The Modular Degree and the Congruence Number of a Weight 2 Cusp Form

Alina Carmen Cojocaru Ernst Kani

January 17, 2004

1 Introduction

Let f be a weight 2 normalized newform on the congruence subgroup Γ_{0}(N) with integral Fourier coefficients. There are two important numerical invariants attached to f: its congruence number and its modular degree.

By definition, the congruence number of f is the largest integer D_{f} such that there exists a weight 2 cusp form on Γ_{0}(N), with integral coefficients, which is orthogonal to f with respect to the Petersson inner product and is congruent to f modulo D_{f}. The modular degree of f is the degree deg ϕ_{f} of the minimal parametrization ϕ_{f} : X_{0}(N) → E of the strong Weil elliptic curve E/ℚ associated to f via the Shimura construction. Here X_{0}(N)/ℚ denotes the modular curve defined by Γ_{0}(N).

It turns out that the two quantities D_{f} and deg ϕ_{f} are closely related. On the one hand we have:

Theorem 1.1 In the above setting, deg ϕ_{f}|D_{f}.

This theorem appears already in [Za], p. 381 (in the case that N is a prime) and also in [AU], p. 278, but the existing proofs are not complete, as we will point out in more detail in the following section. The purpose of our paper is to give a detailed and complete proof of this result.

In the opposite direction, it has been announced by Mazur and Ribet (cf. [Mur]) that the only primes dividing D_{f}/ deg ϕ_{f} are prime divisors of N; their proof has not yet been published. Incidentally, the two invariants need not be equal, as has been shown by Agashe/Stein[AS].

The quantities deg ϕ_{f} and D_{f} are closely linked to several deep conjectures in number theory. For example, in [Fr1], Frey formulated the degree conjecture which asserts that deg ϕ_{f} = O (N^{2+ε}) for any ε > 0. A striking result due to Frey and Mai-Murty is that the degree conjecture is equivalent to the famous ABC conjecture; cf. [Fr2], p. 544 and [Mur], p. 180.

On the other hand, the prime divisors of D_{f}, called the congruence primes of f, have been studied extensively by Doi, Hida, Mazur, Ribet, and others; cf. e.g. [Ri2] and the references therein. The question of bounding the congruence primes of f is partially related to a question raised by Mazur, which is discussed in [Mur], p. 181. This, in turn, is related to another conjecture of Frey (Conjecture 5 of [Fr2]), which is equivalent to the Asymptotic Fermat Conjecture; cf. [Fr2], p. 547.

Notation In addition to the above notation, let S_{2}(Γ_{0}(N)) denote the space of cusp forms of weight 2 on Γ_{0}(N). Moreover, if R ⊂ ℂ is a subring, then S_{R} denotes the submodule of S_{2}(Γ_{0}(N)) consisting of the forms g(z) = ∑_{n≥1} a_{n}(g)e^{2πinz} whose Fourier coefficients a_{n}(g) lie in R. As in [DDT], we let 𝕋_{R} ⊂ End_{R}(S_{R}) denote the R-Hecke algebra. Recall that 𝕋_{R} acts on the right on S_{R}; this action is written as g|t for g ∈ S_{R} and t ∈ 𝕋_{R}.

For two abelian varieties A, B defined over ℚ, let Hom(A, B) = Hom_{ℚ}(A, B) denote the group of homomorphisms defined over ℚ and let Hom^{0}(A, B) := Hom(A, B) ⊗_{ℤ} ℚ. Note that Hom^{0}(A, B) is a finite dimensional ℚ-vector space which contains Hom(A, B) as a lattice.

As usual, the dual abelian variety of A is denoted by Â. Recall that for each α ∈ Hom(A, B) we have a dual map α̂ : B̂ → Â between the dual abelian varieties and that the assignment α ↦ α̂ extends to an isomorphism Hom^{0}(A, B) ≃ Hom^{0}(B̂, Â).

If L ⊂ V is a lattice in a ℚ-vector space V, then the denominator den α = den_{L} α of α ∈ V with respect to L is the smallest positive integer d such that dα ∈ L. (Thus, if n is an integer such that nα ∈ L, then den α|n.) In particular, if α ∈ Hom^{0}(A, B), then den(α) denotes the denominator of α with respect to the lattice Hom(A, B).

