On the Manin constants of modular elliptic curves

Bas Edixhoven

December 24, 1998

1 Definitions and results

For M a positive integer, let X_{0}(M)_{Q} be the modular curve over Q classifying elliptic curves with a given cyclic subgroup of order M and let J_{0}(M)_{Q} be the jacobian of X_{0}(M)_{Q}. An elliptic curve E over Q is said to be modular if it is an isogeny factor (isogenies over Q) of some J_{0}(M)_{Q}; the smallest M for which this happens is then called the level of E. The Shimura-Taniyama conjecture states that every elliptic curve E over Q is modular, and that the level of E equals the conductor of E. A modular elliptic curve of level M is called strong if there exists a closed immersion E ↪ J_{0}(M)_{Q}. It follows from the multiplicity one principle for modular forms that such an immersion is unique up to sign.

Let E be a strong modular elliptic curve of level M. The corresponding strong modular parametrization ϕ : X_{0}(M)_{Q} → E is obtained as follows:

J_{0}(M)_{Q} → E
↑  ↗_{ϕ}
X_{0}(M)_{Q}

where J_{0}(M)_{Q} → E is the dual of one of the two closed immersions E ↪ J_{0}(M)_{Q} and where X_{0}(M)_{Q} → J_{0}(M)_{Q} is the standard immersion sending the cusp ∞ to 0. Let E be the Néron model over Z of E. Then the Z-module of translation invariant differentials on E is free of rank one; let ω be one of the two generators. We get the differential form ϕ^{∗}ω on X_{0}(M)_{Q}. Now there is also a second differential form on X_{0}(M)_{Q} related to E: the normalized newform ∑ a_{n}q^{n} \frac{dq}{q} corresponding to E (normalized means that a_{1} = 1). The multiplicity one principle gives:

ϕ^{∗}ω = c ∑_{n≥1} a_{n}q^{n} \frac{dq}{q}

for some c ∈ Q^{∗}. This number c is called the Manin constant of E (see [7], [8], [9] and [1]).

Conjecture 1 (Manin) c = ±1.

In [7] Mazur showed that c ∈ Z[\frac{1}{2m}]^{∗}, where m is the largest square dividing M. His proof uses results of Raynaud on specialization of finite group schemes (this explains why 2 plays a special rôle) and the description of Deligne and Rapoport [3] of the reduction of X_{0}(M)_{Q} over Z[\frac{1}{m}]. The fact that X_{0}(M)_{Q} has stable reduction over Z[\frac{1}{m}] is crucial. It seems [14] [10] that in a letter to Mestre (Febr. 1985), Raynaud has stated general results implying that, for 2 not dividing m, one has v_{2}(c) = 0 or 1, where v_{2}(c) is the valuation of c at 2. In this article we will be concerned with the primes p dividing m. First we prove the following proposition.

Proposition 2 The number c is an integer.

Proof. Let X_{0}(M) be the compactified coarse moduli scheme \overline{M}([Γ_{0}(M)]) (cf. [6] (8.6)). Let X_{0}(M)^{0} be the open part of X_{0}(M) where the projection to Spec(Z) is smooth. Let E be the Néron model of E over Z. By the Néron property ϕ extends to ϕ : X_{0}(M)^{0} ⟶ E. It follows immediately from [6] Thm.8.11.10 that the formal completion of X_{0}(M) along the (unramified) cusp ∞ is Spf(Z[[q]]). We see that ∞ ∈ X_{0}(M)^{0}(Z) and that ϕ^{∗}ω_{Z} is a differential form (without poles!) in a neighborhood of ∞. If c would not be integral, then ϕ^{∗}ω_{Z} would have poles along the fibers of X_{0}(M)^{0} in characteristics dividing the denominator of c.

