TORSION POINTS ON X_{0}(N)

Robert Coleman, Bruce Kaskel and Kenneth A. Ribet

University of California, Berkeley

To professor Goro Shimura

1. Introduction

Let X be a complex algebraic curve of genus > 1. Choosing a base point x_{0} ∈ X, we view X as a subvariety of its Jacobian J = J(X) via the Albanese map which sends each x ∈ X to the class of the divisor (x) − (x_{0}). Let J_{tor} be the group of torsion points on J. The Manin-Mumford conjecture states that the intersection

S := X ∩ J_{tor}

is finite.

This conjecture was studied first by S. Lang [18], who showed that is implied by a statement about the action of Gal(\overline{K}/K) on J_{tor}, where K is a finitely generated subfield of C over which X and x_{0} are defined. Lang’s hypothesis states that the image of Gal(\overline{K}/K) in Aut J_{tor} contains an open subgroup of the homothety group Ẑ^{∗} ⊆ Aut J_{tor}. The statement remains unproved, despite partial results by Bogomolov [1] and Serre [32]. (See also [14], which may be regarded as a sequel to [18].)

Meanwhile, the Manin-Mumford conjecture was proved by M. Raynaud in a 1983 article [28]. Subsequently, a second proof of the conjecture was given by R. Coleman in [4]. Generalizations and strengthenings of the Manin-Mumford conjecture have been proposed in various quarters. For example, a recent article by E. Ullmo [34] studies points on X which map to points of J with small canonical height. (See also [24].)

This article concerns the intersection X ∩ J_{tor} in the special case where X is a modular curve. More specifically, we limit our attention to the curve X = X_{0}(N) where N is a prime number. The requirement that X have genus > 1 becomes the condition N ≥ 23, which we now impose. We take x_{0} to be the standard “cusp at ∞,” which we call C_{∞}.

A general theorem of Manin-Drinfeld implies that S contains the set of cusps on X. There are two cusps, the standard cusp C_{∞}, which maps to 0 in J_{0}(N), and a second cusp, C_{0}, which maps to a point on J_{0}(N) of order n := num \frac{N−1}{12}. (See [21, Ch. II, §11] for a discussion of the group C generated by C_{0} on J_{0}(N); this group is called the cuspidal subgroup of J_{0}(N). Theorem 1.2 of [21, Ch. III] states that C is the torsion subgroup of the Mordell-Weil group of J.) The inclusion {C_{0}, C_{∞}} ⊆ S is the first piece of information about S at our disposal. Because of it, we describe S as the “cuspidal torsion packet” on X. (As the reader may recall, Coleman uses the term “torsion packet” to refer to X ∩ J_{tor} in [2, 4].)

This work was partially supported by grants from the National Science Foundation. The authors thank M. Baker for helpful feedback to preliminary drafts of our manuscript and for compiling the list of errata to [4] which appears at the end of this article.

A second constraint on S arises from the fixed points of the hyperelliptic involution of X if X = X_{0}(N) is a hyperelliptic curve:

Proposition 1.1. Suppose that X is hyperelliptic, and let P be a fixed point of the hyperelliptic involution of X. Then we have P ∈ S if and only if N is different from 37. In this latter case, 2P belongs to the cuspidal subgroup of J.

Proof. Let h be the hyperelliptic involution. Since h operates on J = J_{0}(N) as −1, the images in J of the divisors (P) − (C_{∞}) and (h(C_{∞})) − (P) are equal. Adding (P)−(C_{∞}), we find that 2((P)−(C_{∞})) and (h(C_{∞}))−(C_{∞}) are equal on J. Hence we have P ∈ S if and only if the point h(C_{∞}) has finite order in J. A theorem of Ogg states that X is hyperelliptic if and only if N is either 37 or one of the seven primes 23, 29, 31, 41, 47, 59 and 71. (See [26, Th. 2] and the article [27].) For the last seven primes on the list, h is the Atkin-Lehner involution w = w_{N} of X and hence h(C_{∞}) is the cusp C_{0}. In particular, h(C_{∞}) has finite order on J, so that P belongs to S. Further, the point 2P on J is the class of the divisor (C_{0}) − (C_{∞}), which is a generator of the cuspidal group C. For the exceptional case N = 37, Mazur and Swinnerton-Dyer [23, §5] have observed that h(C_{∞}) has infinite order on J. Hence P is not an element of S. ■

Let S_{0} be the set which is defined as follows: If X is not hyperelliptic or if N = 37, S_{0} = {C_{0}, C_{∞}}; if X is hyperelliptic and N ≠ 37, S_{0} is the set consisting of the two cusps of X and the hyperelliptic fixed points on X. We then have in all cases

S_{0} ⊆ S.

It seems plausible to us that this inclusion is an equality, i.e., that one has

S =^{?} S_{0}

for all N ≥ 23^{∗}. Despite the somewhat ad hoc definition of S_{0}, the equality S = S_{0} has a simple restatement:

Proposition 1.2. We have S = S_{0} if and only if 2P is a rational point of J for all P ∈ S.

Proof. Assume that S = S_{0}, and let P be an element of S. If P is one of the two cusps of X_{0}(N), then P is rational. If P is not a cusp, then P is a hyperelliptic branch point of X_{0}(N) and N is different from 37. We have seen in this case that 2P lies in the cuspidal group C; in particular, 2P is rational. Assume now that 2P is rational for each P ∈ S. Take P ∈ S and suppose first that P is not itself a rational point. Then there is a g ∈ Gal(\overline{Q}/Q) such that gP ≠ P. The difference gP − P, computed on J, is then a 2-division point because 2P is rational. This means that the non-zero divisor 2(gP) − 2(P) is the divisor of a rational function

∗ Through the efforts of Matt Baker we now know this equality is true for N a prime number in the set {23, 29, 37, 41, 47, 59}. Note that X_{0}(N) is hyperelliptic in each of these cases. on X_{0}(N). This function defines a degree-two map X_{0}(N) → P^{1} for which P is a ramified point. We find then that P is a hyperelliptic branch point of X_{0}(N). Since P lies in S, we must have N ≠ 37; thus P is a point in S_{0}. The second possibility is that P is a rational point of X_{0}(N). In that case, P lies in the intersection C ∩X_{0}(N), in view of Mazur’s theorem [21, p. 33] that C is the torsion subgroup of the Mordell-Weil group of J_{0}(N). We need to know that this intersection consists of the two cuspidal points C_{0} and C_{∞} of J. One way to proceed is to cite the wellknown theorem of Mazur [22, Th. 7.1] which states for all but four prime numbers N ≥ 23 that X_{0}(N)(Q) = {C_{0}, C_{∞}}. The four exceptional primes are N = 37, 43, 67, and 163. For the first of these primes, the equality X_{0}(N) ∩ C = {C_{0}, C_{∞}} is proved by Mazur and Swinnerton-Dyer [23, §5, Prop. 2]. For the primes 43, 67, and 163, there is a unique non-cuspidal point of X_{0}(N), which arises from elliptic curves with complex multiplication by Q(\sqrt{N}). By uniqueness, this point is fixed by the Atkin-Lehner involution w of X_{0}(N). Since w operates on C by multiplication by −1 [21, p. 99], and since n = num \frac{N−1}{12} is odd when N = 43, 67 and 163, the desired statement follows. ■

In this article, we present an assortment of results about S. These include theorems proved by the p-adic techniques of the first author [2, 4] and an inequality (proved in §4) which is presented “in the spirit of Lang [18].”

To illustrate the scope of our results, we will state a theorem which is valid for all N ≥ 23 and a complement which concerns the special case N = 37. Before presenting these results, we introduce some notation: First, we let T be the ring of endomorphisms of J_{0}(N), i.e., the ring of Hecke operators acting on the space of weight-two cusp forms on Γ_{0}(N). (See [21, Ch. II, Prop. 9.5].) Further, we write P_{p} for the “p-primary part” of P, whenever P is a torsion point on J and p is a prime number. Thus P is the sum of its p-primary parts (as p ranges over the set of primes) and P_{p} has p-power order. Using [21, Ch. III, Th. 1.2], we may restate Proposition 1.2 as follows: the equality S = S_{0} holds if and only if 2P_{p} lies in the cuspidal group C for all primes p and all P ∈ S.

Theorem 1.3. Let P be an element of S, and let p ≠ 2, 3 be a prime for which P_{p} does not belong to the cuspidal group C. Then at least one of the following holds:

(i) p = N; (ii) p satisfies 5 ≤ p < 2g, X_{0}(N) does not have ordinary reduction at p, and p is ramified in T in the sense that T/pT is not a product of fields. Further, suppose that p = N and that N is not ramified in T. Then there is some ℓ ≠ N such that P_{ℓ} ∉ C.

Theorem 1.4. The intersection X_{0}(37) ∩ J_{0}(37)_{tor} consists of the two cusps C_{0} and C_{∞}.

We prove the latter theorem by combining results such as Theorem 1.3 with some facts which are specific to the case and N = 37. In particular, we lean heavily on results which were proved by B. Kaskel in [15]; these involve a detailed description of the image of Gal(\overline{Q}/Q) in Aut J_{0}(37)_{tor}. To obtain this description, Kaskel employs many of the techniques pioneered by Serre [31] and Lang-Trotter [20] in the case where X_{0}(N) has genus one. To the best of our knowledge, no analogous description is available in general for J_{0}(N); it would certainly be of considerable interest to find such a description.

