Multiplicities of p-finite mod p Galois representations in J_{o}(Np)

Kenneth A. Ribet

Department of Mathematics University of California Berkeley, CA 94720 USA

This article was prepared while the author was visiting the I.H.E.S., and is based partially on the author’s lecture at I.M.P.A. during an algebraic geometry workshop in April, 1990. It is a pleasure to thank both institutes for their hospitality, as well as Professor B. Mazur for helpful suggestions. The author was partially supported by NSF contract DMS 88-06815.

1. Introduction. Let M ≥ 1 be an integer. Let J_{o}(M) be the Jacobian Pic^{o}(X_{o}(M)) of the modular curve X_{o}(M)_{Q}. For all n ≥ 1, let T_{n} be the n^{th} Hecke correspondence on X_{o}(M). Write again T_{n} for the endomorphism T_{n}^{∗} of J_{o}(M) induced by the correspondence T_{n}. Let T_{M} = Z[…, T_{n}, …] be the subring of End(J_{o}(M)) generated by the T_{n} with n ≥ 1. The ring T_{M} may be identified with the ring generated by the Hecke operators T_{n} acting on the complex vector space of weight-2 cusp forms on Γ_{o}(M).

Suppose that 𝔭 is a maximal ideal of T_{M}. The residue field T_{M}/𝔭 is a finite field k, whose characteristic will be denoted p. Attached to 𝔭 is a semisimple continuous representation

ρ_{𝔭}: Gal(\overline{Q}/Q) → GL(2, k),

unramified outside Mp, with the property that ρ_{𝔭}(φ_{r}) has characteristic polynomial X^{2} − T_{r}X + r (mod 𝔭) for each prime number r prime to Mp. (Here, φ_{r} denotes a Frobenius element of Gal(\overline{Q}/Q) for the prime number r.) The representation ρ_{𝔭} is well defined up to isomorphism.

Consider the group of p-division points

J_{o}(M)[𝔭] := \{ x ∈ J_{o}(M)(\overline{Q}) | λx = 0 for all λ ∈ 𝔭 \},

i.e., the “kernel of 𝔭 on J_{o}(M).” Assume: i. The representation ρ_{𝔭} is irreducible over k. ii. The prime number p is odd.

Then ρ_{𝔭} is an absolutely irreducible representation of Gal(\overline{Q}/Q), and a theorem of Boston, Lenstra, and the author [1] shows that J_{o}(M)[𝔭] is a direct sum of representations isomorphic to ρ_{𝔭}. The associated multiplicity μ_{𝔭} is the number of summands in the direct sum:

μ_{𝔭} = \frac{1}{2} dim_{k} J_{o}(M)[𝔭].

Since J_{o}(M)[𝔭] is non-zero, μ_{𝔭} is a positive integer. Thus J_{o}(M)[𝔭] is “multiplicity free” (in the sense that μ_{𝔭} ≤ 1) if and only if we have μ_{𝔭} = 1.

Using ideas of Mazur [4], the author proved that μ_{𝔭} = 1 whenever M is prime to p ([10], Theorem 5.2). In [5], Mazur and the author considered the case where p exactly divides M, i.e., where M = Np for some integer N ≥ 1 which is prime to p. The principal result of that article states that μ_{𝔭} = 1 provided that ρ_{𝔭} is not finite at p (in the sense of Serre [12], p. 189), or equivalently ([10], Theorem 6.1) provided that ρ_{𝔭} is not modular of level N. These results were motivated by applications to Serre’s conjectures [12], and specifically are used in showing that certain mod p Galois representations which are known to be modular of some level are in fact modular of the conjectured level (see [10]).

In the present article, we explore the situation when p exactly divides M but ρ_{𝔭} is allowed to be finite. Although this situation does not appear in applications to Serre’s conjectures, the author hopes that the present paper will shed light on the general problem of computing μ_{𝔭}.