The analogous result for parametrizations involving the modular curve X_{1}(M) is proved in the paper [14] by Stevens. In that paper Stevens studies parametrizations of Weil curves by the modular curves X_{1}(M). He conjectures (see [14] Conj. I) that every Weil curve E (not just the strong ones) admits a parametrization by X_{1}(M) (where M is the conductor of E) with Manin constant equal to 1. He shows this conjecture to be compatible with quadratic twists over fields that are unramified at the primes where E has additive reduction. He also shows, using results of Rubin on special values of L-functions, that a weak form of his conjecture holds for certain elliptic curves with complex multiplication.

Using the classification by Mazur [7] of the rational torsion points of elliptic curves over Q, one can easily show that up to the primes 2, 3, 5, and 7, Stevens’s version of Manin’s conjecture is equivalent to Manin’s conjecture. For this one remarks that the difference between parametrizations by X_{1}(M)_{Q} and X_{0}(M)_{Q} lies in the so-called Shimura subgroup of J_{0}(M)_{Q}, which is the Cartier dual of a constant group. It follows that the Manin constant c of a strong Weil curve E is not divisible by primes p > 7 where E has reduction type I_{0}^{∗} or I_{ν}^{∗} (we use Kodaira symbols to denote the possible reduction types of elliptic curves, see [15]).

The aim of this article is to attack Manin’s conjecture for the remaining reduction types, these are: II, III, IV, IV^{∗}, III^{∗} and II^{∗}. Our method to do this is in principle the same as Mazur’s [7]: to study the valuation of c at a prime p by reducing ϕ modulo p. As remarked above, it is crucial to have stable models for X_{0}(M) and E at p. This makes it necessary not to work with the model of X_{0}(M) over Z as constructed in [6], but over a suitable extension of Q over which X_{0}(M) and E acquire stable reduction. For a number of reasons, we suppose that p > 7. It has been proved by Carayol [2], that for modular elliptic curves the level and the conductor coincide. The conductors of the curves that we want to study are of the form p^{2}N, where p does not divide N. A stable model of X_{0}(p^{2}N) at p is constructed in [4]. The application of these stable models to the study of Manin constants is given in [5] 4.6. The result that we state here is slightly stronger than [5] Thm.4.6.3.

Theorem 3 Let ϕ : X_{0}(M)_{Q} → E be a strong modular parametrization, let c be its Manin constant and let p > 7 be a prime. Then p does not divide c, except possibly when E has potentially ordinary reduction at p of type II, III or IV. In that case, p divides c at most once.

Of course, the cases where E has reduction type I_{0}, I_{ν}, I_{0}^{∗} or I_{ν}^{∗} are already proved by Mazur and Stevens. In the proof of the theorem for curves with complex multiplication we rely heavily on the work of Stevens. Combining our results with those of Stevens on complex multiplication curves, one can even show that elliptic curves over Q with complex multiplication by Q(\sqrt{−p}), p > 3, have Manin constants (in the original sense as well as in the sense of Stevens) equal to 1 or 2. A lot of details of the proof of the theorem above are not given here; for them we refer to [5]. Finally I want to thank Ling San for giving his comments on a preliminary version of this article.

2 Stable models

The standard method to try to obtain a stable model of a curve over the spectrum S of a discrete valuation ring is as follows. One starts with a regular model X over S. After blowing up repeatedly in closed points of the special fibre one may assume X is a regular model in which the special fibre is a divisor with normal crossings. Now let n be the least common multiple of the multiplicities in this divisor and let π_{0} be a uniformizer on S. Let T be S[π], with π^{n} = π_{0}, thus we have T → S totally ramified of degree n. Let X̃_{T} be the normalization of the pullback X_{T} of X to T. Let t be the closed point of T. If n is invertible on S, then by easy explicit computations with the complete local rings of X, one can show that X̃_{T} is a stable model in the sense that the only singularities of X_{t} are ordinary double points.