2 Proof of Theorem 1.1

2.1 The idea of proof

The basic idea of the proof of Theorem 1.1, due to Hida, Ribet and others (cf. [Za], p. 381) is to construct an idempotent e of the ring End^{0}(J_{0}(N)), where J_{0}(N) denotes the Jacobian of X_{0}(N), satisfying the following three properties: (i) the denominator of e in End_{ℚ}(J_{0}(N)) is deg ϕ; (ii) e ∈ 𝕋_{ℚ}; (iii) the denominator of e with respect to 𝕋_{ℤ} divides D_{f}. Since 𝕋_{ℚ} ⊆ End_{ℚ}(J_{0}(N)), the theorem follows.

A proof of Theorem 1.1 based on the above idea appears first in [Za], pp. 381-382, but only for the case N a prime. Since in this case, by strong results of Mazur, 𝕋_{ℚ} is isomorphic to End_{ℚ}(J_{0}(N)), one actually has that deg ϕ = D_{f}. Zagier’s parenthetic generalization of the result to arbitrary N is stated incorrectly and the suggested proofs of (ii) and (iii) have gaps. A more detailed proof of Theorem 1.1 appears in [AU], pp. 278-279. This proof does not provide an explanation for why property (ii) holds (cf. Remark 2.2) and, also, the proof of (iii) has some gaps. In this exposition we shall unravel the details behind these properties and thus present complete proofs.

2.2 Review of the Shimura construction

Let j : X_{0}(N) → J_{0}(N) denote the injection of X_{0}(N) into J_{0}(N) induced by the map P ↦ cl(P − (∞)), where (∞) ∈ X_{0}(N)(ℚ) is the rational point corresponding to the cusp at infinity. Then, if Ω_{X} = H^{0}(X, Ω_{X}^{1}) denotes the ℚ-vector space of holomorphic differentials on a variety X/ℚ, the pullback j^{∗} induces a canonical identification Ω_{J_{0}(N)} ≃ Ω_{X_{0}(N)} ≃ S_{ℚ}.

Since f = ∑_{n≥1} a_{n}(f)q^{n} ∈ S_{ℤ} is a normalized newform and hence a 𝕋_{ℚ}- eigenform, there exists by the Shimura construction an (essentially unique) abelian quotient p = p_{f} : J_{0}(N) → A_{f} = E (defined over ℚ) such that Ker(p) is connected and such that p^{∗}Ω_{E} = W := ℚf ⊂ Ω_{J_{0}(N)} (via the above identifications); cf. [Sh2], Theorem 1 and/or [Ka2], Example 5.2. In particular, dim E = dim_{ℚ} W = 1, i.e. E is an elliptic curve over ℚ. Moreover, there is an injective ring homomorphism τ = τ_{f} : K_{f} := ℚ({a_{n}(f)}_{n≥1}) ↪ End^{0}(E) := End_{ℚ}(E) ⊗ ℚ such that

(1) τ(χ_{f}(t)) ∘ p = p ∘ t, for all t ∈ 𝕋_{ℚ} ⊂ End^{0}(J_{0}(N)),

where χ_{f} : 𝕋_{ℚ} → K_{f} = ℚ is the character defined by f, i.e. f|t = χ_{f}(t)f, for all t ∈ 𝕋_{ℚ}.

2.3 Construction of the idempotent e

Let ϕ = ϕ_{f} := p_{f} ∘ j : X_{0}(N) → E, which is the associated (minimal) modular parametrization of the elliptic curve E. By functoriality, ϕ induces a homomorphism ϕ^{∗} : J_{E} → J_{0}(N), where J_{E} denotes the Jacobian of E. (Note that the polarization on E induces an isomorphism ρ : E →^{∼} J_{E}.) In addition, by duality we have a homomorphism ϕ_{∗} : J_{0}(N) → J_{E}, given by the formula

ϕ_{∗} = θ_{E}^{−1} ∘ (ϕ^{∗})^{}ˆ ∘ θ, (2)

where θ : J_{0}(N) → Ĵ_{0}(N) and θ_{E} : J_{E} → Ĵ_{E} are the canonical polarizations on J_{0}(N) and on J_{E}, respectively, and (ϕ^{∗})^{}ˆ : Ĵ_{0}(N) → Ĵ_{E} denotes the dual map of ϕ^{∗}. Note that ϕ^{∗} is also the “dual” of ϕ_{∗} in the sense that we have

