PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 8, August 1997, Pages 2255–2263 S 0002-9939(97)03874-4

ON THE RATIONAL CUSPIDAL SUBGROUP AND THE RATIONAL TORSION POINTS OF J_{0}(pq)

SENG-KIAT CHUA AND SAN LING

(Communicated by William W. Adams)

Abstract. For two distinct prime numbers p, q, we compute the rational cuspidal subgroup C(pq) of J_{0}(pq) and determine the ℓ-primary part of the rational torsion subgroup of the old subvariety of J_{0}(pq) for most primes ℓ. Some results of Berkovič on the nontriviality of the Mordell-Weil group of some Eisenstein factors of J_{0}(pq) are also refined.

1. Introduction

Let p, q be two distinct prime numbers, and let X_{0}(p), X_{0}(q) and X_{0}(pq) denote the classical modular curves over ℚ. Let J_{0}(p), J_{0}(q) and J_{0}(pq) denote the corresponding Jacobian varieties, also defined over ℚ.

On X_{0}(p) (resp. X_{0}(q)), there are two (rational) cusps P_{1} and P_{p} (resp. P_{1} and P_{q}). It is well-known that the cuspidal subgroup C(p) (resp. C(q)), generated by the class of the cuspidal divisor P_{1} − P_{p} (resp. P_{1} − P_{q}), is cyclic of order \frac{p−1}{(p−1,12)} (resp. \frac{q−1}{(q−1,12)}) (cf. [7]).

There are exactly four cusps, denoted P_{1}, P_{p}, P_{q} and P_{pq}, on X_{0}(pq), and they are all rational. Let C_{p} = P_{1} − P_{p}, C_{q} = P_{1} − P_{q} and C_{pq} = P_{1} − P_{pq} be the cuspidal divisors of degree 0. For a degree-0 divisor D on X_{0}(pq), we shall denote the divisor class it defines by \overline{D}.

The cuspidal subgroup C(pq) of J_{0}(pq) is clearly generated by \overline{C}_{p}, \overline{C}_{q} and \overline{C}_{pq}. Manin [5] has shown that \overline{C}_{p}, \overline{C}_{q} and \overline{C}_{pq} are of finite order. For each prime ℓ, we shall let r_{ℓ}, s_{ℓ} and t_{ℓ} be the non-negative integers such that ℓ^{r_{ℓ}} (resp. ℓ^{s_{ℓ}}, ℓ^{t_{ℓ}}) is the exact power of ℓ dividing the order of \overline{C}_{p} (resp. \overline{C}_{q}, \overline{C}_{pq}). For an integer x, let v_{ℓ}(x) be the valuation of x at ℓ, i.e., ℓ^{v_{ℓ}(x)} is the exact power of ℓ dividing x. Then we also set, for each prime ℓ,

m_{ℓ} =^{def} min (v_{ℓ} (\frac{p − 1}{(p − 1, 12)}) , v_{ℓ} (\frac{q − 1}{(q − 1, 12)})) ,
M_{ℓ} =^{def} max(v_{ℓ}(p − 1), v_{ℓ}(q − 1)).

In this article, we compute the rational cuspidal subgroup C(pq) of J_{0}(pq). Here, for a finite group G, the notation G_{ℓ} represents the ℓ-primary part of G.

\begin{matrix}c \\ ◯ \end{matrix}1997 American Mathematical Society

Received by the editors September 8, 1995 and, in revised form, March 10, 1996.

1991 Mathematics Subject Classification. Primary 11G18, 11F03, 11F20, 14H40.

The authors would like to thanks Ken Ribet for private communication. We are also grateful to the referee for comments which helped improve the presentation of the paper.

Theorem 1. (i) Let p, q ≥ 5 be two distinct primes. For any prime ℓ, the ℓ-primary part of the cuspidal subgroup C(pq) of J_{0}(pq) is given by:

\begin{matrix}C(pq)_{ℓ} ≅ ℤ/ℓ^{r_{ℓ}}ℤ × ℤ/ℓ^{s_{ℓ}}ℤ & if & ℓ |̸ (\frac{p−1}{2}, \frac{q−1}{2}) , \\ C(pq)_{ℓ} ≅ ℤ/ℓ^{r_{ℓ}}ℤ × ℤ/ℓ^{M_{ℓ}}ℤ × ℤ/ℓ^{m_{ℓ}}ℤ & if & ℓ | (\frac{p−1}{2}, \frac{q−1}{2}) , ℓ ≥ 5, \\ C(pq)_{ℓ} ≅ ℤ/ℓ^{r_{ℓ}}ℤ × ℤ/ℓ^{M_{ℓ}−1}ℤ × ℤ/ℓ^{m_{ℓ}}ℤ & if & ℓ | (\frac{p−1}{2}, \frac{q−1}{2}) , ℓ = 2, 3. \end{matrix}

