 1 by considering a series of examples. As the reader will observe, these examples show in particular that the “error term” (D, p, q, M) is not necessarily divisible by all primes for which A[] is reducible.

For the first example, take M = 1, D = 1, and pq = 14. Thus N = 14, so that the curves A and A′ lie in the unique isogeny class of elliptic curves over Q with conductor 14. [There is a unique weight-two newform on Γ₀(14).] According to the tables of Antwerp IV, there are six curves in this isogeny class [ref. 13, p. 82]. The curve A is identified as [14C] in the notation of ref. 13. We have A = J₀(14); and A′ = J₀¹⁴(1), so that δ¹(14) = δ¹⁴(1) = 1. Since c₂ = 6 and c₇ = 3, Theorem 1 yields the pair of equalities

There are five similar examples of products pq for which J₀^pq(1) has genus one, namely 217, 223, 35, 37, and 311. [See, e.g., the table of Vignéras (ref. 15, p. 122).] In each of the five cases, we shall see that A′ = J₀^pq(1) can be determined as a specific elliptic curve of conductor pq with a small amount of detective work. [We suspect that this detective work was done 15 years ago by J.-F. Michon (see refs. 16 and 17).]

To begin with, we note that in each case there is a single weight-two newform on Γ₀(pq) with integral coefficients, i.e., a single isogeny class of elliptic curves of conductor pq. The strong modular elliptic curve A of conductor pq is identified in ref. 13. Knowing this curve, we have at our disposal c_p and c_q. Further, the integer δ¹(pq) is available from Cremona’s table (ref. 6, pp. 1247–1250).

In the two cases pq = 35 and 37, A coincides with the Jacobian J₀(pq). In this circumstance, an easy argument based on Proposition 1 below shows that the local invariants of A and A′ = J₀^pq(1) are “flipped”—we have c′_p = c_q and c′_q = c_p. After glancing at p. 82 of ref. 13, one sees that A′ = [15C] in the first of the two cases and A′ = [21D] in the second.

Let us now discuss the remaining three cases, 217, 223, and 311, where Cremona’s table gives the values 2, 5, and 3 (respectively) for δ¹(pq). Using Theorem 1 and the value δ^pq(1) = 1 in each case, we obtain equations which express c′_p and c′_q as products of known rational numbers and unknown square integers. These are enough to determine A′. Indeed, when pq = 34, we have

In the six examples we have discussed so far, an alternative approach would have been to read off the numbers c′_p and c′_q from a formula of Jordan and Livné (section 2 of ref. 18; see Theorem 4.3 of ref. 8). As we have seen, A′ is determined in each case by these local invariants.

For an example with a different flavor, we take f to be the modular form associated with the curve A = [57E] of ref. 13. This curve is isolated in its isogeny class; i.e., A[] is irreducible for all . In particular, A′ = A. Because A[] is irreducible for all , the theorem gives (1, 3, 19, 1) = 1. Hence

Next, we consider the elliptic curves of conductor N = 714 = 23717, which are tabulated in Cremona’s book (19). These curves fall into nine isogeny classes, A–I. Four of these classes, B, C, E and H, contain precisely one element. In other words, the four elliptic curves 714B1, 714C1, 714E1, and 714H1 are isolated in their isogeny classes. For each elliptic curve, Theorem 1 expresses δ⁷¹⁴(1) as well as the six degrees δ^pq(714/pq) for pq|714 in terms of δ¹(714) and the integers c_p for p|714. These numbers are available from refs. 6 and 19. The most striking of the four elliptic curves is perhaps 714H1. For this curve, c₂ = c₃ = c₇ = c₁₇ = 1 and δ¹(714) = 40. Hence δ⁷¹⁴(1) and all degrees δ^pq(714/pq) are equal to 40.

For a final example, we consult further tables of John Cremona which are available by anonymous ftp from euclid.ex.ac.uk in /pub/cremona/data. Let A be the curve denoted 1001C1, which has Weierstrass data [0, 0, 1, −199, 1092]. Its minimal discriminant is −7²11³13². This curve is isolated in its isogeny class, which suggests that (1, 7, 13, 11) = 1. Since 11 is a prime, the second part of Theorem 1 yields no information. However, by Proposition 3 below, (1, 7, 13, 11) divides both c₇ and c₁₃. Each of these integers is 2, so that we may conclude at least that (1, 7, 13, 11) is 1 or 2. Cremona’s tables give the value δ¹(1001) = 1008; hence δ⁷¹³(11) is either 252 or 1008.‡ On the other hand, since c₇ and c₁₁ are relatively prime, we find that (1, 7, 11, 13) = 1. Thus δ⁷¹¹(13) = 1008/6 = 168. Similarly, δ¹¹¹³(7) = 168.

