Mazur’s question on mod 11 representations of elliptic curves ^{∗}

E.J. Kani and O.G. Rizzo

1 Introduction

In 1978, Barry Mazur [14] asked the following question:

Question 1. Do there exist two elliptic curves E_{1}/ℚ, E_{2}/ℚ which are not isogenous /ℚ such that their associated Galois representations

\overline{ρ}_{E_{i},N} : Gal(ℚ̄/ℚ) → Aut(E_{i}[N])

are (symplectically) isomorphic for some N ≥ 7?

In 1992 Kraus and Oesterlé [12] found the first such examples for N = 7, and recently Halberstadt and Kraus [5] exhibited explicit infinite families with this property (for N = 7).

For larger N, Mazur found examples for N = 11 and N = 13. In addition, Frey and his group have found many examples by computer [4].

The purpose of this paper is to prove the following result:

Theorem 2. There exist infinitely many one-parameter families of isomorphism classes of pairs of non-isogenous elliptic curves defined over ℚ with symplectically isomorphic 11-structure.

The main idea of the proof of this Theorem is to study the geometry (and arithmetic) of the modular diagonal quotient surfaces Z_{N,1} (as introduced in [9]) in the special case N = 11. Now the algebraic surface Z = Z_{N,1} has a natural model as a variety over ℚ (cf. §3), and an open subvariety of this turns out to be the coarse moduli space of the moduli functor Z_{N,1} which classifies isomorphism classes of triplets (E_{1}, E_{2}, ψ), where ψ : E_{1}[N] → E_{2}[N] is a Gal(ℚ̄/ℚ)-isomorphism which preserves the Weil pairings. Thus, via this modular interpretation (cf. §4), the above Theorem is essentially a consequence of the following result (cf. Theorem 19):

∗This research was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada (NSERC).

Theorem 3. Let Z̄_{ℚ} denote the minimal model of Z_{11,1}/ℚ. Then the canonical map defines an elliptic fibration f_{can} : Z̄_{ℚ} → ℙ_{ℚ}^{1} which has an infinite number of sections.

Indeed, by using the result of Mazur [14] (as supplemented by Kenku [11]) on rational isogenies, one easily concludes that all except finitely many of these sections S_{i}/ℚ ≃ ℙ_{ℚ}^{1} give rise to infinitely many pairs of non-isogenous elliptic curves.

It is interesting to observe that, although the above proof is constructive “in principle,” it does not allow us to write down even a single pair explicitly. The reason for this is that, while the sections are constructed as the multiples of an explicit point Q of infinite order on the associated elliptic curve E/ℚ(t), the point Q itself does not have a modular interpretation (since it lies in the cuspidal part), and its multiples nQ cannot be interpreted explicitly until the elliptic curve E can be determined.

2 Geometric results

2.1 The geometry of Z̃

We recall some of the terminology and results of [9]. Let N be a positive integer and let X(N) = Γ(N)\ℍ^{∗} be the modular curve of level N, on which G_{N} = SL_{2}(ℤ/Nℤ)/ ± 1 acts naturally. We denote the quotient map by π : X(N) → X(1) = G_{N}\X(N). Furthermore, let Z = Z_{N,1} = Δ\(X(N) × X(N)) denote the (singular) modular diagonal quotient surface; here Δ = {(g, g) : g ∈ G_{N}} denotes the diagonal subgroup. Then (as in [9]) the inclusion of the subgroups {1} ≤ Δ ≤ G_{N} × G_{N} induces natural maps

X(N) × X(N) ⟶^{Φ} Z ⟶^{Ψ} X(1) × X(1)

such that Ψ ∘ Φ = π × π : X(N) × X(N) → X(1) × X(1) is the natural projection.

Consider the points i, exp(2πi/3) and ∞ of ℍ^{∗}, and let P̄_{k} (for k = 0, 1, ∞) be their respective images in X(1).

For the remainder of the section, let N = 11. By Theorem 2.1 of [9], we have:

Proposition 4. If Z = Z_{11,1}, then 1. The set of singularities of Z decomposes into Σ_{0} ∪ Σ_{1} ∪ Σ_{∞}, where

Σ_{k} ⊂ Ψ^{−1}(P̄_{k}, P̄_{k}) for k = 0, 1, ∞. All singularities are cyclic quotient singularities.

2. Σ_{0} consists of 6 singularities of type (2, 1).

3. Σ_{1} consists of 2 singularities of type (3, 1) and of 2 singularities of type (3, 2).

4. Σ_{∞} = {z_{1}, …, z_{5}}, where z_{i} is a singularity of type (11, q_{i}) with q_{i} ≡ (\frac{i}{11})i mod 11 and 1 ≤ q_{i} ≤ 11.

Let σ : Z̃ → Z be the minimal desingularization of Z. If C is a curve on Z, we will denote its proper transform on Z̃ by C̃.

