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

Let f = ∑a _n(f)e ^2πinz be a weight-two newform on Γ₀(N), where N = DM is the product of two relatively prime integers D and M and where D is the discriminant of an indefinite quaternion division algebra over Q. Assume that the Fourier coefficients of f are rational integers, so that f is associated with an isogeny class 𝒜 of elliptic curves over Q. Among the curves in 𝒜 is a distinguished element A, the strong modular curve attached to f. Shimura (1) has constructed A as an optimal quotient of J ₀(N). Thus A is the quotient of J ₀(N) by an abelian subvariety of this Jacobian. Composing the standard map X ₀(N) ↪ J ₀(N) with the quotient ξ: J ₀(N) → A, we obtain a covering π: X ₀(N) → A whose degree δ is an integer which depends only on f.

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 (For a partial converse, see ref. 5.) While δ is easy to calculate in practice (6), it seems more difficult to manage theoretically. Murty (24), has summarized what bounds are known at present.

This note concerns relations between δ and analogues of δ in which J ₀(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 ^∨ ↪ J ₀(N)^∨ which is dual to ξ may be viewed as a homomorphism A → J ₀(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 J ₀(N). The composite ξ○ξ^∨ ∈ 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 X ₀ ^D(M) be the Shimura curve associated with Γ₀ ^D(M) and let J′ = J ₀ ^D(M) be the Jacobian of X ₀ ^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 𝒜. 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: Equivalently, one can consider factorizations N = MpqD, where p and q are distinct prime numbers, D ≥ 1 is the product of an even number of distinct primes, and the four numbers p, q, D, and M are relatively prime. The formula displayed above amounts to the heuristic relation for each factorization N = MpqD. Although simple examples show that Eq. 1 is not correct as stated, we will prove that a suitably modified form of it is valid in many cases.

To state our results, we need to be more precise about the numbers c _p and c _q which appear above. We set: Let ξ : J → A and ξ′ : J′ → A′ be the optimal quotients of J and J′ for which A and A′ lie in 𝒜. (This is a change of notation, since we have been taking A to be an optimal quotient of J ₀(N); the new elliptic curve A is the unique curve isogenous to the original A which appears as an optimal quotient of J.) Let c _p and c _q be defined for A as above; i.e., c _p = ord_pΔ(A) and c _q = ord_qΔ(A). Note that c _p, for instance, may be viewed as the order of the group of components of the fiber at p of the Néron model for A. This group is cyclic. Let c′_p and c′_q be defined analogously, with A′ replacing A. Notice that ord_ℓ c _p = ord_ℓ c′_p and ord_ℓ c _q = ord_ℓ c′_q for each prime ℓ such that A[ℓ] is irreducible. Indeed, the curves A and A′ are isogenous over Q. The irreducibility hypothesis on A[ℓ] implies that any rational isogeny A → A′ of degree divisible by ℓ factors through the multiplication-by-ℓ map on A. Hence there is an isogeny ϕ : A → A′ whose degree is prime to ℓ. If d = deg ϕ, the map ϕ induces an isomorphism between the prime-to-d parts of the component groups of A and A′, both in characteristic p and in characteristic q.

Theorem 1. One has where the “error term” ℰ(D, p, q, M) is a positive divisor of c′_pc_q. Further, suppose that M is square free but not a prime number,† and let ℓ be a prime number which divides ℰ(D, p, q, M). Then the Gal(/Q)-module A[ℓ] is reducible.

In our proof of the theorem, we shall first prove a version of the displayed formula in which ℰ(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” ℰ(D, p, q, M) is not necessarily divisible by all primes ℓ for which A[ℓ] 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 = J ₀(14); and A′ = J ₀ ¹⁴(1), so that δ¹(14) = δ¹⁴(1) = 1. Since c ₂ = 6 and c ₇ = 3, Theorem 1 yields the pair of equalities there are two equalities because there are two choices for the ordered pair (p, q). By Theorem 1, the integers ℰ(1, 2, 7, 1) and ℰ(1, 7, 2, 1) are divisible only by the primes 2 and 3. Indeed, we see (once again from ref. 13) that these are the only primes ℓ for which A[ℓ] is reducible. Looking further at the tables, we see that there is a unique curve A′ in the isogeny class of A for which 3c′₂ is the square of an integer. This curve is [14D]. Thus we have A′ = [14D], as Kurihara determined in ref. 14.

There are five similar examples of products pq for which J ₀ ^pq(1) has genus one, namely 2⋅17, 2⋅23, 3⋅5, 3⋅7, and 3⋅11. [See, e.g., the table of Vignéras (ref. 15, p. 122).] In each of the five cases, we shall see that A′ = J ₀ ^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 = 3⋅5 and 3⋅7, A coincides with the Jacobian J ₀(pq). In this circumstance, an easy argument based on Proposition 1 below shows that the local invariants of A and A′ = J ₀ ^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, 2⋅17, 2⋅23, and 3⋅11, 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 so that 3c′₁₇ is a square. We then must have A′ = [34C]. When pq = 46, 2c′₂₃ is a square, and we conclude A′ = [23B]. When pq = 33, 2c′₁₁ is a square and thus A′ = [33B].

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[ℓ] is irreducible for all ℓ. In particular, A′ = A. Because A[ℓ] is irreducible for all ℓ, the theorem gives ℰ(1, 3, 19, 1) = 1. Hence Now Cremona’s table (ref. 6, p. 1247) yields the value δ¹(57) = 4; also, one has c ₃ = 2, c ₁₉ = 1. Thus we find δ⁵⁷(1) = 2. This relation is confirmed by results of D. Roberts (10), who shows, more precisely, that A is the quotient of X ₀ ⁵⁷(1) by its Atkin–Lehner involution w ₅₇.

Next, we consider the elliptic curves of conductor N = 714 = 2⋅3⋅7⋅17, 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, c ₂ = c ₃ = c ₇ = c ₁₇ = 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 ℰ(1, 7, 13, 11) = 1. Since 11 is a prime, the second part of Theorem 1 yields no information. However, by Proposition 3 below, ℰ(1, 7, 13, 11) divides both c ₇ and c ₁₃. Each of these integers is 2, so that we may conclude at least that ℰ(1, 7, 13, 11) is 1 or 2. Cremona’s tables give the value δ¹(1001) = 1008; hence δ^7⋅13(11) is either 252 or 1008.‡ On the other hand, since c ₇ and c ₁₁ are relatively prime, we find that ℰ(1, 7, 11, 13) = 1. Thus δ^7⋅11(13) = 1008/6 = 168. Similarly, δ^11⋅13(7) = 168.