(ii) For p ≥ 5, the cuspidal subgroup C(3p) of J_{0}(3p) is given by:

C(3p) ≅ \{ \begin{matrix}ℤ/\frac{p−1}{3}ℤ × ℤ/\frac{p^{2}−1}{12}ℤ & if & p ≡ 1 (mod 3), \\ ℤ/(p − 1)ℤ × ℤ/\frac{p^{2}−1}{12}ℤ & if & p ≡ 2 (mod 3). \end{matrix}

(iii) For p ≥ 5, the cuspidal subgroup C(2p) of J_{0}(2p) is given by:

C(2p) ≅ \{ \begin{matrix}ℤ/\frac{p−1}{(p−1,8)}ℤ × ℤ/\frac{p^{2}−1}{24}ℤ & if & p ≢ 1 (mod 8), \\ ℤ/\frac{p−1}{4}ℤ × ℤ/\frac{p^{2}−1}{24}ℤ & if & p ≡ 1 (mod 8). \end{matrix}

and C(6) = 0.

After some general discussion on the generators of C(pq) and establishing some relations among these generators, we prove Theorem 1 (i) in Section 3.3. The proofs of Theorem 1 (ii) and (iii) are left to the reader.

Recall that there are degeneracy maps v_{1}, v_{q} : X_{0}(pq) → X_{0}(p) and v_{1}, v_{p} : X_{0}(pq) → X_{0}(q). They induce via Pic functoriality the maps v_{1}^{∗}, v_{q}^{∗} : J_{0}(p) → J_{0}(pq) and v_{1}^{∗}, v_{p}^{∗} : J_{0}(q) → J_{0}(pq), and via Albanese functoriality the maps (v_{1})_{∗}, (v_{q})_{∗} : J_{0}(pq) → J_{0}(p) and (v_{1})_{∗}, (v_{p})_{∗} : J_{0}(pq) → J_{0}(q). Let γ be the map

γ = (v_{1}^{∗} × v_{q}^{∗}) × (v_{1}^{∗} × v_{p}^{∗}) : J_{0}(p)^{2} × J_{0}(q)^{2} → J_{0}(pq). (1)

The image of γ is the old subvariety of J_{0}(pq), which we denote by J_{old}. The group of ℚ-rational torsion points of this old subvariety is denoted by J_{old}(ℚ)_{tor}. We prove in Section 4

Theorem 2. Let ℓ be an odd prime not dividing (\frac{p−1}{(p−1,24)}, \frac{q−1}{(q−1,24)}); then the re-

striction of γ to C(p)_{ℓ}^{2} × C(q)_{ℓ}^{2} defines an isomorphism

C(p)_{ℓ}^{2} × C(q)_{ℓ}^{2} ≅ (J_{old}(ℚ)_{tor})_{ℓ}.

Theorems 1 and 2 can be used to refine some results of Berkovič [1] on the nontriviality of the Mordell-Weil group of some Eisenstein factors of J_{0}(pq).

2. The Dedekind η-functions and cuspidal divisors

2.1. The Dedekind η-functions. Let N be a positive integer, and let δ denote a positive divisor of N. Let r = (r_{δ}) be a family of rational numbers r_{δ} ∈ ℚ indexed by all the positive divisors δ of N. Let

g_{r} = ∏_{δ|N} η_{δ}^{r_{δ}} (2)

be made up from the Dedekind η-functions, where η_{δ}(z) =^{def} η(δz). We recall without proof the following proposition (cf. [2, Proposition 3.2.1] or [3, Proposition 1]):

RATIONAL CUSPIDAL SUBGROUP AND TORSION POINTS OF J_{0}(pq) 2257

Proposition 1. The function g_{r} in (2) is a modular function on the modular curve X_{0}(N) if and only if the following conditions are satisfied:

(0) all the r_{δ} are rational integers;

(1) ∑_{δ|N}r_{δ}δ ≡ 0 (mod 24);

(2) ∑_{δ|N}r_{δ} \frac{N}{δ} ≡ 0 (mod 24);

(3) ∑_{δ|N}r_{δ} = 0;

(4) ∏_{δ|N}δ^{r_{δ}} is the square of a rational number.