Now suppose that n is invertible on S and that O_{S}(S) contains μ_{n}, the n-th roots of unity. Then μ_{n} acts on T by:

∀ζ ∈ μ_{n} : ζ^{#} : π ↦ ζπ,

where ζ^{#} is the automorphism of O_{T} corresponding to the automorphism ζ of T. The quotient for this action is the morphism T → S. We let μ_{n} act on X_{T} = X ×_{S} T via its action on T. It follows that μ_{n} also acts on its normalization X̃_{T}. Since the morphism X̃_{T} → T is equivariant and t is fixed (i.e., μ_{n} acts trivially on k(t)), μ_{n} acts k(t)-linearly on the special fibre X̃_{T}_{,t}. The μ_{n}-action on the complete local rings of X̃_{T}_{,t} is easy to compute.

We will now apply the method described above to the modular curve X_{0}(p^{2}N). Let p > 3 be a prime, and let N be a positive integer that is not divisible by p. We start with the model over Z as contructed by Katz and Mazur in [6] 8.6. In their terminology, we let X_{0}(p^{2}N) be the coarse compactified moduli scheme over Z associated to the moduli problem Γ_{0}(p^{2}N).

Figure 1: The stable reduction of X_{0}(p^{2}N) at p.

[figure: ∞; (2, 0); (0, 2); Igusa curves; horizontal; horizontal; horizontal]

Let S be the spectrum of the ring of Witt vectors of \overline{F}_{p}, and let X_{0}(p^{2}N)_{S} be the pullback of X_{0}(p^{2}N) to S. Then X_{0}(p^{2}N)_{S} is a curve over S with a smooth geometrically irreducible generic fibre. The two-dimensional scheme X_{0}(p^{2}N) is regular, except for some possible quotient singularities. There are two ways to get rid of these singularities. One way is to work with the minimal resolution of X_{0}(p^{2}N) (see [4]). The other way is to introduce a suitable additional level structure (e.g., [Γ(3)^{can}]); then to apply the method described above to the associated moduli scheme, and to obtain the stable model of X_{0}(p^{2}N) as a quotient. The details of this computation can be found in [4]. It turns out that X_{0}(p^{2}N)_{S} acquires stable reduction over T = S[π], with p = π^{n} and n = \frac{p^{2}−1}{2}. We denote the stable model by X̃_{T}. The details of the computation of the action of μ_{n} on X̃_{T}_{,t} can be found in [5]. The description that one finds of X̃_{T}_{,t} together with its μ_{n}-action is as follows. Figure 1 contains a picture of X̃_{T}_{,t}.

The outer two vertical components are the (2,0) and (0,2) components (in the terminology of [6]) from X_{0}(p^{2}N)_{s}. These curves are both isomorphic to X_{0}(N)_{s}, where s denotes the closed point of S. The two central vertical components are isomorphic to the curve \overline{M}([Γ_{0}(N)], [Ig(p)/\{±1\}]) (see [6] 12 for the details of the Igusa moduli problem). The supersingular points on the central vertical components correspond bijectively to the supersingular points on X_{0}(N)_{s}. The four vertical components are glued to the horizontal components at their supersingular points. The horizontal component corresponding to a supersingular point x of X_{0}(N)_{s} is the smooth complete hyperelliptic curve given by the equation:

y^{2} = x^{p+1} + 1,

if the automorphism group of x is {±1} (by this we mean that the pair (E, G), corresponding to x via the moduli interpretation, has automorphism group {±1}). In general, the horizontal component at x is the quotient of this curve by some action of Aut_{s}(x). Each horizontal component is glued to the outer two vertical components at the points with x = ∞, y = ∞. At the points x = 0, y = ±1 it is glued to the central vertical components.

The inertia group μ_{n} preserves the horizontal components; the action on each horizontal component is given by:

∀ζ ∈ μ_{n} : ζ^{#} : \{ \begin{matrix}x & ↦ & ζ^{− \frac{p−1}{2}} x \\ y & ↦ & ζ^{\frac{n}{2}} . \end{matrix}

The μ_{n}-action on the vertical components is as follows. The action is trivial on the outer two components. The two Igusa curves are interchanged. The stabilizer of either of them, μ\frac{n}{2} , acts on the Igusa curve via its action on the moduli problem [Ig(p)/{±1}].