Recall that the minimal desingularization of a cyclic quotient singularity of type (n, q) is a (−n_{1}, …, −n_{r})-chain, where r and n_{1}, …, n_{r} are determined by the continued fraction expansion of n/q (cf. [1, § III.2]). Here, by a (−n_{1}, …, −n_{r})-chain we mean an open chain C = C_{1} + ⋯ + C_{r} of smooth rational curves C_{i} such that C_{i}^{2} = −n_{i}, for i = 1, …, r.

Let Ψ_{i} = pr_{i}∘Ψ : Z → X(1) ≃ ℙ^{1}, where i = 1, 2 and pr_{i} is the projection on the i-th factor of X(1) × X(1). Let Ψ̃_{i} = Ψ_{i} ∘ σ.

Proposition 2.5 of [9] implies that:

Proposition 5. For k = 0, 1, ∞, let E_{k} denote the reduced divisor on Z̃ whose support is σ^{−1}(Σ_{k}), and let E_{∞,i} be the reduced divisor on Z̃ whose support is σ^{−1}(z_{i}). Then

1. E_{0} consists of six (disjoint) (−2)-curves;

2. E_{1} consists of two (−2)-curves and of two (−2, −2)-chains;

3. E_{∞} = E_{∞,1} + ⋯ + E_{∞,5}, where

E_{∞,1} is a (−11)-curve, E_{∞,2} is a (−4, −3)-chain, E_{∞,3} is a (−3, −2, −2, −2, −2)-chain, E_{∞,4} is a (−2, −2, −2, −2, −3)-chain, E_{∞,5} is a (−3, −4)-chain.

Furthermore, if C̃_{k,i} denotes the proper transform of Ψ_{i}^{∗}(P̄_{k}) in Z̃, then each chain in E_{k} joins C̃_{k,1} to C̃_{k,2}, where k = 0, 1, ∞.

2.2 The elliptic fibration

Notation. Let n ≢ 0 mod 11 be a positive integer which is a square modulo 11. Fix a k ∈ 𝔽_{11}^{×} such that k^{2}n ≡ 1 mod 11 and let τ_{k} = ( \begin{matrix}1/k & 0 \\ 0 & k \end{matrix} ) ∈ G_{11}. Let T_{n} = Φ ((τ_{k}^{−1} × 1)T′(1, n)), where T′(1, n) is the Hecke correspondence of X(11) × X(11) as defined in §3.3 of [16]. We call such a curve on Z a Hecke curve.

Remark 6. T_{n} is the curve T_{n,k} of [9] or one of the curves F_{n}^{(i)} of [6].

Proposition 7. a) The (lifted) Hecke curves T̃_{1}, T̃_{3} and T̃_{4} are exceptional (−1)-curves on Z̃. Moreover, T̃_{1} meets precisely three components of E_{0} + E_{1}+E_{∞}: a (−2)-curve Γ_{0} ≤ E_{0}, a (−3)-curve Γ_{1} ≤ E_{1} and E_{∞,1}. In addition, T̃_{1} meets each of these components transversely.

b) Let Z̄ be the surface obtained by blowing down T̃_{1}, Γ_{0}, Γ_{1}; T̃_{3}; T̃_{4}. Then Z̄ is minimal.

Proof. The assertions of part (a) follow from Claims 3 and 4 of [9] and Remark 4.9 of [8]—but see also [6]. Part (b) is Claim 8 of [9].

Notation. If C (or C̃) is a curve on Z̃, denote its image in Z̄ by C̄.

Proposition 8. Let S_{0} (resp. S_{1}) be the (−4)-component of E_{∞,2} (resp. E_{∞,5}). Then their images S̄_{0} and S̄_{1} on Z̄ satisfy S_{0}^{2} = S_{1}^{2} = −3 and S_{0} · S_{1} = 1.

Proof. See the proof of Claim 4 of [9].

Theorem 9. Let f_{can} be the canonical map of Z̄. Then

1. f_{can} : Z̄ → ℙ^{1} is an elliptic fibration with no multiple fibres.

2. S̄_{0} and S̄_{1} are sections of f_{can}.

3. Let E be the elliptic curve over ℂ(t) corresponding to (Z̄, S̄_{0}) and let Q be the point of E corresponding to S̄_{1}. Then Q has infinite order on E.

Proof. We know from (the proof of) [9], Proposition 2.14, that Z̄ is a minimal smooth surface with geometric genus p_{g}(Z̄) = 2, Euler–Poincaré characteristic χ(Z̄) = 3 and Kodaira dimension κ(Z̄) = 1. By Proposition 8, S̄_{0} and S̄_{1} are two rational smooth curves on Z̄ of self-intersection −3 meeting transversally. By the following two lemmata, this suffices to prove the theorem.

Lemma 10. Let E be a smooth minimal compact surface with p_{g}(E) = 2, χ(E) = 3 and κ(E) = 1. Suppose that there is a smooth irreducible rational curve C of self-intersection −3 lying on E. Then:

1. The canonical map gives an elliptic fibration f_{can} : E → ℙ^{1};

2. C is a section of f_{can};

3. f_{can} has no multiple fibre.