It strikes us that the general results at our command, when augmented by the description of [15], seem to represent overkill: they allow us to prove Theorem 1.4 in several, closely related, ways. Because of our interest in the general case, we will present three proofs of the theorem in §6.

2. Ramification

Let P be a torsion point on J = J_{0}(N). We wish to study the set of primes which ramify in the extension Q(P)/Q. First, we record the following statement, which follows directly from the main results of [4] and [2, Th. 5.5].

Theorem 2.1. Suppose that P lies in S. Then the extension Q(P)/Q can be ramified only at the bad prime N, the two very small primes 2 and 3, and the primes p satisfying 5 ≤ p < 2g. Further, if Q(P)/Q is ramified at p = 3, then the ramification at 3 is tame. Finally, if Q(P)/Q is ramified at p and p satisfies 5 ≤ p < 2g, then X does not have ordinary reduction at p.

For each prime number p, we again write P_{p} for the “p-primary component of P.” Further, we again use the symbol T to denote the ring of endomorphisms of J_{0}(N), i.e., the ring of Hecke operators acting on the space of weight-two cusp forms on Γ_{0}(N). Finally, we recall that a prime number p is said to be Eisenstein if it divides the order of the cuspidal subgroup C of J.

Theorem 2.2. Suppose that the field extension Q(P)/Q is unramified at p. Then either p = 2, or P_{p} belongs to the cuspidal group C.

Proof. Assume, to the contrary, that p is an odd prime and that P_{p} is not contained in the cuspidal group C. Let M be the T[Gal(\overline{Q}/Q)]-submodule of J_{0}(N)(\overline{Q})_{tor} which is generated by P_{p}. Then M is a finite abelian group of p-power order which is not contained in C. As a module for the Galois group Gal(\overline{Q}/Q), M is unramified at p.

As explained in [21, Ch. II, §14], the Jordan-Hölder constituents of M can only be of the following three forms: (i) the constant group Z/pZ; (ii) the group μ_{p}; (iii) an irreducible module of the form J_{0}(N)[𝔪], where 𝔪 is a non-Eisenstein maximal ideal of T with residue characteristic p. Types (i) and (ii) occur only if p is an Eisenstein prime, i.e., if p divides the integer n introduced above. However, since p is odd, the modules of types (ii) and (iii) are visibly ramified at p: note that the determinant of J_{0}(N)[𝔪] is the mod p cyclotomic character. In view of our supposition that M is unramified at p, M is a successive extension of copies of Z/pZ, so that p is a divisor of n. In particular p and N are distinct.

We view M as an extension of Q := M/M_{C} by M_{C}, where M_{C} is the intersection M ∩C. Since M is not contained in C, Q is non-zero. Let Q′ ≅ Z/pZ be a minimal sub-module of Q, and let M′ be the inverse image of Q′ in M. Thus we have a tautological exact sequence

0 → M_{C} → M′ → Q′ → 0

in which the groups M_{C} and Q′ are trivial Gal(\overline{Q}/Q)-modules.

We claim that M′ is unramified at N. This claim follows from an argument due to Mazur [21, Ch. II, Lemma 16.5], which we shall now recapitulate. Let σ be an element of an inertia group for the prime N in Gal(\overline{Q}/Q), and let m be an element of M′. Let p^{r} be the order of m. Grothendieck’s theory of semistable reduction (together with the Deligne-Rapoport theorem [11] that J_{0}(N) has multiplicative reduction at N) shows that σm − m belongs to the “toric part” of J_{0}(N)[p^{r}](\overline{Q}_{N}).

The displayed exact sequence forces σm − m to be an element of C. However, reduction mod N induces an isomorphism between C and the group of connected components of the Néron model for J_{0}(N) in characteristic N; this yields in particular that the intersection of C and the toric part of J_{0}(N)[p^{r}](\overline{Q}_{N}) is zero. Hence σm − m = 0, which proves what was claimed.

Now M′ can be ramified a priori only at p and at N. We have just seen that M′ is unramified at N, and the hypothesis to the theorem implies that it is unramified at p. It is therefore everywhere unramified, so that the action of Gal(\overline{Q}/Q) on M′ must be trivial. In other words, we have M′ ⊆ J_{0}(N)(Q)_{tor}. On the other hand, one of the main results of [21] asserts that C = J_{0}(N)(Q)_{tor}. We thus have M′ = M_{C} and Q′ = 0, which is a contradiction. ■

Corollary 2.3. Let P be an element of S, and assume that P_{p} does not belong to the cuspidal group C. Then at least one of the following holds: (i) p = 2 or p = 3;

(i) p = N; (ii) p satisfies 5 ≤ p < 2g and X does not have ordinary reduction at p.

This corollary results directly from the two results above. We have stated it in order to record our progress toward proving Theorem 1.3.

Theorem 2.4. Suppose that P ∈ S and that the order of P is even but not divisible by N. Assume that N is not congruent to 1 mod 8. Then P belongs to S_{0}.

Proof. It will be convenient to write the order of P as 2d. Let R be the 2-division point dP. We claim that the extension Q(R)/Q is ramified at the prime N.

For this, let M be the T[Gal(\overline{Q}/Q)]-submodule of J_{0}(N)(\overline{Q})_{tor} which is generated by R; further, let V be a minimal submodule of M. Thus V is a constituent of J_{0}(N)[2] which belongs to a maximal ideal 𝔪 of T of residue characteristic 2. (See [21, Ch. II, §14] for the terminology.) In view of the hypothesis N ≢ 1 mod 8, 𝔪 cannot be an Eisenstein ideal; this means that V is an irreducible 2-dimensional representation of Gal(\overline{Q}/Q) over T/𝔪. By a theorem of Tate [33], the representation V is ramified at N. This proves the claim.

Equivalently: there is an inertia group I for N in Gal(\overline{Q}/Q) and a σ ∈ I such that σR ≠ R. It follows that σR − R has order 2.

Consider the action of σ on J_{0}(N)[2d]. By Grothendieck’s semistable reduction theorem (and the results of Deligne-Rapoport [11]), σ acts as the sum 1 + A, where 1 represents the identity operator and A an endomorphism satisfying A^{2} = 0. Using the binomial theorem, we find

σ^{d}(P) − P = (1 + dA)P − P = AdP = σ(R) − R.

Hence if g = σ^{d}, then g is an element of Gal(\overline{Q}/Q) such that gP − P has order 2. As we have seen in the proof of Proposition 1.2, it follows that P lies in S_{0}. ■

3. Ramified torsion points revisited

As we indicated above, this article concludes with a list of errata to [4]. In particular, we point out that the proofs of cases (ii) and (iii) of [4, Theorem 20] are incorrect. These cases concern the primes p = 2 and p = 3. Although those statements are not needed for the main theorem of [4], we hope that appropriately modified versions of them may be quite useful. Accordingly, we provide in this section some results for the primes p = 2 and p = 3. In addition, we will complete the proof of Theorem 1.3.

Fix a prime p, and let C_{p} be the completion of an algebraic closure of Q_{p}. Let v be the valuation of C_{p} such that v(p) = 1. Let K^{ur} be the completion of the maximal unramified extension of Q_{p} and let K be a complete subfield of K^{ur}. Let R be the ring of integers of K and let k be the residue field of R. Also let R^{ur} be the ring of integers of K^{ur}, F the residue field of R^{ur} and σ the Frobenius automorphism of K^{ur}.

Let A be an Abelian scheme over R. Define a pairing

( , ) : Ω_{A/R^{ur}}^{1}(A) × A(F) → K^{ur}/pR^{ur}

as follows: Given ω ∈ Ω_{A/R^{ur}}^{1}(A) and P ∈ A(F), let P̃ be a lifting of P to A(K^{ur}) and set

(ω, P) = ∫ P̃ ; 0 ω mod pR^{ur}.

It is easy to check that (ω, P) is independent of the choice of P̃: when P = 0, ∫ P̃ ; 0 ω ∈ pR^{ur}. We note that ( , ) is a Galois-invariant bilinear form and that (ω, P) is trivial when the order of P is prime to p.

Remark 3.1. We can relate ( , ) to the Serre-Tate parameters when A is ordinary. Recall that there is a natural pairing

q: Ta_{p} A(F) × Ta_{p} Â(F) → 1 + pR.

(See Katz [17].) There is also a natural map Ta_{p} Â(F) → Ω_{A/R^{ur}}^{1}(A), α ↦ ω_{α}. Then one can show:

Proposition 3.2. Suppose α = (α_{n}) ∈ Ta_{p} Â(F) and β = (β_{n}) ∈ Ta_{p} A(F). Then

(ω_{α}, β_{n}) ≡ \frac{1}{p^{n}} log q(α, β) mod pR^{ur}.

We say that a quadruple (H, W, F, V) is an F -T -crystal over R if H is free R module of finite rank, W is a direct summand of H and F and V are σ and σ^{−1}-linear endomorphisms of H, respectively, such that

F V = V F = p, V H = W + pH.

(This was called an F-crystal in [4].) If S is a smooth complete scheme of finite type over R, the first de Rham cohomology of S over R has a natural structure of an F - T -crystal where H = H_{DR}^{1}(S/R) and W = H^{0}(S, Ω^{1}) and we call a sub-F -T -crystal of this a de Rham F -T -crystal on S.