The First Assertion of Theorem 1 If V is an abelian variety over Q and is a prime, let Φ(V, ) be the group of components of the fiber at of the Néron model of V. This group is a finite étale group scheme over Spec F, i.e., a finite abelian group furnished with a canonical action of Gal(/F). The association V Φ(V, ) is functorial. For example, as we noted above, if A is an elliptic curve with multiplicative reduction at p, then Φ(A, p) is a cyclic group of order c_p.

The maps ξ and ξ′ induce homomorphisms

Theorem 2. One has

To prove the theorem, we compare the character groups of algebraic tori which are associated functorially to the mod p reduction of J′ and the mod q reduction of J. Recall that the former reduction is described by the well known theory of Cerednik and Drinfeld (20–22), while the latter falls into the general area studied by Deligne and Rapoport (23). [Although Deligne and Rapoport provide only the briefest discussion of the case D > 1, what we need will follow from recent results of K. Buzzard (31).] Our comparison is based on the oft-exploited circumstance that the two reductions involve the arithmetic of the same definite rational quaternion algebra: that algebra whose discriminant is Dq.

To state the result which is needed, we introduce some notation: if V is an abelian variety over Q and is a prime number, let T be the toric part of the fiber over F of the Néron model for V and write (V, ) for the character group Hom(T, G_m). Thus (V, ) is a free abelian group which is furnished with compatible actions of Gal(/F) and End_Q V. At least in the case when V has semistable reduction at , there is a canonical bilinear pairing

The relation between Φ(V, ) and the character groups is as follows (Theorem 11.5 of ref. 25): there is a natural exact sequence

Proposition 1. There is a canonical exact sequence

for all x, y (J′, p

When D = 1, the proposition was proved in ref. 12. (See especially Theorem 4.1 of ref. 12.) The case D > 1 can be handled in an analogous way, thanks to K. Buzzard’s analogue (31) of the Deligne–Rapoport theorem (23). This theme is explored in the work of Jordan and Livné (26) and L. Yang (27).

Let be the “f-part” of (J, q), defined for example as the group of characters x (J, q) such that T_nx = a_n(f)x for all n prime to N. [Recall that a_n(f) is the nth coefficient of f.] It is not hard to check that is isomorphic to Z and that in fact it is contained in (J′, p), viewed as a subgroup of (J, q) via ι. Indeed, consider the decomposition of J as a product up to isogeny of simple abelian varieties over Q. One of the factors is A, which occurs with multiplicity 1, and the other factors are non-f: they correspond to newforms of level dividing N whose nth coefficients cannot coincide with the a_n(f) for all n prime to N. Hence Q is the tensor product with Q of the character group of the toric part of A_Fq; this shows that has rank 1. A similar computation shows that ∩ (J′, p) has rank 1, since A occurs up to isogeny exactly once in J′ and since A has multiplicative reduction at p. The image of in (J", q) × (J", q) is thus finite; it is zero since (J", q) is torsion free.

Fix a generator g of and set τ = u_J(g, g). An arbitrary nonzero element t of may be written ng, where n is a nonzero integer. We then have u_J(t, t) = n²τ.

By the theorem of Grothendieck (25) that was cited above, c_q may be interpreted as u_A(x, x), where x is a generator of (A, q) and where u_A is the monodromy pairing arising from the mod q reduction of A. Meanwhile, the map ξ : J → A induces by pullback a homomorphism ξ* : (A, q) → (J, q) and the dual of ξ induces similarly a homomorphism ξ_* : (J, q) → (A, q). The two homomorphisms are adjoint with respect to the monodromy pairings:

We emerge with the preliminary formula

Proposition 2. Let ξ_* and ξ′_* be the homomorphisms Φ(J, q) → Φ(A, q) and Φ(J′, p) → Φ(A′, p) which are induced by ξ and ξ′ on component groups. Then ( : (A, q)) = #coker ξ_* and ( : (A′, p)) = #coker ξ′_*.