ϕ^{∗} = θ^{−1} ∘ (ϕ_{∗})^{}ˆ ∘ θ_{E}; (3)

cf. e.g. [Ka1], p. 46. By the autoduality property of the Jacobian, the map ϕ_{∗} : J_{0}(N) → J_{E} is related to p : J_{0}(N) → E via the formula

ρ ∘ p = ϕ_{∗}. (4)

In particular, the kernel Ker(ϕ_{∗}) = Ker(p) of ϕ_{∗} is connected and hence ϕ^{∗} is injective; cf. e.g. [Ka1], pp. 47-48. Furthermore, we have

ϕ_{∗} ∘ ϕ^{∗} = deg(ϕ) id_{J_{E}}; (5)

cf. e.g. [Ka2], p. 46. From this it is immediate that the element

e = e_{ϕ} := \frac{1}{deg ϕ}ϕ^{∗} ∘ ϕ_{∗} ∈ End^{0}(J_{0}(N))

is an idempotent, i.e. satisfies the equation e^{2} = e. For later purposes it is useful to note that e is symmetric with respect to the Rosati involution f ↦ f′ := θ^{−1} ∘ f̂∘ θ of End^{0}(J_{0}(N)) (cf. [Mu], p. 189); indeed, by (2) and (3) we have e′ = \frac{1}{deg ϕ}θ^{−1} ∘ (ϕ_{∗} ∘ ϕ^{∗})^{}ˆ ∘ θ = \frac{1}{deg ϕ}θ^{−1} ∘ (ϕ^{∗})^{}ˆ ∘ (ϕ_{∗})^{}ˆ ∘ θ = e. In addition, we observe that its trace on the space Ω_{J_{0}(N)} is tr(e^{∗}|Ω_{J_{0}(N)}) = dim ϕ^{∗}Ω_{J_{E}} = 1.

2.4 Calculation of the denominator of e in End(J_{0}(N))

For this, we shall use the following simple facts about denominators of elements in Hom^{0}(A, B), where A and B are abelian varieties.

Lemma 2.1 If α ∈ Hom^{0}(A, B) and if β : B → C, γ : C → A are homomorphisms of abelian varieties, then we have

(6)  den(α̂) = den(α);
(7)  den(β ∘ α) = den(α), if β is injective;
(8)  den(α ∘ γ) = den(α), if γ̂ is injective.

Proof. Let n := den(α). Then nα ∈ Hom_{ℚ}(A, B) and hence nα̂ = { \atop (nα)} ∈ Hom_{ℚ}(B̂, Â). Thus den(α̂) divides n = den(α), and hence also den(α̂) divides den((α̂)^{}ˆ ) = den(α) (because we have a functorial identification of κ_{A} : A →^{∼} (Â) ). This proves (6).

Moreover, since n(β ∘ α) = β ∘ (nα) ∈ Hom(A, C), we see that m := den(β ∘ α) divides n = den(α). Write α_{1} := nα ∈ Hom(A, B) and α_{2} := m(β ∘ α) ∈ Hom(A, C). Now since \frac{n}{m}α_{2} = β ∘ α_{1}, we see that Ker(\frac{n}{m} id_{A}) ≤ Ker(β ∘ α_{1}) = Ker(α_{1}). Thus α_{1} = α_{3} ∘ \frac{n}{m} id_{A} for some α_{3} ∈ Hom(A, B), and so mα = \frac{m}{n}α_{1} = α_{3}. This means that n = den(α)|m, and so m = n, which verifies (7).

Finally, the last assertion follows immediately from the first two because den(α ∘ γ) = den(γ̂ ∘ α̂) = den(α̂) = den(α) by (6), (7) and (6), respectively.

From the above lemma it follows easily that

den(e_{ϕ}) = deg(ϕ). (9)

Indeed, since ϕ^{∗} is injective, we have by (7) that den(e_{ϕ}) = den(\frac{1}{deg ϕ}ϕ_{∗}). Moreover, since (ϕ_{∗})^{}ˆ is also injective by (3), it follows from (8) that den(\frac{1}{deg ϕ}ϕ_{∗}) = den(\frac{1}{deg ϕ} id_{J_{E}}) = deg ϕ, and so (9) follows.

2.5 Show that e ∈ 𝕋_{ℚ}

To prove that e ∈ 𝕋_{ℚ}, we shall use Atkin-Lehner theory to construct another idempotent ε ∈ 𝕋_{ℚ} and then show that in fact ε = e.