Suppose (H, W) is a de Rham F -T -crystal on A. Let P be a point of A(F). For a coset B of pR^{ur} in C_{p} we set v(B) = min{v(x): x ∈ B}. Let

M_{P} (H) := 1 − min{v(ω, P): ω ∈ W }.

Let H be the first de Rham cohomology group of A over R thought of as an F -T crystal. Suppose P ∈ A(F). We have in general,

M_{pP} (H) + 1 ≥ M_{P} (H) ≥ M_{pP} (H)  (1)

and if M_{P} (H) ≥ 1

M_{pP} (H) + 1 = M_{P} (H).  (2)

It follows, from (1) and (2), that if M_{pP} (H) ≥ 1

M_{pP} (H) + 1 = M_{P} (H).  (3)

Lemma 3.3. Suppose H = H_{DR}^{1}(A, R). Let P ∈ A(F). Then M_{P} (H) = 0 if and only if there is a torsion point of A(K^{ur}) lying over P.

Proof. If there is such a torsion point, \overline{P}, then

∫ \overline{P} ; O ω = 0,

for all ω ∈ W. It follows that v(ω, P) = 1 for all ω ∈ W and so M_{P} (H) = 0.

Now suppose M_{P} (H) = 0. Let U(P) be the residue class above P. Let ω_{1}, …, ω_{g} be a basis for W over R. Let ω = (ω_{1}, …, ω_{g}). Define a function on U(P) into C_{p}^{g} by

F (X) = ∫_{0}^{X} ω,

for X ∈ U(P). Now let z_{1}, …, z_{g} be a set of local parameters on U(P) defined over R and let Z = (z_{1}, …, z_{g}) be the corresponding multivariable. We may use Z to identify U(P) with B(0, 1)^{g}. Having done this the R^{ur}-valued points of U(P) become identified with p(R^{ur})^{g}. Also we may now write F as a series in Z,

F (Z) = (∑_{I} A_{1,I}Z^{I}, …, ∑_{I} A_{g,I}Z^{I}),

where I runs over N^{g}. Since dF = ω, it follows that

G(Z) := (F (pZ) − F (0))/p ∈ R[[Z]].

Also since ω, …, ω_{g} is a basis for W, G′(0) is an invertible matrix. Since F is algebraically closed, it follows that as a function from (R^{ur})^{g} to itself, G is surjective. Now M_{P} (H) = 1−max{1, v(F (0))}, so since M_{P} (H) = 0, F (0) ∈ pR^{g}. Thus, there exists a z ∈ (R^{ur})^{g} such that G(z) = −F (0)/p and so there exists an x ∈ U(P)(R^{ur}) such that F (x) = 0. It follows from [2, Theorem 2.11] that x is a torsion point. ■

Corollary 3.4. Suppose P ∈ A(F) and k ∈ Z_{>0}. Then M_{P} (H) = k if and only if there exists a torsion point in U(p^{k}P)(R^{ur}) but not in U(p^{k−1}P)(R^{ur}).

Proof. Suppose M_{P} (H) = k. Then it follows from (2) that M_{p^{k−1}P} (H) = 1 and M_{p^{k}P} (H) = 0 and so from the lemma there exists a torsion point in U(p^{k}P) but not in U(p^{k−1}P). Now assume there exists a torsion point in U(p^{k}P)(R^{ur}) but not in U(p^{k−1}P)(R^{ur}). Then it follows from the lemma that M_{p^{k}P} (H) = 0 and M_{p^{k−1}P} (H) ≠ 0. It follows from (1) that M_{p^{k−1}P} (H) = 1. Now apply (3) repeatedly to deduce the corollary. ■

Now suppose Y is a smooth complete curve over R and that (H, W) is a de Rham F -T -crystal on Y, which we may equally well think of as a de Rham F -T crystal on its Jacobian J. We will use a great deal of the notation of [4, §3].

We will say that H is special if it is either ordinary or superspecial in the language of [4]. In the ordinary case, as explained in [4] we have bijective σ^{−1} and σ-linear operators operators C and C^{∗} on \overline{W} := W/pW and \overline{W}^{∗} := Hom(\overline{W} , k) satisfying

(Cw, v)^{σ} = (w, C^{∗}v)

for w ∈ \overline{W} and v ∈ \overline{W}^{∗}, where ( , ) is the natural pairing. (Caution: in [4, §3], the superscript “^{∗}” has a different meaning.) In the superspecial case, there are analogous operators, again called C and C^{∗}, with similar properties. In both cases, we obtain an operator on P(\overline{W}) which we call P(C).

Let O ∈ Y (K) and suppose (H, W) is a special de Rham F -T -crystal on Y. Let q = p if H is ordinary and p^{2} if H is superspecial. Let

λ_{ν}(Q) = ∫_{O}^{Q} ν.

That H is special implies that for P ∈ Y (F) there exists g_{ν} ∈ T O_{Y} (U(P)) such that

dg_{ν} ≡ ν mod T^{q−1}Ω_{Y} (U(P)).

Here, T is a uniformizing parameter for U(P). Also, \overline{g}_{ν} is the reduction of g_{ν} modulo p. Set

k_{P} (H) = min\{q, ord_{P} \overline{g}_{ν}: ν ∈ W \}.

We note that k_{P} (H) ≤ 1+min{ord_{P} ν̄: ν ∈ W } if the minimum is strictly less than q, with equality if it is strictly less than p. The following two propositions refine Proposition 15 of [4].

Proposition 3.5. Suppose H is special. Let P ∈ Y (F), U = U(P) and T be a uniformizing parameter on U over K^{ur}. Assume k_{P} (H) < q. Suppose b ∈ U(C_{p}) is a common zero of Λ = {λ_{ν} | ν ∈ W }. Let k = k_{P} (H) and M = M_{P} (H) and v = v(T (b)). Then one of the following holds:

(i) M = 0, v = ∞ or r/s where r and s are positive integers and s ≤ k or (ii) M > 0, v = 1/kq^{M} or (iii) v = 1/(kq^{n}(q − 1)) for some integer n such that q^{n}(q − 1) > q^{M}.

For S a uniformizing parameter on U and let h_{S} be the element of \overline{W}^{∗}

ω → \frac{ω}{dS} (P)

and let j(P) be the image of h_{S} in P(\overline{W}). (This is independent of the choice of S.) If M > 0, we also define e(P) to be the image in P(W) of the element d(P) in W^{∗} which is given by ω ↦ p^{M−1}(ω, P).

Proposition 3.6. In addition to the hypotheses of the last proposition suppose k_{P} (H) = 1 (i.e., that P is not a base point of W). Then we have:

In case (i), b ∈ U(K^{ur}).

In case (ii)^{∗}, every differential in W which vanishes at P vanishes at least q − 2 times. Moreover, if q ≥ 3,

P(C)(j(P)) = j(P) = e(P).  (∗)

If q = 2, choose a 2^{M}th root of 2, let r be the image of T (b)/2^{2^{−M}} in the residue field of K(b) and let h_{b} = rh_{T} ∈ \overline{W}^{∗}. Then

C^{∗}^{M+1}(h_{b}) + C^{∗}^{M}(h_{b}) = d(P).

∗ In fact, using the results proven below, Matt Baker has shown that when q ≥ 3 this case can only occur if dim W = 1.

In case (iii), every differential in W which vanishes at P vanishes at least q − 2 times if n = M and q − 1 times if n > M; further,

P(C)(j(P)) = j(P).

We note that the element h_{b} defined above is independent of choices.

Proof of Propositions 3.5 and 3.6. The main ingredient of the proof of these propositions, as discussed in the proof of Proposition 15 of [4], is the fact that for each ω ∈ W there exists f_{ω,m}(T) ∈ T R^{ur}[[T]] such that

df_{ω,m} ≡ C^{m}ω mod (p, T^{q}dT/T),  (4)

and for x ∈ U,

λ_{ω}(x) = c_{ω} + f_{ω,0}(T (x)) + \frac{f_{ω,1}^{σ}(T (x)^{q})}{p} + ⋯ + \frac{f_{ω,m}^{σ^{m}}(T (x)^{q^{m}})}{p^{m}} + ⋯, (5)

where c_{ω} = λ_{ω}(T^{−1}(0)). Call the series on the right L_{ω}(T (x)). What we do is consider the lower convex hull of the Newton polygons of the series L_{ω}(T) for ω ∈ W. This is the lower convex hull of the set of points

S ∪ {(kq^{i}, −i)}_{i≥0},

where S is {(0, 1 − M)} if M > 0 and if M = 0 is some set of points of the form (x, y) with x-coordinate a non-negative integer at most k − 1 and y-coordinate an integer at least 1. From this and Lemma 14 of [4], Proposition 3.5 follows.

Since, the proof of parts (i) and (iii) of Proposition 3.6 is contained in that of Proposition 15 of [4] and in that proposition the same assertion as the first sentence in (ii) was claimed, but not proved, with q−2 replaced by q−1 and the first equation of (∗) as well as the statement for q = 2 are new, we give the proof of part (ii) now.

Assume M > 0, k = 1.