In order to state our results, we introduce the p-old and p-new subvarieties A and B of J_{o}(M). We assume that M = Np, and that p is prime to N. Recall [8] that the abelian varieties J_{o}(N) and J_{o}(Np) are connected by a natural “degeneracy map”

d: J_{o}(N) × J_{o}(N) ⟶ J_{o}(Np),

whose kernel Σ is the finite group consisting of all pairs (σ, −σ), where σ runs over the Shimura subgroup of J_{o}(N), i.e., the kernel of the map J_{o}(N) → J_{1}(N) induced by the standard covering X_{1}(N) → X_{o}(N). The p-old subvariety of J_{o}(Np) is the image A of d. The p-new subvariety of J_{o}(Np) is the “orthogonal complement” B to A under the natural autoduality J_{o}(Np) ˜→ J_{o}(Np)^{∨}. In other words, the image of B in J_{o}(Np)^{∨} under the isomorphism J_{o}(Np) ˜→ J_{o}(Np)^{∨} is the kernel of the map J_{o}(Np)^{∨} → A^{∨} which is dual to the inclusion of A in J_{o}(Np). One has J_{o}(Np) = A + B, and the group Δ := A ∩ B is finite; this group was calculated in [8].

In the following discussion, we will write simply J for J_{o}(Np). Also, we will let A[𝔭] and B[𝔭] be the analogues of J_{o}(Np)[𝔭] for A and B, respectively. When (i) and (ii) are satisfied, A[𝔭] and B[𝔭] are each direct sums of representations isomorphic to ρ_{𝔭}. We say that A[𝔭] or B[𝔭] is multiplicity free if the corresponding direct sum has at most one term. Thus B[𝔭], for example, is multiplicity free if it is either zero, or else of dimension two over k. Finally, we write J_{o}(N)^{2} for the product J_{o}(N) × J_{o}(N).

Assume that the above hypotheses (i) and (ii) are satisfied.

Proposition 1. The representation ρ_{𝔭} is finite at p if and only if A[𝔭] is non-zero.

Theorem 1. If A[𝔭] = 0, then μ_{𝔭} = 1.

Proposition 1 results easily from known facts; its proof will be given in §2. Using the Proposition, we see immediately that Theorem 1 is a restatement of the result of [5] which was introduced above.

In this article, we shall prove the following complement:

Theorem 2. Suppose that (i) and (ii) are satisfied. Then the group J[𝔭] is multiplicity free if and only if the group B[𝔭] is multiplicity free.

As mentioned above, the condition that J[𝔭] be multiplicity free is equivalent to the equality μ_{𝔭} = 1. Hence Theorem 2 provides a means of verifying this equality even when the condition A[𝔭] = 0 of Theorem 1 is not necessarily satisfied.

Corollary. Suppose that (i) and (ii) are satisfied and that B is an elliptic curve. Then J[𝔭] is multiplicity free.

The Corollary follows directly from the Theorem. Indeed, suppose that B is an elliptic curve. If B[𝔭] is non-zero, then 𝔭 ⊂ T_{pN} generates a non-trivial ideal in the image of T_{pN} in End B. Since this image is isomorphic to Z, we have B[𝔭] = B[p], and the residue field of 𝔭 is the prime field F_{p}. Hence B[𝔭] is either 0, or else of dimension two over F_{p}. In particular, B[𝔭] is multiplicity free.

Theorem 2 is obtained as a consequence of two results proved below. Namely, in §2, we prove, under the hypotheses of Theorem 2, that A[𝔭] is multiplicity free (Theorem 3). In §3, we show that B[𝔭] = J[𝔭] whenever B[𝔭] is non-zero (Theorem 4). These together imply Theorem 2 by an elementary argument, in which we consider separately the case where B[𝔭] is 0 and the case where it is non-zero. Indeed, suppose first that B[𝔭] = 0. Then B[𝔭] is certainly multiplicity free, and Theorem 2 states that J[𝔭] is multiplicity free. However, Δ[𝔭] = 0, from which it follows that the p-divisible group ∪_{i}J[𝔭^{i}] is the direct sum of the analogous p-divisible groups for A and B. This gives, in particular, J[𝔭] = A[𝔭] ⊕ B[𝔭], so that J[𝔭] = A[𝔭], because B[𝔭] is 0. Theorem 3 then implies that J[𝔭] is multiplicity free, as desired. On the other hand, if B[𝔭] is non-zero, then Theorem 4 states that J[𝔭] and B[𝔭] are equal, so that one is multiplicity free if and only if the other is.