Proof: The two formulas are analogous; we shall prove only the assertion relative to ξ_*. Because of the assumption that ξ : J → A is an optimal quotient, the map ξ : A → J is injective. One deduces from this the surjectivity of the map on character groups ξ_* : (J, q) → (A, q). Consider the commutative diagram with exact rows

It is clear that the order of coker(Hom(ξ*, Z)) coincides with the order of the torsion subgroup of coker(ξ*). Since (J, q)/ is torsion free by the definition of , we obtain first the formula

The Second Assertion of Theorem 1 We assume from now on that N is square free and that is a prime for which A[] is irreducible. We should mention in passing that the irreducibility hypothesis holds for one A if and only if it holds for all A . Indeed, the semisimplification of the mod Galois representation A[] depends only on . At the same time, A[] is irreducible if and only if its semisimplification is irreducible.

Lemma 2. There is a prime r|N for which does not divide c_r.

Proof: Suppose to the contrary that divides c_r for all r|N. Then the mod Gal(/Q)-representation A[] is finite at all primes (section 4.1 of ref. 28). If = 2, this contradicts a theorem of Tate (29). If > 2, a theorem of the first author (Theorem 1.1 of ref. 12) implies that A[] is modular of level 1 in the sense that it arises from the space of weight-two cusp forms on SL(2, Z). Since this space is zero, we obtain a contradiction in this case as well.

In order to prove the second assertion of Theorem 1, which concerns the “-part” of (D, p, q, M), we will consider varying decompositions N = DpqM. In these decompositions, the isogeny class , and the integer N in particular, are understood to be invariant. We view the prime as fixed, and recall the hypothesis that A[] is irreducible. (If this irreducibility hypothesis holds for one A , then it holds for all A.) Set

Proposition 3. If N = DpqM, then e(D, p, q, M) is the order of the -primary part of the cokernel of

Proof: In view of Theorem 2, the first statement means that the -primary part of the image of ξ_* : Φ(J, q) → Φ(A, q) is trivial. For each prime number r which is prime to N, let T_r be the rth Hecke operator on J. It is a familiar fact that Φ(J, q) is Eisenstein in the sense that T_r acts on Φ(J, q) as 1 + r for all such r. This was proved by the first author in case D = 1 (see Theorem 3.12 of ref. 12 and ref. 30), and the result can be extended as needed in view of results of Buzzard (31) and Jordan and Livné (26).

It follows from the Eichler–Shimura relation that the image of ξ_* is annihilated by a_r(f) − r − 1 for all r. One deduces from this that the -primary part of the image is trivial: If not, then a_r(f) r + 1 mod for all r, and this implies that the semisimplification of A[] is reducible; cf. Theorem 5.2(c) of ref. 12.

To prove the second statement, we begin by noting that e(D, p, q, M) divides c′_p. As we pointed out earlier, there is an isogeny A → A′ of prime-to- degree. Indeed, A and A′ are isogenous over Q; on the other hand, the hypothesis on A[] implies that any rational isogeny A → A′ of degree divisible by factors through the multiplication-by- map on A. Hence the -primary components of Φ(A, p) and Φ(A′, p) are isomorphic, so that the largest powers of in c_p and c′_p are the same. Thus e(D, p, q, M) divides c_p. Also, since an analogous reasoning shows that c_q and c′_q have the same valuations at , e(D, p, q, M) depends symmetrically on p and q, as asserted. Finally, e(D, p, q, M) divides both c_p and c_q, since it divides c_p and depends symmetrically on p and q.

Corollary. If N = dpqrsm, where p, q, r, and s are primes and d is the product of an even number of primes, then

Proof: Each of the two integers in the displayed equality may be calculated as the order of the -primary part of the cokernel of ξ′ : Φ(J^dpqrs(m), p) → Φ(A′, p). This coincidence gives the first equality. To obtain the second from the first, we note that both e(rsd, p, q, m)² e(d, r, s, pqm)² and e(qsd, p, r, m)² e(d, q, s, prm)² are equal to the -part of the quantity δ^dpqrs(m)c_pc_qc_rc_s/δ^d(pqrsm).

To finish the proof of Theorem 1, we assume from now on that M is not prime. To prove that e(D, p, q, M) = 1, it suffices to show that e(D, p, q, M) divides c_r for each r|N. If r = p or r = q, this divisibility is included in the statement of Proposition 3. Assume, next, that r is a divisor of D, and write D = rsd, where s is a prime. We have

It is a pleasure to thank J. Cremona, H. Darmon, and D. Roberts for helpful conversations and correspondence. This article was supported in part by National Science Foundation Grant DMS 93-06898.