By applying [4, Proposition 2] to the case N = pq, we have a bijection Λ from

{∏_{δ|pq}η_{δ}^{r_{δ}} : r_{δ} ∈ ℚ and ∑_{δ|pq}r_{δ} = 0} to {∑_{d|pq}m_{d}P_{d} : m_{d} ∈ ℚ and ∑_{d|pq}m_{d} = 0}. Iden-

tifying the first set with

\{( \begin{matrix}r_{1} \\ r_{p} \\ r_{q} \\ r_{pq} \end{matrix} ) ∈ ℚ^{4} : r_{1} + r_{p} + r_{q} + r_{pq} = 0\}

and identifying the second set with

\{( \begin{matrix}m_{1} \\ m_{p} \\ m_{q} \\ m_{pq} \end{matrix} ) ∈ ℚ^{4} : m_{1} + m_{p} + m_{q} + m_{pq} = 0\} ,

the map Λ may be represented as a matrix

Λ = \frac{1}{24} ( \begin{matrix}pq & q & p & 1 \\ q & pq & 1 & p \\ p & 1 & pq & q \\ 1 & p & q & pq \end{matrix} ) ,

so

Λ^{−1} = \frac{24}{(p^{2} − 1)(q^{2} − 1)} ( \begin{matrix}pq & −q & −p & 1 \\ −q & pq & 1 & −p \\ −p & 1 & pq & −q \\ 1 & −p & −q & pq \end{matrix} ) .

2.2. Orders of cuspidal divisors. Using the identification above, we have

C_{p} = ( \begin{matrix}1 \\ −1 \\ 0 \\ 0 \end{matrix} ) and Λ^{−1}C_{p} = \frac{24}{(p − 1)(q^{2} − 1)} ( \begin{matrix}q \\ −q \\ −1 \\ 1 \end{matrix} );

C_{q} = ( \begin{matrix}1 \\ 0 \\ −1 \\ 0 \end{matrix} ) and Λ^{−1}C_{q} = \frac{24}{(p^{2} − 1)(q − 1)} ( \begin{matrix}p \\ −1 \\ −p \\ 1 \end{matrix} );

C_{pq} = ( \begin{matrix}1 \\ 0 \\ 0 \\ −1 \end{matrix} ) and Λ^{−1}C_{pq} = \frac{24}{(p^{2} − 1)(q^{2} − 1)} ( \begin{matrix}pq − 1 \\ p − q \\ q − p \\ 1 − pq \end{matrix} ) .

Proposition 1 can then be used to determine the orders of the divisor classes \overline{C}_{p}, \overline{C}_{q} and \overline{C}_{pq}. We summarize the results of our computations as follows:

order of \overline{C}_{p}  order of \overline{C}_{q}  order of \overline{C}_{pq}

\begin{matrix}q = 2 \\ p ≥ 5 \end{matrix} \frac{p−1}{(p−1,8)} if p ≢ 1 (mod 8)

\frac{p−1}{4}  if p ≡ 1 (mod 8)

\frac{p2−1}{24} \frac{p2−1}{8} if p ≡ 1 (mod 3)

\frac{p2−1}{24} if p ≡ 2 (mod 3)

\begin{matrix}q = 3 \\ p ≥ 5 \end{matrix} \frac{p−1}{3}  if p ≡ 1 (mod 3)

p − 1 if p ≡ 2 (mod 3)

\frac{p2−1}{12} \frac{p2−1}{12} if p ≡ 3 (mod 4)

\frac{p2−1}{6} if p ≡ 1 (mod 4)
\begin{matrix}q ≥ 5 \\ p ≥ 5 \end{matrix}  \frac{(q2−1)(p−1)}{24}  \frac{(p2−1)(q−1)}{24}  \frac{(p2−1)(q2−1)}{m}

where m =^{def} ((p^{2} −1)(q^{2} −1), 24(pq −1), 24(p−q), 12(p−1)(q +1), 12(q −1)(p+1)).

3. Relations among the generators \overline{C}_{p}, \overline{C}_{q}, \overline{C}_{pq}

3.1. The general strategy. To establish relations among the generators \overline{C}_{i} of C(pq) we first apply the maps v_{1}^{∗}, v_{q}^{∗} : J_{0}(p) → J_{0}(pq) and v_{1}^{∗}, v_{p}^{∗} : J_{0}(q) → J_{0}(pq) to translate any known relations in J_{0}(p) and J_{0}(q) to some relations in J_{0}(pq). Next, we note that a relation ∑ λ_{i}\overline{C}_{i} = 0 exists if and only if Λ^{−1}(∑ λ_{i}C_{i}) satisfies the conditions of Proposition 1. Using this criterion, one can easily check the validity of any relation. In addition, we can simplify the checking process by applying the maps (v_{1})_{∗}, (v_{q})_{∗} : J_{0}(pq) → J_{0}(p) (resp. (v_{1})_{∗}, (v_{p})_{∗} : J_{0}(pq) → J_{0}(q)) to translate a relation in J_{0}(pq) to a relation in J_{0}(p) (resp. J_{0}(q)).

In this section, we will apply this strategy to establish the relations among the generators \overline{C}_{p}, \overline{C}_{q}, \overline{C}_{pq}. Note that a similar strategy has been used in [4, Section 2] to establish the relations among the generators of C(p^{r}), where p ≥ 5 is a prime. Consequently, we will only state the key lemmas in Section 3.2 since most of the details are similar to those in [4, Section 2].