From this description, it is easy to make a list of all the points in X̃_{T}_{,t} that are stabilized by μ\frac{n}{2} , together with the μ\frac{n}{2} -actions on the cotangent spaces at those points. This list is very useful in the study of the differential forms on X̃_{T}_{,t}.

Since we want to study modular parametrizations of elliptic curves by looking at their stable reductions, we have to study the stable reduction of elliptic curves too. This is done by the same method that we used for the modular curves. The results of these computations are summarized in the following proposition.

Proposition 4 Let S be the spectrum of a complete discrete valuation ring, with algebraically closed residue field not of characteristic 2 or 3. Let E be an elliptic curve over the generic point of S. We attach a number n and a character χ_{ω} : μ_{n} → μ_{n} to E according to its reduction type as follows (χ : μ_{n} → μ_{n} denotes the identity):

\begin{matrix}type & I_{0} & I_{ν} & II & III & IV & I_{0}^{∗} & I_{ν}^{∗} & IV^{∗} & III^{∗} & II^{∗} \\ n & 1 & 1 & 6 & 4 & 3 & 2 & 2 & 3 & 4 & 6 \\ χ_{ω} & χ & χ & χ & χ & χ & χ & χ & χ^{−1} & χ^{−1} & χ^{−1} \end{matrix}

Let π_{0} be a uniformizer on S. Then E acquires stable reduction over T := S[π], where π^{n} = π_{0}. The Galois group of T over S is identified with μ_{n} by: ζ^{#}π = ζπ for all ζ ∈ μ_{n}. Let t be the closed point of T and Ẽ_{T} the Néron model of E_{T}. The the action of μ_{n} on the tangent space at 0 of Ẽ_{T}_{,t} is given by χ_{ω}.

3 Modular parametrizations

Let ϕ : X_{0}(p^{2}N) → E be a strong modular parametrization where p > 3 is a prime that does not divide the positive integer N. We suppose that E has potentially good reduction at p (this means that we exclude the reduction type I_{ν}^{∗}). Let S := Spec W (\overline{F}_{p}), n := \frac{p^{2}−1}{2}, U := S[π], where π^{n} = p. In the previous section we have seen that both X_{0}(p^{2}N) and E acquire stable reduction over U. We denote these stable models by X̃_{U} and Ẽ_{U}, respectively. The parametrization ϕ gives, by pullback, a morphism (still denoted ϕ) between the generic fibres of X̃_{U} and Ẽ_{U}.

Proposition 5 This ϕ extends to a morphism ϕ : X̃_{U} → Ẽ_{U}.

Proof. This follows from the facts that Ẽ_{U} is an abelian scheme and X̃_{U} a stable curve.

The fact that the map ϕ : X̃_{U}_{,u} → Ẽ_{U}_{,u} (where u denotes the closed point of U) is μ_{n}-equivariant already gives a lot of information. For example, let us consider the orders of the μ_{n}-action on the various components of X̃_{U}_{,u}: on a horizontal component, this order divides p + 1; on the two outer vertical components it is 1 and on the two central vertical components together, it is p − 1. Let d be the order of the μ_{n}-action on Ẽ_{U}_{,u} (see the table in proposition 4). It follows that all the components of X̃_{U}_{,u} on which the order of the μ_{n}-action is not divisible by d will be contracted by ϕ. This leads us to the following proposition.

Proposition 6 The morphism ϕ : X̃_{U}_{,u} → Ẽ_{U}_{,u} is constant on the two outer vertical components. On the other components we have:

type

p hor. components central vert. comps.

I_{0}^{∗}  contracted
II, IV, IV^{∗}, II^{∗} p ≡ 1(3) contracted
p ≡ −1(3)  contracted
III, III^{∗}  p ≡ 1(4) contracted
p ≡ −1(4)  contracted

Remark. In order to prove that each horizontal component C is contracted in the case where E has reduction type I_{0}^{∗}, one has to use that the quotient of C by μ\frac{n}{2} has genus 0.