It is perhaps worth stressing that our results prove that μ_{𝔭} = 1 whenever one of A[𝔭] and B[𝔭] vanishes and the hypotheses (i) and (ii) are satisfied. Indeed, suppose this to be the case, and assume first that A[𝔭] vanishes. Then an argument analogous to that just given shows that J[𝔭] = B[𝔭], so that Theorem 2 is trivially true (as is Theorem 4). By Theorem 1, we have μ_{𝔭} = 1; i.e., J[𝔭] = B[𝔭] is multiplicity free. Next, assume that B[𝔭] vanishes. Then, as noted above, we have A[𝔭] = J[𝔭]. Also, A[𝔭] is multiplicity free by Theorem 3. Hence μ_{𝔭} = 1 in this case as well.

Assume now that both A[𝔭] and B[𝔭] are non-zero, and that (i) and (ii) are satisfied. Then our results give the relations

0 ⊂ A[𝔭] ⊆ B[𝔭] = J[𝔭],

and show that A[𝔭] is multiplicity free. We have μ_{𝔭} = 1 in this situation if and only if all three groups A[𝔭], B[𝔭] and J[𝔭] coincide. By the Corollary above, these groups do in fact coincide if B is an elliptic curve, although perhaps not in general. (The author knows of no example where they fail to coincide.)

We close this Introduction with a numerical example. Take N = 11 and p = 3. The dimensions of J_{o}(11) and J_{o}(33) are respectively 1 and 3. Thus, A is isogenous to a product of two copies of the elliptic curve J_{o}(11), while B is an elliptic curve of conductor 33. By the Corollary to Theorem 2, J[𝔭] is multiplicity free for each prime 𝔭 of T_{33} which divides 3.

One can verify that there are in fact two 𝔭’s, and that ρ_{𝔭} is finite for each 𝔭. (In other words, Theorem 1 applies for neither of them.) To describe the two 𝔭, we let E = J_{o}(11) and use the degeneracy map d to identify A[3] with the product E[3] × E[3]. The Hecke operator T_{3} of J induces the operator (x, y) ↦ (−x, −x) on E[3] × E[3]. Accordingly, the diagonal image V_{1} of E[3] in E[3] × E[3] is T_{33}-stable, and T_{33} acts on V_{1} via a homomorphism ϕ_{1} : T_{33} → F_{3} which maps T_{3} to −1. If 𝔭_{1} = ker ϕ_{1}, then Theorem 4 implies that B[𝔭_{1}] = J[𝔭_{1}] = A[𝔭_{1}] = V. Similarly, if V_{2} is the subspace 0 ⊕ E[3] of A[3], then T_{33} acts on V_{2} via a homomorphism ϕ_{2} : T_{33} → F_{3} which maps T_{3} to 0. If 𝔭_{2} = ker ϕ_{2}, then B[𝔭_{2}] = 0, and we have J[𝔭_{2}] = A[𝔭_{2}] = V′.

2. Study of A[𝔭]. Let p be an odd prime, and suppose that N is a positive integer prime to p. Write T for the ring T_{Np}. The p-old subvariety A of J_{o}(pN) is T-stable. We define the p-old quotient of T to be the image T_{A} of T in End(A). Similarly, we define a maximal ideal 𝔭 of T to be p-old if and only if it arises by pullback from a maximal ideal of T_{A}. It is clear that 𝔭 is p-old if and only if A[𝔭] is non-zero. Thus, Proposition 1 states, for 𝔭|p such that ρ_{𝔭} is irreducible, that ρ_{𝔭} is finite at p if and only 𝔭 is p-old.