Proof. It follows from the Enriques–Kodaira classification of minimal surfaces that E admits an elliptic fibration f : E → B, where B is some smooth curve of genus g (cf. [1], p. 194). By Kodaira’s formula for the canonical divisor of an elliptic surface (see Corollary V.12.3 of [1]),

\begin{aligned}K & ≡ (χ(E) − 2χ(B))F + ∑(m_{i} − 1)F_{i} \\ & = (1 + 2g)F + ∑(m_{i} − 1)F_{i}, \end{aligned}  (1)

where F is any elliptic fibre, the sum is over all singular fibres F_{i} of multiplicity m_{i} and ≡ denotes algebraic equivalence.

By the adjunction formula, C · K = 1; thus, if d = C · F, equation (1) yields

\begin{aligned}1 = C · K & = C · ((1 + 2g)F + ∑(m_{i} − 1)F_{i}) \\ & = d (1 + 2g + ∑(m_{i} − 1)) . \end{aligned}  (2)

Since g, d and m_{i} are all non-negative integers, (2) holds if only if d = 1 and 2g + ∑(m_{i} − 1) = 0. This proves that C is a section for f, that g = 0 and that f has no multiple fibres.

Since the geometric genus of E is 2, the canonical map is a rational map f_{can} : E → ℙ^{1}. Since B = ℙ^{1}, (1) becomes K ∼ F. Thus, f_{can} = f up to an automorphism of ℙ^{1}.

Lemma 11. Let B be a curve defined over a field K of characteristic 0 and let E → B be a relatively minimal elliptic fibration defined over K which has two distinct sections C_{0} and C_{1} that meet. Let E be the elliptic curve over the function field κ(B) of B which corresponds to (the generic fibre of) (E, C_{0}). Then the point Q of E corresponding to the section C_{1} has infinite order.

Proof. Let t_{0} ∈ B be a point over which C_{0} and C_{1} meet, let R be the local ring of t_{0} on B and k its residue field. By Theorem IV.6.1 of [18], the special fibre of the Néron model E′/R of E/K(B) is the curve obtained from the fibre E_{t_{0}} by removing all singular points. In particular, C_{1} belongs to the kernel E_{1} of the reduction map (i.e., the specialization map at t_{0}): E′/R → Ẽ′/k. Now, by Corollary IV.9.2 of [18], E_{1} is isomorphic to the kernel E_{1} of the reduction map of a minimal Weierstrass model of E (see Proposition VII.2.2 of [17]), which, since the ground field has characteristic 0, has no non-trivial torsion by Proposition IV.3.2b of [17]. Therefore, Q has infinite order.

3 Rational models

3.1 The Galois action on X(N)

The purpose of this section is to show that the surfaces Z, Z̃ and Z̄ have natural models defined over ℚ and that the fibration and its sections which were defined in the previous section are actually rational over ℚ.

As a first step, let us recall some facts about the Galois action on X(N), following §6.2 of Shimura [16], particularly Theorem 6.6 and Proposition 6.9.

For each positive integer N, let X_{N} denote the smooth projective curve over ℚ whose function field is the field F_{N} = ℚ(j, f_{r,s}) of modular functions of level N. Here j is the modular j-invariant which induces the isomorphism j : X(1) →^{∼} ℙ_{ℂ}^{1}, and the f_{r,s} = f_{(r,s)} are the Fricke functions for (r, s) ∈ ℤ/Nℤ × ℤ/Nℤ; cf. Lang [13], p. 65ff or Shimura [16], p. 137. If G̃_{N} = GL_{2}(ℤ/Nℤ)/±1, then we have a natural G̃_{N}-action on F_{N} and hence also on X_{N} via the rule

g^{∗}f_{r,s} = f_{(r,s)g}.

The induced action on the subfield ℚ_{N} := ℚ(ζ_{N}) ⊂ F_{N} is via the determinant, i.e. we have g^{∗}_{|ℚ_{N}} = σ_{det(g)}^{∗}, for all g ∈ G̃_{N}. Here, for any a ∈ (ℤ/Nℤ)^{×}, σ_{a}^{∗} ∈ Aut(ℚ_{N}) is the unique automorphism such that σ_{a}^{∗}(ζ) = ζ^{a}, for any N-th root of unity ζ ∈ ℚ_{N}.

Thus, if p_{N} : X_{N} → Spec(ℚ_{N}) denotes the structure map induced by the inclusion ℚ_{N} ⊂ F_{N}, then we have

p_{N} ∘ g = σ_{det(g)} ∘ p_{N}, for all g ∈ G̃_{N}.  (3)

Note that since the fixed field of G̃_{N} is F G̃_{N} ; N = ℚ(j), the quotient of X_{N} with respect to G̃_{N} is

X_{ℚ}(1) := G̃_{N}\X_{N} ≃ ℙ_{ℚ}^{1}.