First suppose q ≥ 3. Let a = T (b). Then the assertion λ_{ω}(b) = 0 implies

c_{ω} + \frac{f_{ω,M−1}^{σ^{M−1}}(a^{q^{M−1}})}{p^{M−1}} + \frac{(f′_{ω,M}(0))^{σ^{M}} a^{q^{M}}}{p^{M}} ≡ 0 mod p^{2−M}.  (∗∗)

As

f′_{ω,M}(0) ≡ (C^{M} ω̄/dT )|_{P} mod p,

we obtain P(C)^{M}(j(P)) = e(P) immediately. Hence, to prove (∗), we only have to establish the first equality. It also follows from (∗∗) that C^{M} ω̄ vanishes at P implies c_{ω} is congruent to 0 modulo a^{q^{M−1}}p^{1−M} and hence modulo p^{2−M} as c_{ω} ∈ K^{ur}. Hence

f_{ω,M−1}(a^{q^{M−1}}) ≡ 0 mod p,

which together with (4) implies C^{M−1}ω̄ vanishes q−2 times at P as v(a^{q^{M−1}}) = 1/q. In particular, using the facts that C is bijective on W̄ and is σ^{−1} linear we see that the hyperplane of differentials which vanish at P is stable under C which is equivalent to the first equation of (∗). The fact that ω vanishes q − 2 times at P follows from the above with ω replaced by an element ν ∈ W such that C^{1−M} ω̄ = ν̄ as we now know such a ν vanishes at P.

Suppose now q = 2. Then we see from (5) that, in the residue field of K(b),

2^{M−1}c_{ω} + (r(\frac{d}{dT} f_{ω,M})(0))^{σ^{M}} + (r(\frac{d}{dT} f_{ω,M+1})(0))^{σ^{M+1}} = 0.

The assertion now follows from (4) and the definitions.

■

If Q ∈ Y (C_{p}) and T is a uniformizing parameter defined over K^{ur} on the residue disk containing Q, let v(Q) = v(T (Q) + pR^{ur}) = min{1, v(T (Q))}. This is independent of the choice of T.

Corollary 3.7. Suppose that we are in the situation of Proposition 3.6. In cases (i) or (ii), Λ has exactly q^{M} common zeroes c such that v(c) = q^{−M} unless q = 2, in which case it has either 2^{M} or 2^{M+1} zeroes; in the latter case, P(C)(j(P)) = j(P). In case (iii), Λ has exactly q^{n}(q−1) common zeroes c such that v(c) = (q^{n}(q−1))^{−1}.

Proof. Lemma 10 of [4] together with the proofs of Proposition 15 of [4] when q ≥ 3 and 3.6 when q = 2, we see, that Λ has at most the numbers of common zeroes claimed. That it has at least this number follows from Galois theory. The statement about j(P) when q = 2 follows by an examination of the proof of Proposition 3.6.

■

The following proposition summarizes most of what we know about p = 2 and 3 in the ordinary case.

Proposition 3.8. Suppose Y is embedded in its Jacobian J so that O is the origin. Suppose Q is a ramified torsion point on Y and Ȳ is ordinary. Then p equals 2 or 3. Moreover:

(i) Suppose p = 2. Then Q̄ either does not lift to an unramified torsion point of J or Ȳ is hyperelliptic and Q̄ is a hyperelliptic branch point.

(ii) Suppose p = 3. Then: (a) P(C)(j(Q̄)) = j(Q̄). (b) If 3|(1/v(Q)), Q̄ does not lift to an unramified torsion point on J. (c) If Ȳ is hyperelliptic, then g(Y) = 2. Moreover, in this case, if y^{2} = f(x) is an equation for Y, where f(x) is a polynomial of degree 6 in R[x] such that (\overline{f}(x), \overline{f′}(x)) = 1 and \overline{f′}(x) has degree 4, then

\overline{f′(Q)} = 0.

(d) If g(Y) = 2 and v(Q) = 1/2, then Q̄ lifts to an unramified torsion point on J. Moreover, there are an even number, at most 16, such torsion points on Y.

Note that, by Corollary 3.7, in situation (ii) of the Proposition, v(Q) is either of the form 1/3^{n}, n > 0 or of the form 1/(2 · 3^{n}), n ≥ 0.

Proof. Suppose p = 2. Then, if MQ̄(H) > 0, Q̄ does not lift to an unramified torsion point by Lemma 3.3. Suppose MQ̄(H) = 0. Then we must be in case (iii), so P(C)(j(Q̄)) = j(Q̄) and we can apply the argument in the proof of Theorem 5.5 of [2] for p = 2.

Now suppose p = 3. Then (a) is contained in Proposition 3.6. If v(Q) = 1/3^{n}, then we are in case (ii) of this proposition and (b) follows from Lemma 3.6. Now suppose v(Q) = 1/2 · 3^{n} and n > 0. Then it follows from this Proposition that every differential which vanishes at Q̄ must vanish at least 2 times. Thus, Ȳ is hyperelliptic and Q̄ is a hyperelliptic branch point. By the argument in the proof of Theorem 5.5 of [2], this is impossible, so also in this case we may conclude that Q̄ does not lift to an unamified torsion point. Now (c) follows from (a) and the proof of Theorem 5.5 of [2]. Finally, if v(Q) = 1/2, then MQ̄(H) = 0, so Q̄ does lift to an unramified torsion point on J. By (c), Q̄ is a zero of a function on Ȳ of degree 8 and by Corollary 3.7 there are 0 or 2 such Q’s in any residue class. ■

Note. Theorem 19 of [4] has the appealing corollary:

Proposition 3.9. Let C be a smooth curve of genus g > 1 with good reduction, embedded in its Jacobian J over R. If p > 2g and P is a non-trivial torsion point in the kernel of reduction of J(C_{p}), then

C(C_{p}) ∩ (P + J(K)) = ∅.

This is, in fact, equivalent to the theorem when the crystalline cohomology of C has no unit root part.

Now fix p and let

H_{1} = ⋂_{n≥0} T_{p}^{n}H_{DR}^{1}(X_{0}(N), Z_{p}), H_{2} = { h ∈ H_{DR}^{1}(X_{0}(N), Z_{p}) | lim_{n→∞} T_{p}^{n}h = 0 }.

Set

W_{1} = H_{1} ∩ H^{0}(X_{0}(N), Ω_{X_{0}(N)/Z_{p}}^{1}), W_{2} = H_{2} ∩ H^{0}(X_{0}(N), Ω_{X_{0}(N)/Z_{p}}^{1}).

If I is an ideal of T, we let T_{I} denote the completion of T at I. We write T_{p} for T_{(p)}.

Proposition 3.10. Assume that p ≠ N. Then the pairs H_{1} := (H_{1}, W_{1}) and H_{2} := (H_{2}, W_{2}) are de Rham F -T -crystals, H_{1} is ordinary and

\begin{aligned}H_{1} + H_{2} & = H_{DR}^{1}(X_{0}(N), Z_{p}) \\ W_{1} + W_{2} & = H^{0}(X_{0}(N), Ω_{X_{0}(N)/Z_{p}}^{1}). \end{aligned}

Further, suppose that T is unramified at p, or more generally that T_{p} is divisible by p in T_{𝔪} for each maximal ideal 𝔪 of T such that 𝔪|p and T_{p} ∈ 𝔪. Then H_{2} is superspecial.

Proof. Let V and F be the Verschiebung and Frobenius endomorphisms acting on H_{DR}^{1}(X_{0}(N), Z_{p}). Since these endomorphisms commute with T_{p}, it follows immediately that (H_{i}, W_{i}) is a sub-F -T -crystal. Since T_{p} = V + F we see that H_{1} is ordinary. Because T_{p} is Gorenstein, we have an isomorphism

W_{2} ≈ ⊕T_{𝔪}

of T_{p}-modules in which the sum runs over those maximal ideals 𝔪 of T which lie over p and contain T_{p}. Since F W_{2} ≡ 0 mod p, it follows from the hypotheses of the proposition that V W_{2} ≡ 0 mod p. The fact that T_{p} is self-adjoint and W_{2} is self-orthogonal with respect to the cup product implies that h := rk H_{2} = 2 rk W_{2}. We also know V H_{2} = W_{2} +pH_{2}. Let V̄ and F̄ be the reductions of V and F mod p, rk V̄ = h/2. Thus the sequence

0 → W_{2}/pW_{2} → H_{2}/pH_{2} →^{V̄} W_{2}/pW_{2} → 0

is exact. Together with the equation V F = p, this implies that F H_{2} ⊆ W_{2} + pH_{2} and rk F̄ = h − rk V̄ = h/2. We conclude that F H_{2} = W_{2} + pH_{2}, i.e., that H_{2} is superspecial. ■

Remarks. (i) It is easy to see that H_{2} is not superspecial if T does not satisfy the hypothesis of the proposition which concerns local divisibilities of T_{p} by p. (ii) This hypothesis fails to be satisfied in the particular case N = 37, p = 2. In this case T_{2} lies in the unique maximal ideal 𝔪 of T which divides 2, but T_{2} does not lie in 2T_{𝔪}.

Lemma 3.11. Suppose that ω is an element of W_{2} and that P lies in J_{0}(N)(F). Then we have (ω, P) = 0. In particular, M_{P} (H_{2}) = 0.

Proof. This follows from the identity

(ω, T_{p}Q) = (T_{p}ω, Q),

for ω ∈ Ω_{J_{0}(N)/R^{ur}}^{1}(J_{0}(N)) and Q ∈ J_{0}(N)(F), which is evident from the definitions and the fact that the T_{p} acts bijectively on the p-power torsion points of J_{0}(N)(F).