The remainder of this section will be concerned with the degree of inseparability of ϕ : X̃_{U}_{,u} → Ẽ_{U}_{,u}. The next proposition shows that there will always be at least one component of X̃_{U}_{,u} on which ϕ is either separable or the composition of a separable morphism with the Frobenius endomorphism of Ẽ_{U}_{,u}. Let J̃_{U}^{0} denote the connected component of the Néron model of J_{0}(p^{2}N) over U. Since X̃_{U} is a stable curve, J̃_{U}^{0} is the Picard scheme Pic_{_{X_{U}} /U}^{0} (see [12] 6.1.4, 6.1.6 and 8.2.1).

Proposition 7 Let G be the kernel of ϕ^{∗} : Ẽ_{U}_{,u} → J̃_{U} _{,u}^{0}. If p > 7, then either G = 0 or G = ker(V : Ẽ_{U}_{,u} → Ẽ_{U}_{,u}), the kernel of the Verschiebung.

Proof. Since ϕ : X_{0}(p^{2}N)_{Q} → E is strong, we have that ϕ^{∗} : Ẽ_{U} → J̃_{U}^{0} is injective on the generic fibre. It follows from the semistability of Ẽ_{U} and J̃_{U}^{0} that G is finite and consists of p-power torsion. Now we want to apply a result of Raynaud on the specialization of finite group schemes [13] Corollaire 3.3.6.1 (see also [7] Proposition 1.1) to study the kernel of ϕ^{∗} : Ẽ_{U} → J̃_{U}^{0}. We can not use this result of Raynaud directly, because the absolute ramification index of U is n = \frac{p^{2}−1}{2} ≥ p − 1.

Therefore let T → S be the minimal base change such that E acquires good reduction (see proposition 4, hence T → S is of degree 2, 3, 4 or 6). Taking the quotient of ϕ : X̃_{U} → Ẽ_{U} by the action of the subgroup Gal(U/T) of μ_{n} gives us a morphism ϕ : X̃_{T} → Ẽ_{T}. By Picard functoriality we get: ϕ^{∗} : Ẽ_{T} → J̃_{T}^{0}, where J̃_{T}^{0} is defined to be Pic_{_{X_{T}} /T}^{0} (Pic_{_{X_{T}} /T}^{0} is representable by a smooth group scheme by [12] 8.2.1, we do not know whether Pic_{_{X_{T}} /T}^{0} is the Néron model). Since the absolute ramification index of T is less than p − 1 (this explains the hypothesis “p > 7” in this proposition), we can now apply the result of Raynaud mentioned above to ϕ^{∗} : Ẽ_{T}[p] → H, where H is the image of Ẽ_{T}[p] in J̃_{T}^{0}. The conclusion is that ϕ^{∗} : Ẽ_{T} → J̃_{T}^{0} is a closed immersion.

By pullback to U we get a commutative diagram:

Ẽ_{T}_{,U} ↪ J̃_{T} _{,U}^{0}
↓  ↓
Ẽ_{U} → J̃_{U}^{0}

where Ẽ_{T}_{,U} → Ẽ_{U} is an isomorphism (since E has already good reduction over T) and where J̃_{T} _{,U}^{0} → J̃_{U}^{0} exists because of the Néron property of J̃_{U}^{0}. It follows that:

G = Ẽ_{T}_{,t} ∩ ker(J̃_{T} _{,t}^{0} → J̃_{U} _{,u}^{0}).

Using our knowledge of X̃_{T} and X̃_{U} it is not hard to show that ker(J̃_{T} _{,t}^{0} →

J̃_{U} _{,u}^{0}) is purely additive: it is isomorphic to a direct sum of G_{a,t}’s. The reason that this is true is that the multiplicities of the irreducible components in X̃_{T}_{,t} are less than p (from this it follows that there is an exponential sequence).