Next, consider the subgroup

H_{N} = \{± ( \begin{matrix}1 & 0 \\ 0 & d \end{matrix}) : d ∈ (ℤ/Nℤ)^{×}\}/\{±1\} ≤ G̃_{N},

and let F_{N} = F_{N}^{H_{N}} be its fixed field. Since H_{N} ·G_{N} = G̃_{N} and H_{N} ∩G_{N} = {1} (and since ℚ_{N} is algebraically closed in F_{N}), we have the following diagram of linearly disjoint field extensions:

[figure: ℚ; F_{1}; F_{N}; ℚ(ζ_{N}); F_{1}(ζ_{N}); F_{N}; ℂ; ℂ(j); F_{N} · ℂ; H_{N}; G_{N}]

Thus, if we let

π_{ℚ} = π_{ℚ,N} : X_{ℚ}(N) := H_{N}\X_{N} → X_{ℚ}(1) := G̃_{N}\X_{N}

be the map induced on the quotient curves via the inclusion H_{N} ⊂ G̃_{N}, then the curve X_{ℚ}(N) (called Shimura’s canonical model of X(N)) is absolutely irreducible over ℚ, for we have that X_{ℚ}(N) ⊗ ℂ = X(N) and π_{ℚ} ⊗ ℂ = π. More precisely, the above field diagram translates into the following cartesian diagram of curves:

X_{ℚ}(N) ⟵^{q_{X(N)}}  X_{N}  ⟵ X(N)
π_{ℚ} ↓  ↓ ^{π_{ℚ_{N}}}  ↓ ^{π}
X_{ℚ}(1) ⟵^{q_{X(1)}} X_{ℚ_{N}}(1) ⟵ X(1)
p_{ℚ} ↓  ↓ ^{p_{ℚ_{N}}}  ↓
Spec(ℚ) ⟵^{q} Spec(ℚ_{N}) ⟵ Spec(ℂ)

Remark 12. Note that q, q_{X(1)} and q_{X(N)} are Galois covers with Galois group ≃ (ℤ/Nℤ)^{×} and that π_{ℚ_{N}} is a (ramified) cover with covering group G_{N}. The action of G_{N} does not descend to X_{ℚ}(N), since G_{N} does not normalize H_{N}. On the other hand, the diagonal (Cartan) subgroup of G̃_{N} does normalize H_{N}, so that its action descends to X_{ℚ}(N). In particular, the diamond operator τ_{k} acts on X_{ℚ}(N).

3.2 A ℚ-rational model for Z_{N,1}

We now use Shimura’s canonical model X_{ℚ}(N)/ℚ (and the modular curve X_{N}/ℚ_{N}) to construct a ℚ-rational model Z_{ℚ} of the surface Z = Z_{N,1}. More precisely, we shall construct Z_{ℚ} as a suitable quotient of the surface Y_{N,1} := X_{N} ×_{ℚ_{N}} X_{N} and show that Y_{ℚ}(N) := X_{ℚ}(N) × X_{ℚ}(N) maps to Z_{ℚ}. To this end we first show:

Proposition 13. The surface Y_{N,1} := X_{N} ×_{ℚ_{N}}X_{N} is naturally an irreducible component of Y_{N} := X_{N} ×_{ℚ} X_{N}, and its stabilizer via the G̃_{N} × G̃_{N}-action on Y_{N} is

Stab_{G̃_{N} ×G̃_{N}} (Y_{N,1}) = G := {(g_{1}, g_{2}) ∈ G̃_{N} × G̃_{N} : det g_{1} = det g_{2}}.

Moreover, we have

(p_{N} × p_{N}) ∘ g = σ_{det g_{i}} ∘ (p_{N} × p_{N}), if g = (g_{1}, g_{2}) ∈ G.  (4)

Furthermore, the base change maps q_{Y (N)} : Y_{N,1} → Y_{ℚ}(N) := X_{ℚ}(N) × X_{ℚ}(N) and q_{Y (1)} : Y_{1,1} → Y_{ℚ}(1) := X_{ℚ}(1) × X_{ℚ}(1) are Galois covers with covering groups H := {(h, h) : h ∈ H_{N}} ≤ G and H̄ := G/(G_{N} × G_{N}) ≃ H ≃ H_{N}, respectively.

Proof. Put Y = Y_{N,1} and let pr_{Y,i} (resp. pr_{Y_{N} ,i}) denote the projection onto the i-th factor of Y (resp. Y_{N}). Then there is a unique morphism f = f_{N,1} : Y → Y_{N} such that pr_{Y_{N} ,i} ∘ f = pr_{Y,i}. Clearly, f is a closed immersion. Now Y is irreducible because it is (smooth and) geometrically irreducible over Spec(ℚ_{N}) (because X_{N}/ℚ_{N} is), and hence is a component of Y_{N} since both have dimension 2.

Now let g = (g_{1}, g_{2}) ∈ G̃_{N} × G̃_{N}. Then g ∈ Stab(Y) if and only if g factors over f, i.e. if and only p_{N} ∘ g_{1} = p_{N} ∘ g_{2}. Now by equation (3) we have p_{N} ∘ g_{i} = σ_{det(g_{i})} ∘ p_{N}, and so we see that g factors over f if and only σ_{det(g_{1})} ∘ p_{N} = σ_{det(g_{2})} ∘ p_{N}, and this is equivalent to det(g_{1}) = det(g_{2}), i.e. to g ∈ G. This proves the formula for the stabilizer and equation (4) follows immediately.