3.2. The case p, q ≥ 5. We now determine the relations involving the generators \overline{C}_{p}, \overline{C}_{q} and \overline{C}_{pq} of C(pq) by finding such relations at each prime ℓ. For each prime ℓ, we may write

\frac{(q^{2} − 1)(p − 1)}{24} = ℓ^{r_{ℓ}}ℓ_{1}, \frac{(p^{2} − 1)(q − 1)}{24} = ℓ^{s_{ℓ}}ℓ_{2}, \frac{(q^{2} − 1)(p^{2} − 1)}{m} = ℓ^{t_{ℓ}}ℓ_{3} (3)

where ℓ |̸ℓ_{1}, ℓ |̸ℓ_{2}, ℓ |̸ℓ_{3}. We also set

C_{p,ℓ} = ℓ_{1}ℓ_{2}ℓ_{3}C_{p}, C_{q,ℓ} = ℓ_{1}ℓ_{2}ℓ_{3}C_{q}, C_{pq,ℓ} = ℓ_{1}ℓ_{2}ℓ_{3}C_{pq}.

Since (ℓ_{1}ℓ_{2}, ℓ) = (ℓ_{2}ℓ_{3}, ℓ) = (ℓ_{1}ℓ_{3}, ℓ) = 1, it is clear that \overline{C}_{p,ℓ}, \overline{C}_{q,ℓ} and \overline{C}_{pq,ℓ} generate C(pq)_{ℓ}. By applying the maps v_{1}^{∗}, v_{q}^{∗} : J_{0}(p) → J_{0}(pq) and v_{1}^{∗}, v_{p}^{∗} : J_{0}(q) → J_{0}(pq) as described in Section 3.1, we get

Lemma 1. (i) If ℓ does not divide (\frac{q−1}{(q−1,12)}, \frac{p−1}{(p−1,12)}), then C(pq)_{ℓ} is generated by

\overline{C}_{p,ℓ} and \overline{C}_{q,ℓ}.

(ii) If ℓ divides (\frac{q−1}{(q−1,12)}, \frac{p−1}{(p−1,12)}), then ℓ^{m_{ℓ}} \overline{C}_{pq,ℓ} = α\overline{C}_{p,ℓ}+β\overline{C}_{q,ℓ} for some α, β.

Furthermore, by applying the maps (v_{1})_{∗}, (v_{q})_{∗}: J_{0}(pq) → J_{0}(p) and (v_{1})_{∗}, (v_{p})_{∗}: J_{0}(pq) → J_{0}(q) together with the criterion stated in Section 3.1, we have

Lemma 2. There is no non-trivial relation

ν_{ℓ}\overline{C}_{pq,ℓ} = λ_{ℓ}\overline{C}_{p,ℓ} + μ_{ℓ}\overline{C}_{q,ℓ} (4)

with 0 ≤ λ_{ℓ} < ℓ^{r_{ℓ}}, 0 ≤ μ_{ℓ} < ℓ^{s_{ℓ}} and 0 ≤ ν_{ℓ} < ℓ^{m_{ℓ}}.

To help the reader in reconstructing the proof, we include the following formulas:

v_{1}^{∗}(C_{p}) = −C_{q} + qC_{p} + C_{pq},
v_{q}^{∗}(C_{p}) = −qC_{q} + C_{p} + qC_{pq},
(v_{1})_{∗}(C_{p}) = C_{p},
(v_{1})_{∗}(C_{q}) = 0,
(v_{1})_{∗}(C_{pq}) = C_{p}.

In addition, the two maps v_{1} and v_{q} are related by the formula v_{1} = v_{q} ∘ w_{q} where w_{q} is an Atkin-Lehner involution (see the remark after Theorem 3).

From Lemma 1 (ii) and Lemma 2, if there is any relation

ν_{ℓ}\overline{C}_{pq,ℓ} = λ_{ℓ}\overline{C}_{p,ℓ} + μ_{ℓ}\overline{C}_{q,ℓ}

relating \overline{C}_{pq,ℓ}, \overline{C}_{p,ℓ} and \overline{C}_{q,ℓ}, we may assume v_{ℓ}(ν_{ℓ}) ≥ m_{ℓ}. Together with Lemma 1 (ii), this relation can be reduced to

0 = λ_{ℓ}\overline{C}_{p,ℓ} + μ_{ℓ}\overline{C}_{q,ℓ}. (5)

Therefore, we shall next study relations of this type.

Lemma 3. Suppose (5) is satisfied.

(i) If ℓ |̸(p − 1, q − 1) (hence ℓ ≠ 2), then λ_{ℓ} = μ_{ℓ} = 0.