We now prove this Proposition. Assume, first, that ρ_{𝔭} is finite at p. Theorem 6.1 of [10] then states that ρ_{𝔭} is “modular of level N.” (The hypotheses of that theorem are satisfied because the residue characteristic of 𝔭, which is p, is not congruent to 1 mod p.) In fact, the argument given in [10] proves the apparently stronger fact that 𝔭 is p-old. Namely, under the finiteness assumption, the author constructs a certain subgroup V of the fiber at p of the Néron model for J_{o}(Np). Write J_{F_{p}} for this fiber. The discussion on pp. 470–471 of [10] shows that V cannot be a subgroup of the maximal torus T of J_{F_{p}}. Furthermore, the argument given to prove Lemma 6.3 of [10] shows then that V must map non-trivially to the product J_{o}(N)_{F_{p}} × J_{o}(N)_{F_{p}}, which is the maximal abelian variety quotient of the connected component of the identity in J_{F_{p}}. Since V is annihilated by 𝔭, it follows from [10], Theorem 3.11 that 𝔭 is p-old.

Suppose, conversely, that 𝔭 is p-old. Then A[𝔭] is non-zero. Since A[𝔭] isomorphic to a direct sum of copies of the representation ρ_{𝔭}, we may choose a two-dimensional Gal(\overline{Q}/Q)-stable submodule V of A[𝔭] which is a model for the representation ρ_{𝔭}. To show that ρ_{𝔭} is finite, we must produce a finite flat T/𝔭-vector space scheme V over Z_{p} such that V(\overline{Q}_{p}) and V are isomorphic representations of Gal(\overline{Q}_{p}/Q_{p}). (We identify this latter Galois group with a decomposition group for p in Gal(\overline{Q}/Q).) One has inclusions of Gal(\overline{Q}_{p}/Q_{p})-modules V ⊆ A[𝔭] ⊆ A[p], and the latter group extends to a finite flat group scheme S of type (p, …, p) over Z_{p} because A has good reduction at p. Hence V extends to a finite flat group scheme V of this type: the Zariski closure of V in S. Because p > 2, the action of T/𝔭 on V extends uniquely to V in view of Raynaud [6], 3.3.6. This completes the proof of Proposition 1.

Consider the subring R of T_{N} which is generated by all T_{n} with n prime to p. For each maximal ideal 𝔪 of R, we denote by ρ_{𝔪} the semisimple two-dimensional representation of Gal(\overline{Q}/Q) over R/𝔪 which is characterized by the property analogous to that given above for the ρ_{𝔭}.

Proposition 2. The index of R in T_{N} is finite and prime to p.

To prove the Proposition, we can (and shall) assume that N ≥ 11, since otherwise T_{N} = 0. This enables us to apply a result of N. Katz [3] concerning the operator θ = q \frac{d}{dq} on weightk mod p modular forms, for which the assumption N ≥ 3 is made. The result proved by Katz had been proved by Serre and Swinnerton-Dyer (see [11], [14]) in the case N = 1.

Let S be the space of weight-2 cusp forms on Γ_{o}(N) with coefficients in F_{p}. For each f ∈ S, let ∑_{n≥1} a_{n}(f)q^{n} ∈ F_{p}[[q]] be the q-expansion of f. As is well known (cf. [7], §2), an argument due to Shimura ([13], Chapter 3) shows that the pairing

T_{N}/pT_{N} × S ⟶ F_{p}, (T, f) ↦ a_{1}(f|T)

is perfect. Let A be the subring of T_{N}/pT_{N} generated by the T_{n} with n prime to p. Suppose that f is orthogonal to A in the sense that a_{1}(f|T_{n}) = 0 for all n prime to p. Since a_{1}(f|T_{n}) = a_{n}(f), the power series ∑_{n≥1} a_{n}(f)q^{n} is then annihilated by θ. By the theorem of Katz, the sum ∑_{n≥1} a_{n}(f)q^{n} vanishes. (The point is that ∑_{n≥1} a_{n}(f)q^{n}, viewed as the mod p modular form f, otherwise has filtration w(f) = 2. Since 2 is prime to p, the filtration of θf is then 2 + p + 1. This contradicts the vanishing of θf.) The subspace of S orthogonal to A is then 0, which proves that A = T_{N}/pT_{N}. The Proposition now follows immediately, since T_{N} is free of finite rank over Z, and since A is the image of R in T_{N}/pT_{N}.