Next we observe that since X_{N} = X_{ℚ}(N) ⊗ ℚ_{N}, we have the cartesian diagram

Y_{ℚ}(N) ⟵^{q_{Y (N)}}  Y
p×p ↓  ↓ ^{p_{N} ×p_{N}}
Spec(ℚ) ⟵^{q} Spec(ℚ_{N})

in which p = p_{ℚ} ∘ π_{ℚ} : X_{ℚ}(N) → Spec(ℚ) is the structure map and q_{Y (N)} is the unique morphism such that pr_{Y_{ℚ},i} ∘ q_{Y (N)} = q_{X(N)} ∘ pr_{Y,i}, for i = 1, 2. (Here, pr_{Y_{ℚ},i} : Y_{ℚ}(N) → X_{ℚ}(N) denotes the projection onto the i-th factor of Y_{ℚ}(N).)

Now since q is a Galois cover with group {σ_{a}} ≃ (ℤ/Nℤ)^{×}, the same is true for q_{Y (N)}; more precisely, q_{Y (N)} is Galois with covering group {σ̃_{a}}, where σ̃_{a} ∈ Aut(Y) is the unique automorphism such that q_{Y (N)} ∘ σ̃_{a} = q_{Y (N)} and (p_{N} × p_{N}) ∘ σ̃_{a} = σ_{a} ∘ (p_{N} × p_{N}). We observe:

σ̃_{a} = (h_{a}, h_{a}), where h_{a} = ( \begin{matrix}1 & 0 \\ 0 & a \end{matrix}) ∈ H_{N}.  (5)

Indeed, by (4) we have (p_{N} × p_{N}) ∘ (h_{a}, h_{a}) = σ_{a} ∘ (p_{N} × p_{N}). Moreover, since q_{X(N)} ∘h = q_{X(N)} for all h ∈ H_{N}, we see that pr_{Y_{ℚ},i} ∘q_{Y (N)} ∘(h_{a}, h_{a}) = q_{X(N)} ∘ pr_{Y,i}∘(h_{a}, h_{a}) = q_{X(N)}∘h_{a}∘pr_{Y,i} = q_{X(N)}∘pr_{Y,i}, and so q_{Y (N)}∘(h_{a}, h_{a}) = q_{Y (N)} by the defining property of q_{Y (N)}. Thus, (h_{a}, h_{a}) satisfies the defining property of σ̃_{a}, and so (5) holds. We thus see that Y_{ℚ}(N) is the quotient of Y with respect to the subgroup H.

Clearly, Y_{ℚ_{N}}(1) := X_{ℚ_{N}}(1) ×_{ℚ_{N}} X_{ℚ_{N}}(1) is the quotient of Y with respect to the subgroup G_{N} × G_{N} ≤ G with quotient map π_{ℚ_{N}} × π_{ℚ_{N}}. Now since G_{N} ◁ G̃_{N}, we have that G_{N} × G_{N} ◁ G, and hence H̄ = G/(G_{N} × G_{N}) acts as a group of automorphisms on Y_{ℚ_{N}}(1). Now the quotient map induces an isomorphism H →^{∼} H̄ because H ∩(G_{N} ×G_{N}) = {1} and H ·(G_{N} ×G_{N}) = G, and so by a similar argument as above we see that q_{Y (1)} : Y_{ℚ_{N}}(1) → Y_{ℚ}(1) is the quotient map with covering group H̄.

Theorem 14. Let Δ̃ = Δ · H = {(g, g) : g ∈ G̃_{N}} ≤ G, where as before Δ = {(g, g) : g ∈ G_{N}}. Then the quotient variety Z_{ℚ} := Δ̃\Y_{N,1} is a ℚrational model of Z, and we have morphisms

X_{ℚ}(N) × X_{ℚ}(N) ⟶^{Φ_{ℚ}} Z_{ℚ} ⟶^{Ψ_{ℚ}} X_{ℚ}(1) × X_{ℚ}(1)  (6)

such that the base change of Φ_{ℚ} and Ψ_{ℚ} with ℂ is Φ and Ψ. In particular, Ψ_{ℚ} ∘ Φ_{ℚ} = π_{ℚ} × π_{ℚ}.

Proof. As before, write Y = Y_{N,1}. Furthermore, let

π_{Δ} : Y → Z_{ℚ_{N}} := Δ\Y and πΔ̃ : Y → Z_{ℚ} := Δ̃\Y

denote the associated quotient maps and spaces; these exist (as varieties) since we are dealing with quotients of (quasi-) projective varieties by finite groups.