(ii) If ℓ |(p − 1, q − 1) and ℓ ≠ 2, then λ_{ℓ} and μ_{ℓ} satisfy

λ_{ℓ}ℓ_{2}q ≡ −μ_{ℓ}ℓ_{1} ≡ λ_{ℓ}ℓ_{2} (mod ℓ^{r_{ℓ}}) (ℓ ≥ 3)

and λ_{ℓ} ≡ μ_{ℓ} ≡ 0 (mod ℓ^{M_{ℓ}}) (ℓ ≥ 5) or λ_{3} ≡ μ_{3} ≡ 0 (mod 3^{M_{3}−1}).

(iii) If at least one of p, q ≡ 3 (mod 4), then λ_{2} = μ_{2} = 0.

(iv) If p ≡ q ≡ 1 (mod 4), then λ_{2} and μ_{2} satisfy

λ_{2}ℓ_{2}q ≡ −μ_{2}ℓ_{1} ≡ λ_{2}ℓ_{2} (mod 2^{r_{2}}) and λ_{2} ≡ μ_{2} ≡ 0 (mod 2^{M_{2}−1}).

3.3. Proof of Theorem 1 (i). To prove Theorem 1 (i), note that if ℓ does not divide (\frac{p−1}{2}, \frac{q−1}{2}), then ℓ does not divide (\frac{p−1}{(p−1,12)}, \frac{q−1}{(q−1,12)}). Therefore C(pq)_{ℓ} is generated by \overline{C}_{p,ℓ} and \overline{C}_{q,ℓ}. By Lemma 3 (i) and (iii), there is no relation between \overline{C}_{p,ℓ} and \overline{C}_{q,ℓ} in this case. The orders of \overline{C}_{p,ℓ} and \overline{C}_{q,ℓ} are given by (3). In the remaining case, Lemma 1 (ii) and Lemma 2 show that

C(pq)_{ℓ} ≅ ⟨\overline{C}_{p,ℓ}, \overline{C}_{q,ℓ}⟩ × ℤ/ℓ^{m_{ℓ}}ℤ.

If ℓ divides (\frac{p−1}{2}, \frac{q−1}{2}), then ℓ divides (p − 1, q − 1).

If ℓ ≥ 5, Lemma 3 (ii) shows that ⟨\overline{C}_{p,ℓ}, \overline{C}_{q,ℓ}⟩ ≅ ℤ/ℓ^{r_{ℓ}}ℤ × ℤ/ℓ^{M_{ℓ}}ℤ.

If ℓ = 3, Lemma 3 (ii) shows that ⟨\overline{C}_{p,3}, \overline{C}_{q,3}⟩ ≅ ℤ/3^{r_{3}}ℤ × ℤ/3^{M_{3}−1}ℤ.

If ℓ = 2, Lemma 3 (iv) shows that ⟨\overline{C}_{p,2}, \overline{C}_{q,2}⟩ ≅ ℤ/2^{r_{2}}ℤ × ℤ/2^{M_{2}−1}ℤ.

This completes the proof of Theorem 1 (i).

3.4. Additional remarks on Theorem 1. Consider the following cuspidal divisors (cf. [8]):

divisor  order of divisor class
D^{+,−} = P_{1} + P_{p} − P_{q} − P_{pq}  \frac{(p+1)(q−1)}{((p+1)(q−1),24)}
D^{−,+} = P_{1} − P_{p} + P_{q} − P_{pq}  \frac{(p−1)(q+1)}{((p−1)(q+1),24)}
D^{−,−} = P_{1} − P_{p} − P_{q} + P_{pq}  \frac{(p−1)(q−1)}{((p−1)(q−1),24)}

Note that we have the inclusions of groups

⟨2\overline{C}_{p}, 2\overline{C}_{q}, 2\overline{C}_{pq}⟩ ⊆ ⟨\overline{D^{+,−}}, \overline{D^{−,+}}, \overline{D^{−,−}}⟩ ⊆ C(pq) = ⟨\overline{C}_{p}, \overline{C}_{q}, \overline{C}_{pq}⟩.

If we are only interested in the odd part of C(pq), we may as well consider the odd part of ⟨\overline{D^{+,−}}, \overline{D^{−,+}}, \overline{D^{−,−}}⟩. Using techniques employed in the earlier sections, we can show that there are no non-trivial relations among \overline{D^{+,−}}, \overline{D^{−,+}} and \overline{D^{−,−}} (even at the 2-primary part). Therefore the odd part of C(pq) is isomorphic to that of ⟨\overline{D^{+,−}}⟩ × ⟨\overline{D^{−,+}}⟩ × ⟨\overline{D^{−,−}}⟩.