It remains to find out what the intersection of Ẽ_{T}_{,t} with a direct sum of additive groups can be. If Ẽ_{T}_{,t} is an ordinary elliptic curve over t, then its p^{m}-torsion is isomorphic to μ_{p^{m},t} × Z/p^{m} and it follows that the intersection is at most Z/p, the kernel of V. If Ẽ_{T}_{,t} is a supersingular elliptic curve, then its p-torsion subgroup is a group scheme called M_{2} (see [11](15.5)). This M_{2} cannot be embedded in a direct sum of additive groups, hence the intersection is at most the Frobenius kernel (which is isomorphic to α_{p,t}). This completes the proof of the proposition.

4 Manin constants

We are now ready to apply the results of the preceding sections to the study of Manin constants. Let ϕ : X_{0}(p^{2}N) → E be a strong modular parametrization, where p > 7 is a prime that does not divide the positive integer N. We suppose that E has potentially good reduction at p, not of type I_{0}^{∗}. Let S, n, U and T be as in the preceding section: S = SpecW (\overline{F}_{p}), n = \frac{p^{2}−1}{2}, U = S[π] with π^{n} = p and T = S[π^{\frac{n}{d}}], where d is as in the table in proposition 4 (T → S is the minimal base change after which E acquires good reduction). Let ϕ : X̃_{U} → Ẽ_{U} be the strong modular parametrization extended to the stable models over U.

Let ω_{S} be a Néron differential of E over S. Then ϕ^{∗}ω_{S} is a meromorphic differential form on X_{0}(p^{2}N)_{S}. The valuation at p of the Manin constant of E is the multiplicity of the (2, 0)-component (this is the component on which the cusp ∞ lies) in the divisor of ϕ^{∗}ω_{S}. By explicit computations using minimal Weierstrass equations, it is easy to compute a Néron differential ω_{U} of E over U. It turns out that ω_{U} = π^{a}ω_{S}, where a is given in the following table:

\begin{matrix}type & II & III & IV & IV^{∗} & III^{∗} & II^{∗} \\ a & \frac{1}{6} · n & \frac{1}{4} · n & \frac{1}{3} · n & \frac{2}{3} · n & \frac{3}{4} · n & \frac{5}{6} · n \end{matrix}

Since the parametrization ϕ can be extended over the stable models (see proposition 5), ϕ^{∗}ω_{U} is a differential form on X̃_{U}. In particular, ϕ^{∗}ω_{U} is a section of the invertible sheaf of regular differentials on X̃_{U}_{,u}. We now have the following formula for the valuation v_{p}(c) at p of the Manin constant c of E:

v_{p}(c) = \frac{1}{n}(v_{C}(ϕ^{∗}ω_{U}) − a),

where v_{C}(ϕ^{∗}ω_{U}) denotes the valuation of ϕ^{∗}ω_{U} (as a regular differential) along the (2, 0)-component C of X̃_{U}_{,u}.

Proposition 8 If ϕ : X̃_{U}_{,u} → Ẽ_{U}_{,u} is separable on some component of X̃_{U}_{,u}, then p does not divide c.

Sketch of proof. According to the formula for v_{p}(c) above, we have to show that v_{C}(ϕ^{∗}ω_{U}) < n + a. By hypothesis, ϕ^{∗}ω_{U} gives a non-zero differential form on some component D of X̃_{U}_{,u}. Let D′ be another component of X̃_{U}_{,u} meeting D in a point x. Then v_{D′}(ϕ^{∗}ω_{U}) can be estimated in terms of v_{x}(ϕ^{∗}ω_{U}|_{D}), the order of vanishing at x of the restriction of ϕ^{∗}ω_{U} to D. One then proceeds in the same way with (π^{−r}ϕ^{∗}ω_{U})|_{D′}, where r = v_{D′}(ϕ^{∗}ω_{U}). It is clear from figure 1 that, after at most two steps, one arrives at the component C. It turns out that the estimates, if done properly, are sharp enough.

Proposition 9 Under the hypotheses of this section, one has that v_{p}(c) ≤ 1.