Since Δ ◁ Δ̃, and H →^{∼} Δ̃/Δ, we see that πΔ̃ = q_{Z(N)} ∘ π_{Δ}, where q_{Z} : Z_{ℚ_{N}} → Z_{ℚ} is the quotient map with covering group Δ̃/Δ ≃ H. Furthermore, the inclusions of subgroups H ≤ Δ̃ ≤ G induce quotient maps Φ_{ℚ} : Y_{ℚ}(N) = H\Y → Z_{ℚ} = Δ̃\Y and Ψ_{ℚ} : Z_{ℚ} = Δ̃\Y → Y_{ℚ}(1) = G\Y which fit into the following cartesian diagram:

Y_{ℚ}(N) ⟵^{q_{Y (N)}}  Y  ⟵ X(N) × X(N)
Φ_{ℚ} ↓  ↓ ^{π_{Δ}}  ↓ ^{Φ}
Z_{ℚ}  ⟵^{q_{Z}}  Z_{ℚ_{N}}  ⟵  Z
Ψ_{ℚ} ↓  ↓  ↓ ^{Ψ}
Y_{ℚ}(1) ⟵^{q_{Y (1)}} Y_{ℚ_{N}}(1) ⟵ X(1) × X(1)
↓  ↓  ↓
Spec(ℚ) ⟵^{q} Spec(ℚ_{N}) ⟵  Spec(ℂ)

Thus, the base change of Φ_{ℚ} and Ψ_{ℚ} is Φ and Ψ, respectively. Furthermore, we have Ψ_{ℚ} ∘ Φ_{ℚ} = π_{ℚ} × π_{ℚ} because we have equality for these morphisms after (faithfully flat) base change with q, and so they must be equal.

Remark 15. The above proof shows that Z = Z_{N,1} has a canonical ℚ-rational model Z_{ℚ} = Z_{N,1/ℚ}; this is the only case required here. In a similar way, however, one can show that every Z_{N,ε} has a canonical ℚ-rational model Z_{N,ε/ℚ}. Indeed, we can use essentially the same construction if we replace Y = Y_{N,1} by Y_{N,ε} := X_{N} ×_{ℚ_{N} ,σ_{ε}}X_{N}, the fibre product of X_{N} with itself via the morphisms p_{N} : X_{N} → Spec(ℚ_{N}) and σ_{ε} ∘ p_{N}. In addition, the group Δ has to be replaced by the twisted diagonal subgroup Δ_{ε} = (1 × h_{ε}^{−1})Δ(1 × h_{ε}) = {(g, α_{ε}(g)) : g ∈ G_{N}}, and similarly, Δ̃ by Δ̃_{ε} = Δ_{ε}H. (Note that Δ̃_{ε} ≤ G and that the group G still acts on Y_{N,ε}.) Thus, the canonical ℚ-rational model of Z_{N,ε} is

Z_{N,ε/ℚ} = Δ̃_{ε}\Y_{N,ε}.

We further observe that each Y_{N,ε} has a canonical embedding f_{ε} : Y_{N,ε} → Y_{N} := X_{N} ×_{ℚ}X_{N}, which leads to the decomposition of Y_{N} into its irreducible components:

Y_{N} := X_{N} ×_{ℚ} X_{N} = ∐_{ε} Y_{N,ε};  (7)

note that (1 × h_{ε}^{−1}) ∘ f_{1} = f_{ε}, so Y_{N,ε} ≃ Y_{N,1} and G̃ := G̃_{N} × G̃_{N} acts transitively on the components. In fact, since G◁G̃, we see that StabG̃(Y_{N,ε}) = G for all ε and hence g(Y_{N,ε}) = Y_{N,aε}, where a = det(g_{1})/ det(g_{2}) if g = (g_{1}, g_{2}) ∈ G̃_{N} × G̃_{N}.

Proposition 16. Every Hecke curve T_{n} on Z_{ℚ} is defined over ℚ.

Proof. Recall that T_{n} = Φ ((τ_{k}^{−1} × 1)T′(1, n)) , where we have k^{2}n ≡ 1 mod N. By Proposition 7.7 of [16], T′(1, n) is defined over ℚ, i.e. we can view T′(1, n) ⊂ X_{ℚ}(N) × X_{ℚ}(N). By Remark 12, the action of τ_{k} commutes with the action of Gal(ℚ̄/ℚ), i.e. τ_{k} is defined over ℚ, and hence Φ_{ℚ} ((τ_{k}^{−1} × 1)T′(1, n)) is a model for T_{n} over ℚ.

3.3 A model for the elliptic fibration

We now return to the situation of §2, i.e. we assume that N = 11. Here we have:

Proposition 17. The surface Z̃ admits a rational model Z̃_{ℚ}. Moreover, each component of E_{∞} is defined over ℚ.

Proof. By the uniqueness property of the minimal resolution of singularities, the morphism σ : Z̃ → Z descends to a morphism σ_{ℚ} : Z̃_{ℚ} → Z_{ℚ}. This proves the first part.

Since Ψ is defined over ℚ, the set Σ_{∞} is Galois invariant. By Proposition 4.4, we see that each singularity of Σ_{∞} has a unique type. Since the singularity type is a Galois invariant, it follows that every singularity z_{i} of Σ_{∞} is ℚ-rational. Since σ is defined over ℚ, the chain E_{∞,i} is ℚ-rational. It is clear that C̃_{∞,1} is ℚ-rational as well. Therefore, the component of E_{∞,i} which meets C̃_{∞,1} (which is unique by Proposition 5) is ℚ-rational, and hence so are the other components. This proves the second part.