Remark 2.2 In [AU], the authors deduce this fact (implicitly) from their assertion that an isogeny J_{0}(N) ∼ E × A (where A = Ker(ϕ_{∗})) induces a splitting

(10) End^{0}(J_{0}(N)) ≃ End^{0}(E) × End^{0}(A).

This statement, which is asserted without proof on p. 278, is in fact false in general because A may have a factor isogenous to E. For example, this always happens in the case that f ∈ S_{ℤ} is a 𝕋_{ℚ}-eigenform which is not a newform. (Note that for this case the Shimura construction and the assertions of §2.2-2.4 apply as well.) If, however, we assume that f is a newform (as we did throughout), then we do have such splitting (10), but the proof of this fact requires the deep results of Ribet[Ri1] (Corollary 4.2) and more. The following proof shows that such deep results are not necessary here.

Since f ∈ S_{ℚ} is a newform, we have (by Atkin-Lehner theory) a 𝕋_{ℚ}- module decomposition S_{ℚ} = W ⊕ W′, where (as before) W = ℚf. Let e_{W} ∈ End_{𝕋_{ℚ}}(S_{ℚ}) denote the projector onto W; in particular, Im(e_{W}) = W and Ker(e_{W}) = W′. Now since S_{ℚ} is a free 𝕋_{ℚ}-module of rank 1 (cf. [DDT], p. 36), it follows that End_{𝕋_{ℚ}}(S_{ℚ}) = End_{𝕋_{ℚ}}(𝕋_{ℚ}) = 𝕋_{ℚ} and hence there is a unique ε ∈ 𝕋_{ℚ} such that g|ε = e_{W} (g), for all g ∈ S_{ℚ}. Note that ε is necessarily an idempotent and that 𝕋_{ℚ}ε = Ann_{𝕋_{ℚ}}(W′); cf. Bourbaki[Bo], Prop. 1d) of §VIII.1 (p. 8). Thus,

εt = χ_{f}(t)ε, (11)

for if g = cf + w′ ∈ S_{ℚ} with c ∈ ℚ, w′ ∈ W′, then g|εt = cf|εt = cf|t = χ_{f}(t)cf = χ_{f}(t)cf|ε = χ_{f}(t)g|ε, which is (11). In particular, we see that χ_{f}(ε) = 1 (because ε = ε^{2} = χ_{f}(ε)ε).

Recall that we can view 𝕋_{ℚ} as a subring of End^{0}(J_{0}(N)). Now 𝕋_{ℚ} is (in general) not preserved under the Rosati involution, i.e. 𝕋′_{ℚ} ⊄ 𝕋_{ℚ}. However, we shall see presently that ε′ ∈ 𝕋_{ℚ} and that in fact, ε′ = ε. As a first step towards this we show:

f|t′ = χ_{f}(t)f, for all t ∈ 𝕋_{ℚ}. (12)

To see this, recall that the newform f is also a 𝕋′_{ℚ}-eigenform (cf. [Mi], p. 165) and so (since 𝕋_{ℚ} is commutative) there is a character χ′_{f} : 𝕋_{ℚ} → ℚ such that f|t′ = χ′_{f}(t)f, for all t ∈ 𝕋_{ℚ}. Thus, since t′ (viewed as an endomorphism of S_{ℚ}) is the adjoint of t with respect to the Petersson inner product ⟨·, ·⟩ on S_{ℚ} (cf. Shimura[Sh1], p. 76 and p. 171), we have

χ′_{f}(t)⟨f, f⟩ = ⟨f|t′, f⟩ = ⟨f, f|t⟩ = \overline{χ_{f}(t)}⟨f, f⟩,

where \overline{χ_{f}(t)} denotes the complex conjugate of χ_{f}(t). This shows that χ′(f) = \overline{χ_{f}(t)} = χ_{f}(t), which proves (12).

From (12) it follows that

(13) εt′ = χ_{f}(t)ε and tε′ = χ_{f}(t)ε′, for all t ∈ 𝕋_{ℚ}.

Indeed, if g = cf + w′ ∈ S_{ℚ} with c ∈ ℚ, w′ ∈ W′, then by (12) we have g|εt′ = cf|εt′ = cf|t′ = χ_{f}(t)cf = χ_{f}(t)cf|ε = χ_{f}(t)g|ε, which proves the first equation of (13). Moreover, the second equation follows from the first by applying the Rosati involution to both sides.