Corollary. Let 𝔭 be a maximal ideal of T_{N} of residue characteristic p. Let 𝔪 = R ∩ 𝔭. Then the natural inclusion R/𝔪 ↪ T_{N}/𝔭 is an isomorphism, and we have J_{o}(N)[𝔭] = J_{o}(N)[𝔪].

The first statement is an immediate consequence of the Proposition, since the index of R/𝔪 in T_{N}/𝔭 is simultaneously a power of p and a divisor of the index i = (T_{N}: R) of R in T_{N}. For the second, we must show that a point x of J_{o}(N) which is annihilated by 𝔪 is automatically annihilated by 𝔭. Let λ be an element of 𝔭. Then iλ ∈ 𝔪. Hence x in annihilated by iλ, and it is also annihilated by pλ, since p ∈ 𝔪. Thus x is annihilated by λ, since (i, p) = 1.

Proposition 3. Let 𝔪 be a maximal ideal of R of residue characteristic p. Suppose that the associated representation ρ_{𝔪} of Gal(\overline{Q}/Q) is irreducible. Then J_{o}(N)[𝔪] has dimension two over R/𝔪.

By a theorem of Cohen-Seidenberg, we may find a maximal ideal 𝔭 of T_{N} such that 𝔪 = 𝔭 ∩ R. The Proposition follows from the above Corollary, together with the fact that J_{o}(N)[𝔭] has dimension two over T_{N}/𝔭 ([10], Theorem 5.2b).

Remark: Proposition 2 becomes false if we allow the case p = 2. Indeed, if N = 23, then T_{N} is isomorphic to the ring of integers of the quadratic field Q(\sqrt{5}). Let R be the subring of T_{N} which is generated by the T_{n} with n odd. Then the index of R in T_{N} is divisible by 2. For more details concerning this example, see [2], §4. (The author wishes to thank G. Shimura for pointing out this example to him some years ago.)

Recall now that the subring of T_{A} generated by the Hecke operators T_{n} with n prime to p may be identified with the ring R ⊆ T_{N} which appears above. Indeed, T_{N} is a subring of the ring of endomorphisms of J_{o}(N), which acts diagonally on the product J_{o}(N)^{2} = J_{o}(N) × J_{o}(N). This gives a natural action of T_{N} on J_{o}(N)^{2}. This action preserves the finite subgroup Σ of J_{o}(N)^{2}, and so descends to a faithful action of T_{N} on the quotient A. If n is prime to p, the operator T_{n} of T_{N} induces the operator labeled T_{n} in T_{A}. Hence the subring R of T_{N} generated by such T_{n} maps isomorphically onto the indicated subring of T_{A}. Note, however, that the p^{th} Hecke operator in T_{N} maps to an endomorphism of A which is not necessarily an element of T_{A}. Let τ denote this element of T_{N}. Then, with our conventions, the operator T_{p} ∈ T_{A} is induced by the endomorphism (x, y) ↦ (τx+py, −x) of J_{o}(N)^{2}, which we regard as the matrix ( \begin{matrix}τ & p \\ −1 & 0 \end{matrix} ) of endomorphisms of J_{o}(N). Theorem 3. Let 𝔭 be a maximal ideal of T with residue characteristic p. Assume that ρ_{𝔭} is irreducible, and that A[𝔭] is non-zero. Then A[𝔭] is of dimension two over T/𝔪. Since A[𝔭] is non-zero, the maximal ideal 𝔭 is p-old. By abuse of notation, we consider that 𝔭 is a maximal ideal of T_{A}. Set 𝔪 = R ∩ 𝔭. Considering first the action of R on J_{o}(N), we find that J_{o}(N)[𝔪] has dimension two over R/𝔪 (Proposition 3). The kernel Σ of the quotient map J_{o}(N)^{2} → A is Eisenstein [9], and therefore prime to 𝔪 (cf. [10], Theorem 5.2c). Therefore, A[𝔪] may be identified with J_{o}(N)[𝔪]^{2}, and in particular has dimension four over R/𝔪.