Proposition 18. The surface Z̄ admits a ℚ-rational model Z̄_{ℚ}. If f_{can} is the canonical map of Z̄_{ℚ}, then f_{can} : Z_{ℚ} → ℙ_{ℚ}^{1} is an elliptic fibration which admits the sections S̄_{0} and S̄_{1}.

Proof. Although this statement could be proved by applying the Minimal Model Theorem of Shafarevich–Lichtenbaum to the canonical map of Z̃_{ℚ}, we prefer to give the following more explicit proof.

Recall from Proposition 7(b), that Z̄ is obtained from Z̃ by blowing down T̃_{1}, Γ_{0}, Γ_{2}; T̃_{3}; T̃_{4}. We claim that all of these curves are ℚ-rational.

It follows from Propositions 16 and 17 that every T̃_{n} is ℚ-rational. Arguing as in the proof of Proposition 17, we conclude that E_{0} and E_{1} are ℚ-rational. It follows that Γ_{0} and Γ_{1} are ℚ-rational, since they are the unique components of E_{0} and E_{1} meeting T̃_{1}—cf. Proposition 7(a). This proves that Z̄ is defined over ℚ. Therefore, the canonical class of Z̄ is ℚ-rational, and thus f_{can} is defined over ℚ.

Since S_{0} and S_{1} are the unique components of E_{∞} of self-intersection −4, and they meet C̃_{∞,1} and C̃_{∞,2}, respectively, it follows that they are fixed by the Galois action. Thus, their images S̄_{0} and S̄_{1} in Z̄_{ℚ} are ℚ-rational.

Notation. The set M = M(Z_{N,1}) of Mazur’s trivial points of Z = Z_{N,1} is defined by

M(Z_{N,1}) = ⋃_{n} Ψ_{ℚ}^{−1} (ξ_{n}(X′_{0}(n)(ℚ))) ,

where ξ_{n} : X_{0}(n) → T′(1, n) ⊂ X(1) × X(1) is the normalization map and X′_{0}(n) := X_{0}(n) \ {cusps} is the non-cuspidal part of X_{0}(n).

In addition, we put M̃ = σ^{−1}(M) ∩ Z̃_{ℚ}(ℚ) and let M̄ denote the image of M̃ in Z̄_{ℚ}.

Theorem 19. The canonical map f_{can} : Z̄_{ℚ} → ℙ_{ℚ}^{1} admits infinitely many sections {S_{i}} such that S_{i}(ℚ) \ M̄ is infinite for every i.

Proof. By Proposition 18, the elliptic curve E/ℂ(t) of Theorem 9 is actually defined over ℚ, as is the point of infinite order Q. Thus, the multiples of Q correspond to infinitely many rational sections {S_{i}} of f_{can}. Since the genus of X_{0}(n) is ≥ 1 for n ≥ 16, only finitely many sections can be components of some Ψ̄^{∗}ξ_{n}(X_{0}(n)). By restricting the index i, we can assume that no S_{i} is contained in Ψ̄^{∗}ξ_{n}(X_{0}(n)). By a theorem of Mazur and Kenku [11], X′_{0}(n)(ℚ) = ∅ for n > 163. Thus, M̃ is contained in finitely many curves of Z̃, and each section S_{i} has a finite intersection with it. Since S_{i} ≃ ℙ_{ℚ}^{1}, the theorem follows.

4 Modular Interpretation

4.1 Product moduli

Fix a positive integer N, and let X_{N} be the contravariant functor

X_{N} : \underline{Sch}_{/ℚ} ⟶ \underline{Sets}

from the category of schemes over ℚ to the category of sets which is defined as follows. If S is a ℚ-scheme, then let X_{N}(S) be

{⟨E/S, α⟩ : E/S elliptic curve, α:E[N] →^{∼} (ℤ/Nℤ × ℤ/Nℤ)_{/S}},

where ⟨ · ⟩ denotes the class of S-isomorphisms, and if f : T → S is a morphism of ℚ-schemes, then X_{N}(f) : X_{N}(S) → X_{N}(T) is the map induced by base change, i.e. X_{N}(f)(⟨E/S, α⟩) = ⟨E_{(T)}, α_{(T)}⟩, where E_{(T)} = E ×_{S} T and α_{(T)} = α ×_{S} id_{T} : E_{(T)}[N] = E[N] ×_{S} T → (ℤ/Nℤ)_{/T}^{2} = (ℤ/Nℤ)_{/S}^{2} ×_{S} T.

We have a natural forget map π_{N} : X_{N} → X_{1} defined by

π_{N,S}(⟨E/S, α⟩) = ⟨E/S⟩;

here we use the obvious fact that the functor X_{1} can be identified with the functor that classifies isomorphism classes of elliptic curves. Clearly, π_{N} is G̃_{N}-invariant, where the group G̃_{N} = GL_{2}(ℤ/Nℤ)/ ± 1 acts on the left on X_{N} by g · ⟨E/S, α⟩ = ⟨E/S, g ∘ α⟩. (Note that ⟨E/S, −α⟩ = ⟨E/S, α⟩, so the action factors over the quotient group.)