■

Proposition 3.12. Suppose p ≥ 5, p ≠ N, H_{2} is superspecial and Q ∈ S. Then the extension Q(Q) of Q is unramified above p.

Proof. We may work locally at p. Let P be an element of X_{0}(N)(F) and let T be a uniformizing parameter on U(P) defined over R^{ur}. Let H_{i}^{ur} = H_{i} ⊗ R^{ur}. Suppose Q ∈ U(P) and Q is ramified. For k ∈ {1, 2}, let M_{k} = M_{P} (H_{k}^{ur}) and q_{k} = p^{k}. Note that M_{k} = 0 for k = 2 by the Lemma above. Then since P is not a base point for H^{0}(X_{0}(N), Ω_{X_{0}(N)/F}^{1}), it is not a base point for W_{j}⊗F for some j ∈ {1, 2}. Suppose {1, 2} = {i, j}. Then k_{P} (H_{j}^{ur}) = 1, and Proposition 3.6, Proposition 3.5 and the Lemma above imply that either (a) j = 1, M_{1} > 0, v(Q) = 1/p^{M_{1}} and every differential in W_{1} ⊗F which vanishes at P vanishes p−2 times or (b) v(Q) = 1/q_{j}^{n}(q_{j} −1) where n ≥ M_{j} and all differentials in W_{j} ⊗ F which vanish at P vanish at least q_{j} − 2 times (or q_{j} − 1 times if n > M_{j}). If k_{P} (H_{i}^{ur}) ≥ p, every differential in W_{i} ⊗ F vanishes at least p − 1 times at P. If 1 ≤ k_{P} (H_{i}^{ur}) < p then Proposition 3.5 applies, but none of its conclusions are consistent with what we have concluded above. Thus every differential in W_{i} ⊗ F vanishes at least p − 1 times at P.

Thus every differential in H^{0}(X_{0}(N), Ω_{X_{0}(N)/F}^{1}) which vanishes at P vanishes at least p − 2 times. This is impossible for p ≥ 5 since g > 1. ■

In view of Theorem 2.2, Proposition 3.12 completes the proof of Theorem 1.3 for primes p which are different from N. To prove Theorem 1.3, we need only prove that P_{N} ∉ C implies P − P_{N} ∉ C, when T is unramified at N. This question will be addressed below.

The proof of Proposition 3.12 yields:

Corollary 3.13. Suppose p = 2 or 3 and Q(Q) is ramified at some prime 𝔭 above p. Suppose that {1, 2} = {i, j} and the reduction P of Q modulo 𝔭 is not a base point for W_{j}. Then P is a base point for W_{i}. Moreover, if p = 3 every non-zero differential in W_{i} must vanish at least two times at P and if at a prime 𝔭 above 3 j = 2 or v(Q) < (p^{M_{1}}(p − 1))^{−1} and j = 1 then X_{0}(N) mod 3 is hyperelliptic and P is a hyperelliptic branch point.

We next prove:

Proposition 3.14. Suppose Q ∈ S and N divides the order of Q. Then for all embeddings \overline{Q} into \overline{Q}_{N}, the image of Q in X_{0}(N)(\overline{Q}_{N}) reduces to a singular point of the reduction of the Deligne-Rapoport model [11] of X_{0}(N) over Q_{N}.

Proof. Let N be the Néron model of J_{0}(N) and let X be the complement of the singular locus in the minimal regular model of X_{0}(N). The Néron property of N implies that the map from X_{0}(N) to J_{0}(N) extends to a map from X to N. The reduction of X differs from the non-singular locus of the reduction of Deligne-Rapoport model only by at most three components, which correspond to supersingular elliptic curves with complex multiplication by the fourth or third roots of unity. (See [21, Appendix, §1].)

Let M be the Z_{N}-submodule of H^{0}(X_{0}(N), Ω_{X_{0}(N)/Q_{N}}^{1} ) consisting of differentials which correspond to weight-two cusp forms with integral q-expansions at the two cusps of X_{0}(N). Let D be the special fiber of the Deligne-Rapoport model of X_{0}(N) over Z_{N}, and let X^{×} represent the irreducible component of D which contains the cusp C_{∞}, endowed with log-structure at the supersingular points. (See [16].) Let W_{∞} and W_{0} be the wide opens in X_{0}(N) whose points reduce to the irreducible components of D containing C_{∞} and C_{0}. (See [5].) Then the natural map from M into H_{DR}^{1}(W_{∞}, Q_{N}) is an injection. Its image is H := H_{cris}^{1}(X^{×}, Z_{N}). (See [7, §3].) Indeed, let Y_{∞} and Y_{0} be the connected components of C_{∞} and C_{0} in the formal completion of X along its reduction. Then Y_{∞} ⊗ Q_{N} may be identified with the underlying affinoid (ibid.) of W_{∞}, W_{∞} \ W_{0}, and the natural map of Y_{∞} into the formal completion of N along its reduction takes Y_{∞} into the connected component of the origin, N^{0}. Further, the map Ȳ_{∞} → N̄^{0} is an Albanese morphism into the generalized Jacobian of X relative to the modulus SS of supersingular points (Ȳ_{∞} may naturally be identified with X − SS). (See also [13, §8] and Theorem 3.1 and Corollary A2.3 of [7].) Now H has natural endomorphisms F and V so that if W = H, the quadruple (H, W, F, V) is an ordinary F -T -crystal. It follows by a direct translation of the proof of Theorem 20(i) of [4] as g = rk_{Z_{N}} (W) that as soon as g > 1, σ(S) ∩ Y_{∞}(C_{N}) ⊂ Y_{∞}(Q_{N}^{ur}) for any embedding σ of Q̄ into Q̄_{N}. Exploiting the Atkin-Lehner involution, we see similarly that if g > 1, σ(S) ∩ Y_{0}(C_{N}) ⊂ Y_{0}(Q_{N}^{ur}) for any embedding σ of Q̄ into Q̄_{N}.

The proposition follows since every torsion point of order N is ramified at all primes above N in view of Theorem 2.2 and the fact that the order of cuspidal group is prime to N. ■

Our aim now is to prove the following result, which will complete the proof of Theorem 1.3:

Theorem 3.15. Suppose N does not divide the discriminant of the Hecke algebra T. Let P be an element of S, written as usual as the sum ∑ P_{p}. Assume that P_{p} ∈ C for all p ≠ N. Then P_{N} = 0.

Our proof of Theorem 3.15 exploits a recent unpublished theorem of B. Edixhoven which concerns the special fibre \overline{N} of N. In its statement, the horizontal bar denotes the reduction map to characteristic N and the points 0 and ∞ are the cusps C_{0} and C_{∞}:

Theorem 3.16. Let \overline{C} be the image of C in \overline{N}. Then \overline{C} ∩ \overline{X} = \{\overline{0}, \overline{∞}\}.

We recall also a simple result from [29] which concerns the action of Gal(\overline{Q}/Q) on the ℓ-power torsion points of J = J_{0}(N), i.e., the ℓ-adic representation of Gal(\overline{Q}/Q) attached to J. This result has been proved only under the hypothesis that ℓ is different from 2 and 3 and is prime to the discriminant of T. We apply it in the case ℓ = N, which explains the presence of the discriminant hypothesis in the statement of Theorem 3.15. When the hypothesis is satisfied, T_{N} = T ⊗ Z_{N} is the product of discrete valuation rings; thus, Mazur’s theorem [21, Ch. II, §14–§15] to the effect that the N-adic Tate module Ta_{N}(J) of J is free of rank 2 over T_{N} is especially easy to prove. Choosing a basis for Ta_{N}(J) over T_{N}, we view the N-adic representation of Gal(\overline{Q}/Q) defined by J as a continuous homomorphism

ρ_{N}: Gal(\overline{Q}/Q) → GL(2, T_{N}).

We obtain from [29]:

Proposition 3.17. Suppose that N is prime to the discriminant of T. Then the image of ρ_{N} contains the group SL(2, T_{N}).

Here is our proof of Theorem 3.15:

Proof. Using Proposition 3.17, one shows that there is a σ ∈ Gal(\overline{Q}/Q) such that σ(P_{N}) lies in the kernel of the reduction map to characteristic N. Replacement of P by σ(P) preserves the hypothesis that P_{p} ∈ C for all p ≠ N and does not affect the question of whether P_{N} is non-zero. We may thus assume that P and P − P_{N} have the same image in characteristic N. It follows that P is a point on N^{rig}, the generic fiber of the formal completion of N along its reduction. The connected components of this rigid space are in one-to-one correspondence with the connected components of the reduction of the Néron model, which in turn are in one-to-one correspondence, via reduction, with the elements of C. Thus the reduction of P is a point in \overline{C} ∩ \overline{X}. Now it will be shown in [10] that the intersection of N^{rig} with X_{0}(N) is the rigid space associated to the formal completion of the minimal model of this curve along its reduction (this is a general phenomenon that occurs when the reduction of the minimal model is semistable and every edge in its graph is contained in a cycle). Thus it follows from Edixhoven’s theorem that P − P_{N} is either 0 or ∞. But now Proposition 3.14 implies that P_{N} = 0. ■

4. Lang’s Approach

We proceed in the spirit of Lang [18]:

Lemma 4.1. Suppose X is a curve of genus g > 1 embedded in its Jacobian and n an integer such that |n| > 1. Then #(nX ∩ X) ≤ gn^{2}.