4. Rational torsion points of the old subvariety

The goal of this section is to establish Theorem 2. Let γ be the map in (1), let K denote the kernel of γ and let ℓ be an odd prime. Let Σ(p) (resp. Σ(q)) denote the Shimura subgroup of J_{0}(p) (resp. J_{0}(q)); it is a cyclic group of order \frac{p−1}{(p−1,12)} (resp. \frac{q−1}{(q−1,12)}). Let \overline{C(p)} be the image of the anti-diagonal C(p) in J_{0}(pq), i.e.,

\overline{C(p)} = \{x ∈ J_{0}(pq) : x = (v_{1}^{∗} − v_{q}^{∗})(c) for c ∈ C(p)\},

and \overline{C(q)} is similarly defined. It is shown in [9] that

K_{ℓ} ≅ Σ(p)_{ℓ} × Σ(q)_{ℓ} × (\overline{C(p)}_{ℓ} ∩ \overline{C(q)}_{ℓ}). (6)

Moreover, it is known that the group \overline{C(p)} ∩ \overline{C(q)} has order (\frac{p−1}{(p−1,24)}, \frac{q−1}{(q−1,24)}).

Let P ⊆ J_{0}(p)^{2} × J_{0}(q)^{2} be defined as P = γ^{−1}(J_{old}(ℚ)_{tor}). For each odd prime ℓ, we have an exact sequence of Gal(\overline{ℚ}/ℚ)-modules

0 ⟶ K_{ℓ} ⟶ P_{ℓ} ⟶^{γ} (J_{old}(ℚ)_{tor})_{ℓ} ⟶ 0. (7)

Now let r be a prime distinct from p and q. Let T_{r} be the rth Hecke operator on J_{0}(pq). Since (r, pq) = 1, T_{r} acts on J_{0}(p)^{2} × J_{0}(q)^{2} (hence on P) diagonally, i.e., it acts on J_{0}(p) (resp. J_{0}(q)) like the usual Hecke operator T_{r} in 𝕋_{p} = End_{ℚ}(J_{0}(p)) (resp. 𝕋_{q} = End_{ℚ}(J_{0}(q))) (cf. [9], “Formulaire”).

Since Σ(p) and C(p) (resp. Σ(q) and C(q)) are Eisenstein, i.e., they are annihilated by T_{r} − (1 + r) for all primes r ≠ p (resp. r ≠ q), it follows from the isomorphism (6) that (T_{r} −(1+r))K_{ℓ} = 0 for all primes r ≠ p, q. By considering the action of T_{r} on the Néron model of J_{0}(pq) over ℤ and restricting to characteristic r, the Eichler-Shimura relation yields the equation

T_{r} = Frob_{r} + r/Frob_{r}

on J_{0}(pq)/𝔽_{r} where Frob_{r} denotes the Frobenius at r. This implies, in particular, (T_{r} − (1 + r))(J_{old}(ℚ)_{tor})_{ℓ} = 0. Since T_{r} − (1 + r) kills K_{ℓ} and (J_{old}(ℚ)_{tor})_{ℓ}, and T_{r} − (1 + r) commutes with the action of Gal(\overline{ℚ}/ℚ), T_{r} − (1 + r) induces (using the Snake Lemma) a homomorphism δ_{r} : (J_{old}(ℚ)_{tor})_{ℓ} → K_{ℓ}^{Gal(\overline{ℚ}/ℚ)}. When ℓ does not divide (\frac{p−1}{(p−1,24)}, \frac{q−1}{(q−1,24)}), \overline{C(p)}_{ℓ} ∩ \overline{C(q)}_{ℓ} is trivial, and (6) implies that K_{ℓ}^{Gal(\overline{ℚ}/ℚ)} is trivial. Therefore δ_{r} is the trivial homomorphism. This means that T_{r} − (1 + r) annihilates P_{ℓ}, for all primes r ≠ p, q.

Lemma 4 (cf. [11]). Let S be a set of primes of density one that does not contain p. The ideal in 𝕋_{p} = End_{ℚ}(J(p)) generated by {T_{r}−(1+r) : r ∈ S} is the Eisenstein ideal I_{p}.

Proof. Let I denote the ideal in 𝕋_{p} generated by {T_{r} − (1 + r) : r ∈ S} and let 𝔪 denote a maximal ideal of 𝕋_{p} containing {T_{r}−(1+r) : r ∈ S}. By [10, Theorem 5.2], 𝔪 gives rise to a reducible Galois representation ρ_{𝔪} : Gal(\overline{ℚ}/ℚ) → GL(2, 𝕋_{p}/𝔪) satisfying

Tr(ρ_{𝔪}(Frob_{r})) ≡ T_{r} (mod 𝔪), det(ρ_{𝔪}(Frob_{r})) ≡ r (mod 𝔪)

for almost all r. Proposition 14.1 of [6, Chapter II] shows that 𝔪 is Eisenstein. Hence, if 𝔪 ⊂ 𝕋_{p} is a maximal ideal, 𝔪 ⊃ {T_{r} − (1 + r) : r ∈ S} if and only if 𝔪 ⊃ I_{p}.