From the two equations (13) it follows immediately that ε is symmetric. Indeed, by taking t = ε in (13) we obtain εε′ = ε and εε′ = ε′ (recall that χ_{f}(ε) = 1) and so ε = ε′ is symmetric.

We now relate ε to the idempotent e constructed in §2.3 For this, we first observe that by taking t = ε in (1) we obtain p ∘ ε = id_{E} ∘p because χ_{f}(ε) = 1 (and τ is a ring homomorphism). Thus, by (4) we have

ϕ_{∗} ∘ ε = ϕ_{∗} and hence e ∘ ε = e.

Thus, since ε and e are both symmetric we also have e = e′ = (eε)′ = ε′e′ = εe, which means that ε and e commute. This in turn implies that ε − e is an idempotent, for (ε − e)^{2} = e^{2} + ε^{2} − 2eε = e + ε − 2e = ε − e. Thus, its trace is tr(ε − e|S_{ℚ}) = dim Im(ε − e). On the other hand, tr(ε − e|S_{ℚ}) = tr(ε|S_{ℚ}− tr(e|S_{ℚ}) = dim Im(e_{W}) − 1 = 0, and so Im(ε − e) = 0, i.e. e = ε ∈ 𝕋_{ℚ}.

2.6 Show that the denominator of e in 𝕋_{ℤ} divides D_{f}

Let L_{f} := ℤf ⊕ (S_{ℤ} ∩ ⟨f⟩^{⊥}). Then it is not difficult to see that the congruence number D_{f} (as defined in the introduction) is equal to the exponent of the (finite) group S_{ℤ}/L_{f}; cf. e.g. [Za], p. 381.

Moreover, put T := e𝕋_{ℤ} ⊕ (1 − e)𝕋_{ℤ} and \overline{T} := T /𝕋_{ℤ}. Note that \overline{T} is a cyclic group of order d, where d is the denominator of e in 𝕋_{ℤ}. Indeed, by using the surjectivity of χ_{f} : 𝕋_{ℤ} ⟶ ℤ and (11), we see that T = e𝕋_{ℤ} ⊕ (1 − e)𝕋_{ℤ} = e + 𝕋_{ℤ}, and hence \overline{T} = (eℤ + 𝕋_{ℤ})/𝕋_{ℤ} ≃ ℤ/dℤ.

Next, we shall relate \overline{T} to S_{ℚ}/L_{f} and hence obtain that the order of \overline{T} divides D_{f}. For this, consider the exact sequence of ℤ-modules

0 ⟶ 𝕋_{ℤ} ⟶ T ⟶ \overline{T} ⟶ 0,

which induces the exact sequence

0 ⟶ Hom_{ℤ}(\overline{T} , ℤ) ⟶ Hom_{ℤ}(T , ℤ) ⟶^{r} Hom_{ℤ}(𝕋_{ℤ}, ℤ)

⟶ Ext_{ℤ}^{1}(\overline{T} , ℤ) ⟶ Ext_{ℤ}^{1}(T , ℤ) ⟶ Ext_{ℤ}^{1}(𝕋_{ℤ}, ℤ) ⟶ …,

where r denotes the restriction map defined by r(f) = f_{|𝕋_{ℤ}}. Since \overline{T} is a finite cyclic group, we have Hom_{ℤ}(\overline{T} , ℤ) = 0 and Ext_{ℤ}^{1}(\overline{T} , ℤ) ≃ \overline{T} ≃ ℤ/dℤ. Also, since T is a free (hence projective) ℤ−module, Ext_{ℤ}^{1}(T , ℤ) = 0. Therefore we are left with the exact sequence

0 ⟶ Hom_{ℤ}(T , ℤ) ⟶^{r} Hom_{ℤ}(𝕋_{ℤ}, ℤ)⟶ℤ/dℤ ⟶ 0,

which implies that H := Hom_{ℤ}(𝕋_{ℤ}, ℤ)/r(Hom_{ℤ}(T , ℤ)) is a finite cyclic group of order d.

Next we show that H is isomorphic to a quotient of S_{ℤ}/L_{f} and that hence d|D_{f}. To see this, recall that we have the perfect pairing a_{ℂ} : S_{ℂ} × 𝕋_{ℂ} → ℂ given by a_{ℂ}(g, t) = a_{1}(g|t), where a_{1}(g|t) denotes the first Fourier coefficient of g|t, and that this pairing induces the isomorphism (cf. [Ri2], Th. 2.2)

a_{ℤ} : S_{ℤ} ⟶^{∼} Hom_{ℤ}(𝕋_{ℤ}, ℤ).