Proof. Let W_{n} = X + X + ⋯ + X, where the sum is over |n| copies of X. If a ∈ W_{g−2}, then X ⊆ W_{g−1} − a. It follows from Lemma 5.4 of [2] that there exists an a ∈ W_{g−2} such that (W_{g−1} − a) ∩ nX is finite. (Note that condition (ii) in the proof of this lemma should be replaced by the condition: D is not special.) Hence

#(X ∩ nX) ≤ deg(W_{g−1} · nX),

where ( · ) is the intersection pairing. By Theorem 5 of [19, Ch. 4, §3],

deg(W_{g−1} · n(X)) = deg (n^{−1}(W_{g−1}) · X) .

We see from the discussion following Proposition 4 of §3 that n^{−1}(W_{g−1}) is numerically equivalent to n^{2} · W_{g−1}. The result now follows from the fact that deg(W_{g−1} · X) = g. (Proof: We actually compute W_{g−1} · −X. Suppose the divisor Q_{1} + ⋯ + Q_{g} is not special. Let A be the point on J equal to ∑ Q_{i}. Then if P ∈ X and −P lies in W_{g−1} − A, we see that P = Q_{i} for some i.) ■

Proposition 4.2. Suppose X and the embedding into its Jacobian are defined over Q and there exists a σ ∈ Gal(\overline{Q}/Q) which acts on J[M](\overline{Q}) via the homothety n, where n is an integer satisfying |n| > 1. Then

# (X(\overline{Q}) ∩ J[M](\overline{Q})) ≤ gn^{2}.

Proof. The proposition follows immediately from the lemma, since the indicated intersection lies both in X and nX. ■

Proposition 4.2 and the results of [2] can be combined to deduce consequences in other contexts. For example, let p be a prime, and for each triple of integers (a, b, c) with abc prime to p, let

F_{a,b,c}^{p} : y^{p} = (−1)^{c}x^{a}(1 − x)^{b}

denote the indicated quotient of the degree-p Fermat curve F^{p} : X^{p} + Y^{p} + Z^{p} = 0. (See, e.g., [8, §3] for background on Fermat quotients.) The cusps on F^{p} are the images of points on the Fermat curve for which one of the projective coordinates X, Y, Z is 0; the cusps on F_{a,b,c}^{p} are the points where the coordinate z is 0, 1 or ∞. According to well-known results of Rohrlich [30], the cusps of F^{p} lie in a common torsion packet; i.e., their differences form degree-zero divisors whose images on the Jacobian of F_{p} are torsion. The corresponding statement then holds for the quotients F_{a,b,c}^{p}. One can ask whether the cusps form a complete torsion packet on Fermat curves F^{p} and F_{a,b,c}^{p} of genus > 1. (This question is discussed in [3] and especially in [6].) For instance, let x_{0} a cusp of F_{a,b,c}^{p}, and use it to embed F_{a,b,c}^{p} in its Jacobian J. Does the cuspidal torsion packet S := F_{a,b,c}^{p} ∩ J_{tor} contain only the cusps?

To prove statements of this sort, it has proved helpful to show first that S has small cardinality. As an application of Proposition 4.2, we obtain:

Proposition 4.3. Assume that F_{a,b,c}^{p} is not hyperelliptic and that p ≥ 5. Then the cuspidal torsion packet of F_{a,b,c}^{p} has at most (p − 1)/2 · (1 + p)^{2} elements.

Suppose now that p is a prime ≥ 5. Then Coleman, Tamagawa and Tzermias [9] have shown, using the proposition and results of Coleman [6] and Greenberg [12], that the cusps on F^{p} form a complete torsion packet.

5. Galois action on torsion points of J_{0}(37)

We next present some results of [15] about the action of Gal(\overline{Q}/Q) on the group of torsion points of J_{0}(37).

The space of weight-two cusp forms on Γ_{0}(37) is two dimensional; it is spanned by the eigenforms with integral coefficients which arise from the two isogeny classes of rational elliptic curves with conductor 37. The action of the Hecke ring T on these forms induces an inclusion T ⊆ Z × Z. It is well known that the image of T in Z × Z is the order consisting of pairs of integers (a, b) which satisfy a ≡ b mod 2. Using the results of [21, Ch. II], one sees that the adelic Tate module

Ta_{f} J := lim_{⟵} J[n]

of J = J_{0}(37) is free of rank 2 over T⊗Ẑ. Equivalently: for each prime p, the p-adic Tate module Ta_{p} J = Ta_{p} (J(\overline{Q})) is free of rank two over the ring T_{p} := T ⊗ Z_{p}.

The image G of Gal(\overline{Q}/Q) in Aut J(\overline{Q})_{tors} is the image of Gal(\overline{Q}/Q) in

Aut_{T⊗Ẑ} Ta_{f} ≈ GL(2, T ⊗ Ẑ).

Since the determinant of the representation Gal(\overline{Q}/Q) → GL(2, T⊗Ẑ) is the adelic cyclotomic character Gal(\overline{Q}/Q) → Ẑ^{∗}, the image of this representation lies in the group

A := { M ∈ GL(2, T ⊗ Ẑ) | det M ∈ Ẑ^{∗} }.

In other words, we have

G ⊆ A ⊂ GL(2, T ⊗ Ẑ).

For each prime p, let G_{p} and A_{p} be the images of G and A in the group GL(2, T_{p}). One has G ⊆ ∏_{p} G_{p}. More generally, if S is a set of primes (possibly infinite), let G_{S} be the image of G in GL(2, ∏_{p∈S} T_{p}), i.e., in the product ∏_{p∈S} G_{p}. Further, let G^{S} be the image of G in ∏_{p∉S} G_{p}, and make similar definitions with G replaced by A. For each S, one has trivially A = A_{S} × A^{S}, but the analogous decomposition for G is by no means automatic. If p_{1}, …, p_{n} are distinct primes, we sometimes write G(p_{1}, …, p_{n}) in place of G_{{p_{1},... ,p_{n}}} and make similar use of A(p_{1}, …, p_{n}).

The main result of [15] is a description of G as a subgroup of A.

Proposition 5.1. For all p ≥ 5, one has

G_{p} = A_{p} = { M ∈ GL(2, Z_{p} × Z_{p}) | det M ∈ Z_{p}^{∗} }.

Further, let S be the set {2, 3, 37}. The natural inclusion G ↪ G_{S} × G^{S} is an isomorphism.

In the displayed equation, the second equality, which asserts that A_{p} is a GL(2), results from the circumstance that 2 = (Z × Z: T) is prime to p. The first equality is not too difficult to establish, even in the more general case where 37 is replaced by an arbitrary prime number [29]. The equation G = G_{S} ×G^{S} may be summarized by the statement that each group G_{p} with p ≠ 2, 3, 37 is a subgroup of G. One can prove it by the methods of Serre [31, §6.2]. ■

Proposition 5.1 reduces the determination of G to a determination of its quotient G_{S}, where S = {2, 3, 37}. On the other hand, it is the quotient G_{S}, as opposed to the full group G, which intervenes in our proof of Theorem 1.4. The group G_{S} is a subgroup of G_{2} × G_{3} × G_{37} which turns out to have index 3 in the product G_{2}×G_{3}×G_{37}. In order to describe G_{S}, it is convenient to discuss first the individual factors G_{2}, G_{3}, and G_{37}. The last of the three factors has already been determined: it coincides with its overgroup A_{37}. (One can say that G_{37} is “as large as possible.”) The following consequence of this fact will be useful below:

Corollary 5.2. Let P be a torsion point on J_{0}(37) whose order is divisible by 37. Then P has at least 37^{2} − 1 = 1368 conjugates.

We next describe G_{2}. To do this, we shall introduce a certain homomorphism

θ: A_{2} → F_{2}.

View A_{2} as the group

{ (α, β) ∈ GL(2, Z_{2}) × GL(2, Z_{2}) | det(α) = det(β) and α ≡ β mod 2 }.

Let 𝔰𝔩_{2}(F_{2}) be the lie algebra of 2 × 2 matrices over F_{2} with trace 0. The adjoint action (α, β) : M ↦ aMa^{−1}, where a = α mod 2 = β mod 2, makes 𝔰𝔩_{2}(F_{2}) into an A_{2}-module. We regard F_{2} as an A_{2}-module with trivial action. One may verify explicitly that the map

𝔰𝔩_{2}(F_{2}) → F_{2}, ( \begin{matrix}a & b \\ c & a \end{matrix} ) ⟼ a + b + c

is a non-zero map of A_{2}-modules. Meanwhile, it is well known, and not hard to check by direct calculation, that the association

A_{2} → 𝔰𝔩_{2}(F_{2}), (α, β) ↦ \frac{αβ^{−1}−I}{2} mod 2

is a 1-cocycle on A_{2} with values in 𝔰𝔩_{2}(F_{2}). On composing this cocycle with the map 𝔰𝔩_{2}(F_{2}) → F_{2} we have exhibited, we obtain the desired homomorphism θ.

For later use, we define a second non-trivial homomorphism

ϵ: A_{2} → F_{2}

as the composite of the surjection A_{2} → GL(2, F_{2}) defined by (α, β) ↦ α mod 2 and the unique non-zero homomorphism GL(2, F_{2}) → F_{2}. This latter map is the “sign” function S_{3} → {±1}, viewed as taking values in the additive group F_{2}. It is clear that ϵ is non-trivial on G_{2}, since G_{2} maps onto GL(2, F_{2}).