We have T_{A}/𝔭 ⊇ R/𝔪 and A[𝔪] ⊇ A[𝔭]. Also, the dimension of A[𝔭] over T_{A}/𝔭 is a multiple of 2. Hence, this dimension is either 2 or 4, with the latter case occurring if and only if we have both A[𝔭] = A[𝔪] and T_{A}/𝔭 = R/𝔪. However, if these equalities are satisfied, then T_{p} ∈ T_{A} operates on J_{o}(N)[𝔪]^{2} as a scalar, i.e., as an element of R/𝔪. This is impossible, since the operation of T_{p} is given by the matrix ( \begin{matrix}τ & 0 \\ −1 & 0 \end{matrix} ), which is clearly not a scalar because of the −1 in the lower-left corner. (The element τ of T_{N} is not in R; however τ preserves J_{o}(N)[𝔪] and acts on J_{o}(N)[𝔪] as a scalar, in view of the corollary to Proposition 2.) The Theorem is therefore proved. Remark: The example of level 23 mentioned above shows that Theorem 3 becomes false if the prime p = 2 is not excluded. Indeed, suppose that p = 2 and that N = 23. Let A again be the p-old subvariety of J_{o}(pN) = J_{o}(46). The group A[2] is naturally isomorphic to J_{o}(23)[2] ⊕ J_{o}(23)[2], since the kernel of the degeneracy map d has order 11 = num (\frac{22}{12}), which is prime to 2. For n odd, the action of T_{n} on A[2] is the diagonal action of T_{n} coming from the action of T_{23}/2T_{23} on J_{o}(23)[2]. The ring T_{23}/2T_{23} is the field with 4 elements, because 2 remains prime in Q(\sqrt{5}). The T_{n} with n odd map to elements of the prime field F_{2}, since they lie in the order of T_{23} having index 2 in T_{23}. The element T_{2} of T_{46} induces on J_{o}(23)[2] ⊕ J_{o}(23)[2] the map (x, y) ↦ (τx, −x), where τ is the automorphism of J_{o}(23)[2] coming from the Hecke operator T_{2} of T_{23}. It follows that all T_{n} ∈ T_{46} act as elements of F_{2} on 0 ⊕ J_{o}(23)[2], with T_{2} acting as 0. If 𝔭 ⊆ T_{46} is the kernel of the map T_{46} → F_{2} representing this action, then A[𝔭] contains 0 ⊕ J_{o}(23)[2], and therefore has dimension ≥ 4 over T_{46}/𝔭 = F_{2}. The associated 2-dimensional representation ρ_{𝔭} of Gal(\overline{Q}/Q) is irreducible as follows from [4], Chapter II, Proposition 14.1. This representation is an F_{2}-model for J_{o}(23)[2], considered as a representation of Gal(\overline{Q}/Q) over F_{4}.

3. Study of B[𝔭]. In this §, N is again a positive integer, and p a prime which is prime to N. We do not make the assumption p > 2. The p-old and p-new subvarieties of J = J_{o}(Np) are again denoted A and B, respectively. Their intersection is a finite subgroup Δ of J. Consider a maximal ideal 𝔭 of T = T_{Np} for which ρ_{𝔭} is irreducible. (We do not assume that the residue characteristic of 𝔭 is p.)

Theorem 4. If B[𝔭] ≠ 0, then B[𝔭] = J[𝔭].