To show that I = I_{p}, it suffices to demonstrate that I_{𝔪} = I_{p,𝔪} (in 𝕋_{𝔪}) for all Eisenstein primes 𝔪. By [6, Chapter II, Proposition 16.6], the ideal I_{p,𝔪} is a principal ideal in 𝕋_{p,𝔪}, generated by T_{r} − (1 + r) for any “good” prime r. (For the definition of a “good” prime, see [6, Chapter II, Section 16].) The set of “good” primes has Dirichlet density \frac{r−2}{r−1} if r > 2, and \frac{1}{4} if r = 2. In particular, there is a good prime r_{0} in S. Since I_{𝔪} ⊂ I_{p,𝔪} evidently, T_{r_{0}} − (1 + r_{0}) ∈ I shows that I_{p,𝔪} = I_{𝔪}.

Take S to be the set of all primes except p and q, and apply Lemma 4. It follows that P_{ℓ} is Eisenstein, i.e., P_{ℓ} ⊆ J_{0}(p)[I_{p}]_{ℓ}^{2} × J_{0}(q)[I_{q}]_{ℓ}^{2}. In other words (cf. [6, Chapter II, Corollary 16.4]),

P_{ℓ} ⊆ (Σ(p)_{ℓ} ⊕ C(p)_{ℓ})^{2} × (Σ(q)_{ℓ} ⊕ C(q)_{ℓ})^{2}.

Let (σ_{1} + c_{1}, σ_{2} + c_{2}, σ_{3} + c_{3}, σ_{4} + c_{4}) ∈ P_{ℓ} where σ_{1}, σ_{2} ∈ Σ(p), σ_{3}, σ_{4} ∈ Σ(q), and c_{1}, c_{2} ∈ C(p), c_{3}, c_{4} ∈ C(q). Since C(p) and C(q) consist only of ℚ-rational torsion points of J_{0}(p) and J_{0}(q), respectively, we have (σ_{1}, σ_{2}, σ_{3}, σ_{4}) ∈ P_{ℓ}. For any τ ∈ Gal(\overline{ℚ}/ℚ), (σ_{1}^{τ} , σ_{2}^{τ} , σ_{3}^{τ} , σ_{4}^{τ}) ∈ P_{ℓ} and γ(σ_{1}^{τ} , σ_{2}^{τ} , σ_{3}^{τ} , σ_{4}^{τ}) = γ(σ_{1}, σ_{2}, σ_{3}, σ_{4}). Therefore,

(σ_{1}^{τ} − σ_{1}, σ_{2}^{τ} − σ_{2}, σ_{3}^{τ} − σ_{3}, σ_{4}^{τ} − σ_{4}) ∈ K_{ℓ} ≅ Σ(p)_{ℓ} × Σ(q)_{ℓ}

since we are still in the case ℓ |̸ (\frac{p−1}{(p−1,24)}, \frac{q−1}{(q−1,24)}). It follows that (cf. [3, Theorem 1]) σ_{1}^{τ} − σ_{1} = −(σ_{2}^{τ} − σ_{2}) and σ_{3}^{τ} − σ_{3} = −(σ_{4}^{τ} − σ_{4}), i.e., σ_{1} + σ_{2} ∈ Σ(p)_{ℓ}(ℚ) and σ_{3} + σ_{4} ∈ Σ(q)_{ℓ}(ℚ). Therefore, we obtain σ_{1} = −σ_{2} and σ_{3} = −σ_{4}, implying that (σ_{1}, σ_{2}, σ_{3}, σ_{4}) ∈ K_{ℓ}. This show that P_{ℓ} = K_{ℓ} ⊕ (C(p)_{ℓ}^{2} × C(q)_{ℓ}^{2}). Putting this into (7) shows that γ induces an isomorphism

C(p)_{ℓ}^{2} × C(q)_{ℓ}^{2} ≅ (J_{old}(ℚ)_{tor})_{ℓ}.

This completes the proof of Theorem 2.

5. A remark on the nontriviality of certain Eisenstein factors of J_{0}(pq)

Let J^{new} =^{def} J_{0}(pq)/J_{old} be the new quotient of J_{0}(pq). Let 𝕋 be the subring of End_{ℚ}(J^{new}) generated over ℤ by the Hecke operators. The Eisenstein ideal I_{pq} is the ideal in 𝕋 generated by T_{r} − (1 + r), for all primes r ≠ p, q. It is of finite index in 𝕋.