Proposition 5.3. The group G_{2} is the kernel of θ: A_{2} → F_{2}.

For the proof of this Proposition, see [15]. In what follows, we we will present a concrete explanation for the inclusion G_{2} ⊆ ker θ.

View G_{2} as the Galois group of the extension K of Q gotten by adjoining all 2-power division points on the two strong modular elliptic curves of conductor 37. Call these curves E and E′; order them so that the discriminants of E and E′ are Δ = 37 and Δ′ = 37^{3}, respectively. The field K contains Q(\sqrt{Δ}) = Q(\sqrt{Δ′}): this field is the quadratic subfield of the S_{3}-extension of Q gotten by adjoining the 2-division points on E or E′. On the other hand, K contains Q(\sqrt{−1}, \sqrt[4]{Δ}) and Q(\sqrt{−1}, \sqrt[4]{Δ′}): these are subfields of the field of 4-division points of E × E′. In particular, K contains the biquadratic field Q(\sqrt{−1}, \sqrt[4]{Δ′/Δ}). Here we note that Δ′/Δ is a perfect square because of the isomorphism E[2] ≈ E′[2], so that \sqrt[4]{Δ′/Δ} is either rational or a quadratic irrationality. Thus the subfield Q(\sqrt{−1}, \sqrt[4]{Δ′/Δ}, \sqrt{Δ}) of K is a (2, …, 2)-extension of Q whose degree over Q could a priori be as large as 8. In fact, however, the quantities \sqrt[4]{Δ′/Δ} and \sqrt{Δ} are equal, so that the degree of this field over Q is at most 4.

Corollary 5.4. Let P be a non-zero point on J_{0}(37) of 2-power order. Then there is a g ∈ G_{2} such that ϵ(g) = 0 and such that gP − P has order 2.

This result may be deduced from the equality G_{2} = ker θ by straightforward calculation. Alternatively, we can proceed somewhat more intrinsically as in the proof of Theorem 2.4, considering the action of an inertia subgroup I of Gal(\overline{Q}/Q) for the prime 37. Composing ϵ with the projection Gal(\overline{Q}/Q) → G_{2}, we will regard ϵ as a map Gal(\overline{Q}/Q) → F_{2}. We must find a g ∈ Gal(\overline{Q}/Q) in the kernel of ϵ with the property that gP − P has order two.

Suppose first that P itself has order 2. Then it suffices to find a g in the kernel of ϵ which does not fix P. For this, it is helpful to recall the structure of J_{0}(37)[2] as a Gal(\overline{Q}/Q)-module. Let J = J_{0}(37) and let 𝔪 be the unique maximal ideal of T with residue characteristic 2. The kernel J[𝔪] is isomorphic to E[2] (and also to E′[2]), and the quotient J[2]/J[𝔪] is again isomorphic to J[𝔪] (cf. [21, p. 112]). The group Gal(\overline{Q}/Q) acts on E[2] via its quotient GL(2, F_{2}), which is the symmetric group on three letters. The image in Aut E[2] of the kernel of ϵ is the alternating group on three letters, which permutes transitively the three non-zero elements of E[2]. An element g in the kernel of ϵ which maps non-trivially to this alternating group can fix no non-zero element of J[2]. Indeed, such elements g fix no non-zero elements of the quotient E[2] and no non-zero elements of the submodule E[2].

Suppose now that P has order 2^{n}, with n ≥ 2. Then Q = 2^{n−1}P is a point of J_{0}(37) of order 2. As we have noted, the action of Gal(\overline{Q}/Q) on E[2] cuts out an S_{3}-extension of Q. This extension is ramified at 37; indeed, it contains the quadratic field Q(\sqrt{37}). For each non-zero R in E[2], we may find an inertia subgroup I ⊂ Gal(\overline{Q}/Q) for the prime 37 and a σ ∈ I such that σR ≠ R. (Indeed, the image of each I in S_{3} has order 2; some conjugate of this element will move R.) Using again that J[2] is an extension of E[2] by E[2], we may find an inertia group I for 37 and a σ ∈ I such that σQ ≠ Q. Now consider the action of σ on J_{0}(37)[2^{n}]. As in the proof of Theorem 2.4, σ acts as the sum 1 + A, where 1 represents the identity operator and A an endomorphism satisfying A^{2} = 0. Since

σQ = Q + A 2^{n−1}P,

the quantity 2^{n−1}AP has order 2. Also

σ^{2^{n−1}}P = (1 + A)^{2^{n−1}}P = P + 2^{n−1}AP

by the binomial theorem. It thus suffices to take g = σ^{2^{n−1}}: this element is automatically in the kernel of ϵ because it is the square of an element of Gal(\overline{Q}/Q). ■

We now describe

G_{3} = Gal (Q(E[3^{∞}], E′[3^{∞}])/Q) ↪ Aut(Ta_{3} E) × Aut(Ta_{3} E′).

As noted by Serre [31, 5.5.8], G_{3} maps onto the first factor Aut(Ta_{3} E). Let X be the image of G_{3} in the second factor Aut(Ta_{3} E′) ≈ GL(2, Z_{3}). Since E′[3] may be regarded as J_{0}(37)[𝔪], where 𝔪 is the unique Eisenstein prime of T, the module E′[3] is isomorphic to the direct sum μ_{3} ⊕ Z/3Z. In particular, the image of X mod 3 has order 2. Let \overline{X} be the image of X in GL(2, Z/9Z), i.e., in Aut E[3^{2}]. Then the order of \overline{X} is a divisor of 2 · 81 which is divisible by 2. According to [15], \overline{X} has order 54 = 2 · 27; thus, \overline{X} is of index 3 in the inverse image in GL(2, Z/9Z) of the image of \overline{X} mod 3. On the other hand, X is the full inverse image of \overline{X} in Aut(Ta_{3} E′) ≈ GL(2, Z_{3}). Concretely, with respect to a suitable basis of Ta_{3} E′ one has:

X = \{ ( \begin{matrix}1 + 3e & 3f \\ 3g & h \end{matrix} ) ∈ GL(2, Z_{3}) | g ≡ −fh mod 3 \} .

For later use, we introduce the surjective homomorphism

X → Z/9Z, ( \begin{matrix}1 + 3e & 3f \\ 3g & h \end{matrix} ) ↦ e + 3(e^{2} − g^{2}) mod 9.

Composing this map with the projection G_{3} → X, we obtain a surjection

F: G_{3} → Z/9Z.

The group G_{3} is contained in the group of pairs (g, x) ∈ GL(2, Z_{3}) × X which satisfy the condition det g = det x. As Kaskel shows in [15], G_{3} has index 3 in this latter group; this additional constraint on the elements of G_{3} may be traced to the fact that both Q(E[3^{∞}]) and Q(E′[3^{∞}]) contain the field Q(\sqrt[3]{37}).

Proposition 5.5. With respect to suitable bases, G_{3} is the group of pairs

( ( \begin{matrix}a & b \\ c & d \end{matrix} ) , ( \begin{matrix}1 + 3e & 3f \\ 3g & h \end{matrix} )) ∈ GL(2, Z_{3}) × X

which satisfy the determinant condition ad − bc = (1 + 3e)h − 9fg and the supplementary condition

(ac + bd)(c^{2} + d^{2}) ≡ g ≡ −fh mod 3.

Corollary 5.6. The homothety −2 belongs to G_{3}.

We now state without proof a description of the full Galois group G. This description involves the homomorphisms ϵ: G_{2} → Z/2Z and F: G_{3} → Z/9Z which were introduced above.

Theorem 5.7. The group G is the subgroup of ∏_{ℓ} G_{ℓ} consisting of all elements (…, g_{ℓ}, …) which are related by the identity

det(g_{37})^{2} ≡ 2^{20F (g_{3})+18ϵ(g_{2})} mod 37

in F_{37}^{∗}.

The reader will remark that the groups G_{{2,3,37}} and G^{{2,3,37}} are “independent” in the sense that G is the product of these two factors of G; we have already noted this point above. In the crucial quotient G_{{2,3,37}}, the two factors G_{2} and G_{3} are independent of each other: the group G_{{2,3}} is the product of its quotients G_{2} and G_{3}. On the other hand, G_{37} is linked to each of G_{2} and G_{3}. See [31, Prop. 22] for a related observation.

The following consequence of Theorem 5.7 can be seen by inspection.

Corollary 5.8. The homothety −5 belongs to G_{{2,3,37}}.

6. Torsion points on X_{0}(37)

We now prove Theorem 1.4, which states that S = X_{0}(37)∩J_{0}(37)_{tor} contains only the two cusps C_{∞} and C_{0}.

Let P be an element of S. Recall that we may decompose P as the sum of its primary components P_{p}. According to Corollary 2.3, all P_{p} are zero except perhaps for p = 2, p = 3, and p = 37:

P = P_{2} + P_{3} + P_{37}.

We may thus view S as a subset of J[M], where M is the product of suitable powers of 2, 3, and 37. (With the appropriate definition, one can take M = 2^{∞}3^{∞}37^{∞}.) As we have just seen, there is an element of Gal(\overline{Q}/Q) which acts on J[M] as the homothety −5. By Proposition 4.2, we get

#S ≤ 2(−5)^{2} = 50,

since the genus of X_{0}(37) is two. On the other hand, any element of S for which P_{37} is non-zero must have at least 1368 conjugates by Corollary 5.2. Thus each P ∈ S may be decomposed as the sum P_{2} + P_{3}.