Sketch of proof. The principle is the same as in the proof of the preceding proposition, but now one has to start at a component D on which ϕ has degree of inseparability at most 1. An easy computation shows that, in that case, v_{D}(ϕ^{∗}ω_{U}) ≤ n.

In order to get a stronger result, we will now consider E together with E′, the quadratic twist of E over Q(\sqrt{(\frac{−1}{p})p}). This twist acts on the reduction types at p as follows:

original type II III IV IV^{∗} III^{∗} II^{∗}
type of twist IV^{∗} III^{∗} III^{∗} II III IV

It is known that E′ is again a modular elliptic curve, and it is easy to see that the conductor of E′ is also p^{2}N. Let Ẽ be the strong curve in the isogeny class of E′, let ϕ̃ be its strong modular parametrization and let c̃ be its Manin constant. It follows that we have a commutative diagram:

X_{0}(p^{2}N)_{Q} ⟶^{ϕ̃}  Ẽ
ϕ′ ↘  ↙ α
E′

with α a cyclic isogeny. Let f \frac{dq}{q} and f̃\frac{dq}{q} be the normalized newforms of E and Ẽ. Over C, the modular parametrization ϕ is given as follows:

ϕ : X_{0}(M)(C) → C/Λ = E(C) : P ↦ ∫_{∞}^{P} f \frac{dq}{q} ,

where Λ is the period lattice of f \frac{dq}{q}:

Λ = \{∫_{γ} f \frac{dq}{q} ; γ ∈ H_{1}(X_{0}(M)(C), Z)\} .

This gives the following formula for deg(ϕ), the degree of ϕ:

deg(ϕ) = ∥f∥^{2}/vol(C/Λ), where ∥f∥^{2} = \frac{i}{2} ∫_{X_{0}(M)(C)} |f|^{2} \frac{dq ∧ dq̄}{qq̄}.

By a standard computation, see [17], one sees that

δΛ ⊂ Λ̃ ⊂ δ^{−1}Λ,

where Λ̃ is the period lattice of f̃\frac{dq}{q} and δ is the Gauss sum \sqrt{(\frac{−1}{p})p}. Since Λ̃ is proportional to Λ, there are exactly three possibilities: Λ̃ = δΛ, Λ̃ = Λ or Λ̃ = δ^{−1}Λ.

In the second case, E has complex multiplication. As stated in the introduction, a combination of our results and those of Stevens [14] implies that in this case p does not divide c. Now we assume that E does not have complex multiplication. The formula:

∥f^{2}∥ = C · [PSL_{2}(Z) : Γ_{0}(p^{2}N)] · Res_{s=2} ∑_{n=1}^{∞} \frac{|a_{n}|^{2}}{n^{s}}

(where C is some fixed constant) in [17] implies that ∥f∥^{2} = ∥f̃∥^{2}. It now follows that deg(ϕ̃) equals p^{−1} deg(ϕ) or p deg(ϕ) in the first or third case, respectively. By comparing the stable reductions of ϕ and ϕ̃, one can see that either ϕ or ϕ̃ is separable on some component of X̃_{U}_{,u}. More precisely, we are in one of the two cases:

1. Λ̃ = δΛ, deg(ϕ̃) = p^{−1} deg(ϕ), ϕ is inseparable and ϕ̃ is separable.

2. Λ̃ = δ^{−1}Λ, deg(ϕ̃) = p deg(ϕ), ϕ is separable and ϕ̃ is inseparable.

After replacing E, if necessary, by Ẽ, we may assume that E has reduction type II, III or IV at p. Let ω be a Néron differential of E over Z, then ω′ = δ^{−1}ω is a Néron differential on E′. Let c′ be the Manin constant associated to ϕ′: ϕ^{′∗}ω′ = c′f̃\frac{dq}{q}. Then we have the formulas:

\frac{deg(ϕ)}{c^{2}} = \frac{∥f∥^{2}}{vol(E, ω)} \frac{deg(ϕ′)}{(c′)^{2}} = \frac{∥f̃∥^{2}}{vol(E′, ω′)}

where

vol(E, ω) = ∫_{E(C)} \frac{i}{2}ω ∧ \overline{ω} and vol(E′, ω′) = ∫_{E′(C)} \frac{i}{2}ω′ ∧ \overline{ω′}.