To verify the above assertion, it is thus enough to construct a homomorphism

A : L_{f} = ℤf ⊕ (S_{ℤ} ∩ ⟨f⟩^{⊥}) ⟶ Hom_{ℤ}(T , ℤ)

with the property that r ∘ A = (a_{ℤ})_{|L_{f}}, for then it follows immediately that H = Hom_{ℤ}(𝕋_{ℤ}, ℤ)/r(Hom_{ℤ}(T , ℤ)) ≃ ℤ/dℤ is a quotient of Hom_{ℤ}(𝕋_{ℤ}, ℤ)/r(Im(A)) ≃ S_{ℤ}/L_{f}.

To define A, let a ∈ ℤ, g ∈ S_{ℤ} ∩ ⟨f⟩^{⊥}, t_{1}, t_{2} ∈ 𝕋_{ℤ} and put

A(af + g)(et_{1} + (1 − e)t_{2}) := a_{1}(af|t_{1} + g|t_{2}). (14)

We shall see presently that this is a well-defined (injective) group homomorphism^{1}. Note that by taking t_{1} = t_{2} in (14), it then follows that r ∘ A = (a_{ℤ})_{|L_{f}}, as desired.

To see that A is well-defined, write S_{ℚ} = s𝕋_{ℚ} (recall that S_{ℚ} is a free 𝕋_{ℚ}-module of rank 1) and let g ∈ S_{ℤ} ∩ ⟨f⟩^{⊥}. Then there exist t_{f}, t_{g} ∈ 𝕋_{ℚ} such that

(15)  f = χ_{f}(t_{f})s|e,
(16)  g = s|(1 − e)t_{g}.

Indeed, since f ∈ S_{ℚ}, we can write f = s|t_{f} for some t_{f} ∈ 𝕋_{ℚ}, and then f = χ_{f}(e)f = f|e = s|t_{f}e = s|et_{f} = χ_{f}(t_{f})s|e because χ_{f}(e) = 1 (cf. (11)). This proves (15). Moreover, by writing g = s|t_{g} for some t_{g} ∈ 𝕋_{ℚ} and then using (15) in the form s|e = \frac{1}{χ_{f} (t_{f})}f, we obtain g = s|et_{g} + s(1 − e)t_{g} =

\frac{χ_{f} (t_{g})}{χ_{f} (t_{f})}f + s|(1 − e)t_{g}. But g ∈ ⟨f⟩^{⊥}, hence 0 = ⟨g, f⟩ = \frac{χ_{f} (t_{g})}{χ_{f} (t_{f})}⟨f, f⟩ + ⟨s|(1 −

e)t_{g}, χ_{f}(t_{f})s|e⟩ = \frac{χ_{f} (t_{g})}{χ_{f} (t_{f})}⟨f, f⟩, where we have also used that e is a symmetric idempotent. This implies that χ_{f}(t_{g}) = 0, which, in turn, proves (16).

1One can show that the homomorphism A is in fact an isomorphism. This implies that S_{ℤ}/L_{f} ≃ H ≃ ℤ/dℤ is a cyclic group of order d.

Now we can check easily that A is well-defined. Indeed, suppose that t_{1}, t′_{1}, t_{2}, t′_{2} ∈ 𝕋_{ℤ} are such that et_{1} + (1 − e)t_{2} = et′_{1} + (1 − e)t′_{2}. Then et_{1} = et′_{1}, (1 − e)t_{2} = (1 − e)t′_{2} and hence by using (15), (16) we obtain that f|t_{1} = χ_{f}(t_{f})s|et_{1} = χ_{f}(t_{f})s|et′_{1} = f|t′_{1} and g|t_{2} = s|t_{g}(1 − e)t_{2} = s|t_{g}(1 − e)t′_{2} = g|t′_{2}, and so A is well-defined.

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Princeton University, Mathematics Department, Princeton, New Jersey,

08544-1000, USA; cojocaru@math.princeton.edu

Queen’s University, Dept. Mathematics and Statistics, Kingston, Ontario,

K7L 3N6, Canada; kani@mast.queensu.ca