We claim next that P_{2} = 0, i.e., that P = P_{3}. If not, there is a g ∈ Gal(\overline{Q}/Q) such that gP_{2} − P_{2} has order 2 by Corollary 5.4. Because of the independence of G_{2} and G_{3}, we may assume that g fixes P_{3}. Therefore, gP − P has order two on J. As we saw in §1, it follows that P is a hyperelliptic branch point of X_{0}(37). Indeed, the divisor 2((gP) − (P)) is the divisor of a non-constant function, and the point P is forced to ramify in the corresponding degree-two covering of P^{1}. However, we have seen in Proposition 1.1 that the hyperelliptic branch points of X_{0}(37) do not belong to S.

Knowing that S ⊆ J_{0}(37)[3^{∞}], we apply Corollary 5.6 and Proposition 4.2 and deduce the bound #S ≤ 8. To conclude, we consider the degree-two covering π: X_{0}(37) → E which is obtained by dividing X_{0}(37) by its Atkin-Lehner involution. This covering may be viewed as the composite of the Albanese map X_{0}(37) → J_{0}(37) sending C_{∞} to 0 with a homomorphism of abelian varieties J_{0}(37) → E whose kernel is E′ (regarded as an abelian subvariety of J_{0}(37)). As we have remarked, the image of G_{3} in Aut(E[3^{∞}]) is all of GL(2, Z_{3}). Hence any non-zero element of E[3^{∞}] has at least eight conjugates. It follows that the image of S in E[3^{∞}] is the single element 0. Indeed, S contains the two rational points C_{∞} and C_{0} and consists of at most eight elements. Therefore, it can contain no point with more than six conjugates. Thus S is contained in the set π^{−1}({0}); since π has degree two, S can have no more than two elements. ■

With an eye to possible generalizations, we present two variants of the proof of Theorem 1.4. In the first variant, we begin as before, writing P ∈ S as P_{2}+P_{3}+P_{37}, but noting that P_{37} = 0 because of Proposition 4.2. Thus P = P_{2} + P_{3}.

Proposition 6.1. The point P_{3} belongs to the cuspidal subgroup of J_{0}(37).

Proof. We consider the two normalized newforms

\begin{aligned}F_{1}(q) & = q + 0q^{2} + q^{3} + ⋯ \\ F_{2}(q) & = q − 2q^{2} − 3q^{3} + ⋯ \end{aligned}

of weight two on Γ_{0}(37) and let ω_{1} and ω_{2} be the corresponding regular differentials on X_{0}(37).

Lemma 6.2. If p is an odd prime, neither ω_{1} nor ω_{2} has a double zero modulo p.

Proof. Let w be the Atkin-Lehner involution of X_{0}(37). Then

\begin{aligned}w^{∗}ω_{1} & = −ω_{1} \\ w^{∗}ω_{2} & = ω_{2}. \end{aligned}

It follows, from this and the fact that X_{0}(37) is hyperelliptic of genus 2, that both of these differentials are pullbacks of invariant differentials on elliptic curves by degree two morphisms. From this and the Riemann-Hurwitz theorem, the lemma follows. ■

Continuing the proof of Proposition 6.1, we work in the context of §3, taking p = 3. After glancing at the displayed q-expansions, we see that W_{1} = Z_{p}ω_{1} and W_{2} = Z_{p}ω_{2} in that setting. In view of the Lemma we have just proved and Corollary 3.13, we find that the extension Q(P)/Q is unramified at 3. The desired conclusion now follows from Theorem 2.2. ■

Our second proof of Theorem 1.4 now concludes as follows. We claim as before that P_{2} = 0. Indeed, if P_{2} is non-zero, then we may find as before a g ∈ Gal(\overline{Q}/Q) such that gP_{2} −P_{2} has order 2. Because P_{3} lies in J_{0}(37)(Q), g fixes P_{3}. Therefore, gP − P has order two on J, and we have seen that this is not the case. Thus we have P = P_{3} ∈ C, where C is the cuspidal subgroup of J_{0}(37). However, one knows that X_{0}(37) ∩ C consists precisely of the two cusps on X_{0}(37) [23, §5, Prop. 2].

To make a third proof, we again write P ∈ S as the sum P_{2} + P_{3} + P_{37}. As we saw in the second proof, P_{3} belongs to C. Once we know that P_{37} is zero, we can conclude as before. To see that this is the case, we argue by contradiction, supposing that P_{37} is not zero. Then P_{37} cannot belong to C, a group of order three. Thus, by Theorem 1.3, P_{2} must be non-cuspidal, and hence non-zero. By Theorem 5.4, there is a g_{2} ∈ G_{2} with ϵ(g_{2}) = 0 such that g_{2}P_{2} − P_{2} has order two. In view of Theorem 5.7, there is a g = (…, g_{ℓ}, …) ∈ G whose component at 2 is the given g_{2} and whose component at 37 is 1. We then have

gP − P = (g_{2}P_{2} − P_{2}) + (g_{3}P_{3} − P_{3}) + (P_{37} − P_{37}) = (g_{2}P_{2} − P_{2}).

In computing gP − P, we have used that P_{3} is rational, so that g_{3}P_{3} = P_{3} for any choice of g_{3}. (Of course, we could have taken g_{3} = 1.) Thus gP − P has order two, and we have seen that this is impossible.

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Errata for [4]

This list of errata for Coleman’s article “Ramified torsion points on curves” was compiled by Matt Baker:

— p. 615 Conjecture B, part (ii): should read “K/ℚ is unramified at 𝔭.”

— p. 620 line 3: should read “ … we get p^{k}w = F^{n}h_{n} for all n ≥ k.”

— p. 623 fifth line from bottom: should read “ω_{i} = \frac{ϕ^{∗}ω_{i+1}^{σ}}{p} + df_{i}.”

— p. 623 fourth line from bottom: “Let 0 < n_{0} < n_{1} < ⋯ ” should read “Let 0 ≤ n_{0} < n_{1} < · · ·.”

— p. 623 third line from bottom: “ω̃_{i} ∈ Ω̃” should be “[ω_{i}]^{∗} ∈ Ω̃.”

— p. 624 line 2: the p^{a_{i}} in the denominator should be a p.

— p. 624 line 3: should read “Let h_{i} = g_{i} − g_{i}(Q) ∈ A^{0}(Y) ⊗ R^{u}.”

— p. 624 line 5: should read “dL(T) = ν on U.”

— p. 624 line 8 : should read “Let k_{i} = ord_{U} ν̃_{i} + 1.”

— p. 624 Theorem 10: When p = 2 in part (iii), one can conclude from the proof that i = M = 0, k_{i} = 1, Q_{i} = (1, 0), Q_{i+1} = (2, −1).

— p. 625 line 6: should read “s(k_{1}p − h) ≤ (1 + s)(k_{0} − h).”

— p. 625 third line from bottom: p^{h} should be p^{n}.

— p. 627 fifth line from bottom: replace sentence beginning “To finish … ” with “To finish the proof of the proposition, we will eliminate all but one of the above possibilities for D.”

— p. 628 line 7: “n_{i+1} − n_{i} > 1” should be “n_{i+1} − n_{i} ≥ 1.”

— p. 628 line 11: v^{n_{i}} should be V^{n_{i}}.

— p. 628 line 12: “for all η ∈ W̃^{∗}” should be “for all η ∈ W.”

— p. 629 line 4: “k_{0}ω)” should be “k_{0}(ω).”

— p. 630 seventh line from bottom: should read “ω ∈ W̃ ↦ (\frac{ω}{dT̃})(U).”

— p. 630 fourth line from bottom: “let e = e(a)” should be “let e = e(A).”

— p. 631 line 2: should read “b ∈ U(ℂ_{p}).”

— p. 631 line 13: Proposition 10 should be Theorem 10.

— p. 631, Prop. 15, part (ii): q − 1 should be replaced by q − 2.

— p. 631 line 15: “when p = 2, A = 1” should be “when q = 2, A ≤ 1.”

— p. 631 middle: “for some g_{ω,i}T)” should read “for some g_{ω,i}(T).”

— p. 631 fifth line from bottom: replace “g_{ω,i}(0)” by “\frac{d}{dT} g_{ω,i}(0).”

— p. 632 line 3: replace “a_{n} = a^{q^{n}}” by “a_{n} = a^{q^{n−1}}.”

— p. 632 line 4: replace “Theorem 4.2.1 of [C-1]” by “Theorem 4.1 of [C-1].”

— p. 632 middle: replace “_{\frac{1}{p^{n{′ \atop i+1}} −p^{n′_{j}}}} ” by “_{\frac{1}{p^{n{′ \atop j+1}} −p^{n′_{j}}}}.”

— p. 633 line 10: should say “ω ∈ W.”

— p. 633 proof of Theorem 18: reference to Cor. 13.1 should be to Cor. 13.3.

— p. 633 proof of Theorem 19: reference to Prop. 10 should be to Prop. 13.

— p. 634 Theorem 20: when p = 2 or 3, the theorem is incorrect. The present article contains corrected version.

— p. 635 proof of Theorem 20: reference to Prop. 19 should be to Prop. 15.

UC Mathematics Department, Berkeley, CA 94720-3840 USA