Let Λ be a non-trivial ideal of 𝕋 containing I_{pq} and a prime number ℓ. Let a_{Λ} = ∩_{n=1}^{∞}Λ^{n}. The Eisenstein factor J^{(Λ)} of J^{new} associated with Λ is defined as

J^{(Λ)} =^{def} J^{new}/a_{Λ}J^{new}.

In [1], the following theorem is proved:

Theorem 3 (cf. [1, Theorem 4]). Let p and q be distinct prime numbers, ℓ an odd prime with ℓ |(p + 1) but ℓ |̸(q − 1) if ℓ > 3, and 9 |(p + 1)(q − 1) but 9 |̸(q − 1) if ℓ = 3. Then the ideal Λ = ⟨I_{pq}, ℓ, 1 − W_{p}⟩ is a proper ideal of 𝕋, and the group J^{(Λ)}(ℚ) is finite.

Remark. Here, W_{p} is induced by the Atkin-Lehner involution on X_{0}(pq), which is derived from the transformation τ ↦ gτ of the Poincaré upper half plane, where

g = ( \begin{matrix}A & B \\ C & D \end{matrix} ) is any matrix in M_{2}(ℤ) such that p |A, p |D, pq |C and det(g) = p.

A key step in the proof of Theorem 3 in [1] is to produce a point of order ℓ in J^{new}(ℚ) which is annihilated by Λ. Given the conditions on ℓ in Theorem 3, by proving that J_{old} has no ℚ-rational point of order ℓ, Berkovič showed that the image of the divisor class of D^{+,−} = P_{1} + P_{p} − P_{q} − P_{pq} in J^{new}(ℚ) has order divisible by ℓ. A suitable multiple of \overline{D^{+,−}} would produce the point of J^{new}(ℚ) with the desired properties. The constraint ℓ |̸(q − 1) if ℓ > 3 and 9 |̸(q − 1) if ℓ = 3 are necessary for the proof used in [1], because it depends on the following fact: the Galois module J_{0}(p)(\overline{ℚ})_{ℓ} has a Jordan-Holder factor isomorphic to ℤ/ℓℤ or μ_{ℓ} if and only if ℓ |\frac{p−1}{(p−1,12)}. A careful application of Theorems 1 and 2 above allows us to relax some of the constraints on ℓ in Theorem 3.

Theorem 4. Let p and q be distinct prime numbers, ℓ an odd prime with ℓ |(p + 1) if ℓ > 3 and 9 |(p+1)(q−1) and 3 |(p+1) if ℓ = 3. Then the ideal Λ = ⟨I_{pq}, ℓ, 1−W_{p}⟩ is a proper ideal of 𝕋 and the group J^{(Λ)}(ℚ) is finite.

The (relaxed) constraints on ℓ in Theorem 4 imply that ℓ satisfies the conditions in Theorem 2, so (J_{old}(ℚ)_{tor})_{ℓ} is known. It is easy to see that (J_{old}(ℚ)_{tor})_{ℓ} is strictly smaller than C(pq)_{ℓ}, so an element in its complement in the latter yields an element of J^{new}(ℚ) of the desired properties. The details of the proof are left to the reader.

References

1. V. G. Berkovič, The rational points on the Jacobian of modular curves., Math. USSR Sbornik 30 (1976), AMS Translations 478–500.

2. G. Ligozat, Courbes modulaires de genre 1, Bull. Soc. Math. France Mémoire 43 (1975), 5–80. MR 54:5121

3. S. Ling, The old subvariety of J_{0}(pq) and the Eisenstein kernel in Jacobians, Israel J. of Math 84 (1993), 365–384. MR 94h:11047

4. , On the ℚ-rational cuspidal subgroup and the component group of J_{0}(p^{r}), 1996, To appear in Israel J. of Math.

5. Ju. Manin, Parabolic points and zeta function of modular curves., Izv. Akad. Nauk. SSSR 6 (1972), AMS Translations 19–64. MR 47:3396

6. B. Mazur, Modular curves and the Eisenstein ideal., Pub. Math. I.H.E.S. 47 (1978), 33–186.

7. A. P. Ogg, Rational points on certain elliptic modular curves, Proc. Sym. Pure Math, vol. 20, AMS, 1973, pp. 221–231. MR 49:2743

8. , Hyperelliptic modular curves, Bull. Soc. Math. France 102 (1974), 449–462. MR 51:514

9. K. Ribet, The old subvariety of J_{0}(pq), Arithmetic Algebraic Geometry, Birkhäuser, 1989, pp. 293–307. MR 92a:11069

10. , On modular representations of Gal(\overline{ℚ}/ℚ) arising from modular forms, Invent. Math. 100 (1991), 431–476. MR 91g:11066

11. , June 1994, Private communication.

Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Republic of Singapore

  E-mail address: matchua@nus.sg
  E-mail address: matlings@nus.sg