We have an exact sequence

0 → Δ → A × B → J → 0,

where Δ is embedded “diagonally” in A × B, and where this product is mapped to J by (x, y) ↦ x − y. View J[𝔭] as the group R/Δ, where

R = \{ (x, y) ∈ A(\overline{Q}) × B(\overline{Q}) | (λx, λy) ∈ Δ for all λ ∈ 𝔭 \}.

Let π: R → A be the projection (x, y) ↦ x. We may identify ker π with B[𝔭]. Further, since R contains the diagonal image of Δ, the image π(R) of π contains Δ, regarded as a subgroup of A. From this perspective, it is clear that the theorem amounts to the statement that π(R) coincides with Δ in A.

Let I ⊆ T be the annihilator of B in T, i.e., the kernel of the natural map T → End B. The hypothesis B[𝔭] ≠ 0 is equivalent to the statement that 𝔭 contains I. For λ ∈ I and (x, y) ∈ R, we then have λx = 0, since (λx, λy) is necessarily the 0-element of Δ. In other words, we have Δ ⊆ π(R) ⊆ A[I]. To conclude the proof, we will show that the quotient A[I]/Δ involves only “Eisenstein primes” of T. This gives the desired equality Δ = π(R), since π(R)/Δ is killed by 𝔭 and since ρ_{𝔭} is irreducible.

Consider the element γ := T_{p}^{2} − 1 of T. By (3.7) and (3.10) of [10], T_{p} coincides with the negative of the Atkin-Lehner involution on B. Therefore, γ = 0 on B, so that γ ∈ I. Accordingly, we have A[I] ⊆ A[γ]. Let τ again denote the p^{th} Hecke operator on J_{o}(N). Since the restriction to A of T_{p} is induced by the endomorphism ( \begin{matrix}τ & p \\ −1 & 0 \end{matrix} ) of J_{o}(N)^{2}, γ

restricts to the endomorphism of A induced by the product ( \begin{matrix}−1 & τ \\ 0 & −1 \end{matrix} ) ( \begin{matrix}1 + p & τ \\ τ & 1 + p \end{matrix} ). Abusing notation, we will again refer to this product as γ.

The main theorem of [8] relates Δ to the kernel Ω of ( \begin{matrix}1 + p & τ \\ τ & 1 + p \end{matrix} ) on J_{o}(N)^{2}. Firstly, Ω contains the kernel Σ of the quotient J_{o}(N)^{2} → A. Secondly, we have Δ = Δ̃/Σ, where Δ̃ is a subgroup of Ω such that the quotient Σ^{∗} := Ω/Δ̃ is naturally dual to Σ. Both groups Σ and Σ^{∗} are Eisenstein in the sense that the relation T_{l} = 1 + l holds on each of them, for all primes l prime to pN. The group Ω is also the kernel of γ on J_{o}(N)^{2}, since ( \begin{matrix}−1 & τ \\ 0 & −1 \end{matrix} ) is visibly an automorphism of J_{o}(N)^{2}.

We claim that Q := A[γ]/Δ is an extension of Σ by Σ^{∗}. To see this, apply the Snake Lemma to the diagram

0 → Σ → J_{o}(N)^{2} → A → 0
↑ 0 ↑ γ ↑ γ
0 → Σ → J_{o}(N)^{2} → A → 0

to obtain a 4-term exact sequence 0 → Σ → Ω → A[γ] → Σ → 0. Equivalently, this gives a 3-term exact sequence 0 → Ω/Σ → A[γ] → Σ → 0. Then Q maps to Σ = A[γ]/(Ω/Σ) with kernel (Ω/Σ)/(Δ̃/Σ). The latter group is Ω/Δ̃ = Σ^{∗}.

The claim implies that the support of Q contains only primes which divide the ideal generated by all T_{l} − (1 + l). It follows that 𝔭 is not in the support of Q, cf. [10], 5.2c. A fortiori, 𝔭 is not in the support of A[I]/Δ. As noted above, this proves that π(R) = Δ, and concludes the proof of the Theorem.

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