From these it follows directly that:

deg(ϕ̃) = p deg(ϕ)\frac{(c′)^{2}}{c^{2}} \frac{1}{deg(α)}.

Since we are either in case 1 or case 2, we see that deg(α) is a square. It follows from the classification of cyclic isogenies over Q [7] [16] that p does not divide deg(α). This implies that v_{p}(c̃) = v_{p}(c′). We arrive at:

v_{p}(deg ϕ̃) = v_{p}(deg ϕ) + 1 + 2v_{p}(\frac{c̃}{c}).

In case 2 above, it follows that v_{p}(c̃) = v_{p}(c) = 0. In case 1 above, v_{p}(c) = 1 and v_{p}(c̃) = 0. Of course, if case 1 occurs, then Manin’s conjecture is false.

If E has potentially supersingular reduction at p, then one can prove that only case 2 occurs. The reason for this is that then the μ_{n}-action on Ẽ_{U}_{,u} does not commute with the Frobenius endomorphism, giving a parity condition on the degree of inseparability of ϕ. If E has potentially ordinary reduction at p, then these two do commute, and we cannot prove that case 1 does not occur.

References

[1] B.J. Birch and H.P.F. Swinnerton-Dyer. Elliptic curves and modular functions. Modular functions of one variable IV, 2–32. Springer Lecture Notes in Mathematics 476 (1975).

[2] H. Carayol. Sur les représentations ℓ-adiques associées aux formes modulaires de Hilbert. Ann. scient. Éc. Norm. Sup., série 4, t.19, 409-468 (1986).

[3] P. Deligne and M. Rapoport. Les schémas de modules des courbes elliptiques. Modular functions of one variable II, 143–316. Springer Lecture Notes in Mathematics 349.

[4] S.J. Edixhoven. Minimal resolution and stable reduction of X_{0}(N). Preprint Utrecht November 1986, accepted for publication in the Annales de l’Institut Fourier.

[5] S.J. Edixhoven. Stable models of modular curves and applications. Thesis Utrecht June 1989, to be published.

[6] N.M. Katz and B. Mazur. Arithmetic moduli of elliptic curves. Annals of Mathematics Studies 108, Princeton University Press (1985).

[7] B. Mazur. Rational isogenies of prime degree. Invent. Math. 44, 129–162 (1978).

[8] B. Mazur. Courbes elliptiques et symboles modulaires. Séminaire Bourbaki juin 1972. Springer Lecture Notes in Mathematics 317.

[9] B. Mazur and H.P.F. Swinnerton-Dyer. Arithmetic of Weil curves. Invent. Math. 25, 1–16 (1974).

[10] J.-F. Mestre and J. Oesterlé. Letter to B. Gross, May 1985.

[11] F. Oort. Commutative group schemes. Springer Lecture Notes in Mathematics 15 (1966).

[12] M. Raynaud. Spécialisation du foncteur de Picard. Publications Mathématiques de l’I.H.E.S. 38 (1970).

[13] M. Raynaud. Schémas en groupes de type (p,...,p). Bull. Soc. Math. France 102, 241-280 (1974).

[14] G. Stevens. Stickelberger elements and modular parametrizations of elliptic curves. To be published.

[15] J. Tate. Algorithm for determining the type of a singular fibre in an elliptic pencil. In Modular Functions of One Variable IV, 33-52. Springer Lecture Notes in Mathematics 476 (1975).

[16] M. Kenku. On the number of Q-isomorphism classes of elliptic curves in each Q-isogeny class. J. of Number Theory 15, 199-202 (1982).

[17] D. Zagier. Modular parametrizations of elliptic curves. Canad. Math. Bull. 28(3), 372-384 (1985).

Bas Edixhoven Department of Mathematics University of California at Berkeley Berkeley, CA 94720 U.S.A.