It is well known (cf. [2], [3]) that the affine curve X′_{N} = X_{N} \ {cusps}, the non-cuspidal part of the curve X_{N}/ℚ introduced in subsection 3.1, represents the functor X_{N} when N ≥ 3, i.e. for every ℚ-scheme S we have a bijection

P_{X_{N} ,S} : X_{N}(S) ⟶^{∼} X′_{N}(S),

which is compatible with base change. Moreover, this bijection is G̃_{N}-equi-variant, i.e. we have P_{X_{N} ,S}(g⟨E/S, α⟩) = gP_{X_{N} ,S}(⟨E/S, α⟩).

For N ≤ 2 the curve X′_{N} only coarsely represents X_{N}: we then still have the maps P_{X_{N} ,S}, but they are only bijections when S = Spec(K), K an algebraically closed field. In the case N = 1 and S = Spec(K), K any field, P_{X_{1},K} can be given explicitly; it is the unique map such that the function j ∈ κ(X_{1}) ≃ ℚ(j) evaluates to the j-invariant j(E/K) of the elliptic curve:

j(P_{X_{1},K}(⟨E/K⟩)) = j(E/K).

Note that the above G̃_{N}-equivariance of P_{X_{N}} implies the compatibility relation

P_{X_{1},K} ∘ π_{N,K} = π_{N} ∘ P_{X_{N} ,K},

where π_{N} : X_{N} → X_{1} ≃ G̃_{N}\X_{N} denotes the quotient morphism.

Recall from subsection 3.1 that X_{N} and hence also X′_{N} come equipped with the structure maps

p_{N} : X_{N} → Spec(ℚ_{N}), p′_{N} : X′_{N} → Spec(ℚ_{N}).

Viewing Spec(ℚ_{N}) as the scheme which represents the functor μ_{N}^{prim} that associates to each scheme S/ℚ its set of primitive N-th roots of unity (i.e. the set of morphisms Spec(ℚ_{N}) → S), we see that the morphism p′_{N} represents the morphism of functors p′_{N} : X_{N} → μ_{N}^{prim} defined by

p′_{N,S}(⟨E/S, α⟩) = e_{N}(α^{−1}(1, 0), α^{−1}(0, 1)) ∈ μ_{N}^{prim}(S),  (8)

in which e_{N} = e_{N}^{E} : E[N] × E[N] → 𝔾_{m}_{/S} denotes the e_{N}-pairing (cf. [15] or [10]).

Next, consider the product functor Y_{N} = X_{N} × X_{N}, as well as the subfunctor Y_{N,ε} of Y_{N} defined for ε ∈ (ℤ/Nℤ)^{×} by

Y_{N,ε}(S) = {⟨E_{1}/S, α_{1}; E_{2}/S, α_{2}⟩ ∈ Y_{N}(S) :
e_{N} ∘ ((α_{2}^{−1} ∘ α_{1}) × (α_{2}^{−1} ∘ α_{1})) = e_{N}^{ε}};

like X_{N}, these are functors from \underline{Sch}_{/ℚ} to \underline{Sets}. We have:

Proposition 20. If N ≥ 3, the ℚ-schemes Y′_{N} = X′_{N} ×_{ℚ} X′_{N} and Y′_{N,1} = X′_{N} ×_{ℚ_{N}} X′_{N} represent the functors Y_{N} and Y_{N,1}, respectively; i.e. we have natural isomorphisms of functors

P_{Y_{N}} : Y_{N} →^{∼} h_{Y′_{N}} and P_{Y_{N,1}} = P_{Y_{N} |Y_{N,1}} : Y_{N,1} →^{∼} h_{Y′_{N,1}}.

Proof. Since Y_{N} is just the product functor, the first assertion is obvious. For the second notice that the condition that e_{N}^{E_{1}} = e_{N}^{E_{2}} ∘(α_{2} ×α_{2})^{−1} ∘(α_{1} ×α_{1}) is by equation (8) equivalent to the condition that p′_{N}(⟨E_{1}, α_{1}⟩) = p′_{N}(⟨E_{2}, α_{2}⟩), which means that Y_{N,1} = X_{N} ×_{ℚ_{N}} X_{N} is the fibre product of X_{N} with itself over Spec(ℚ_{N}) via p′_{N}, and hence is represented by the fibre product Y′_{N,1} = X′_{N} ×_{ℚ_{N}} X′_{N} over Spec(ℚ_{N}) (viewed as a ℚ-scheme).

Remark 21. The same argument shows that Y_{N,ε} = X_{N} ×_{ℚ_{N} ,σ_{ε}} X_{N} is the fibre product of X_{N} with itself via the maps p′_{N} and σ_{ε} ∘ p′_{N}, and hence is represented by the scheme Y′_{N,ε} = X′_{N} ×_{ℚ_{N} ,σ_{ε}} X′_{N} (cf. Remark 15). Thus, P_{Y_{N}} restricts to an isomorphism

P_{Y_{N,ε}} = P_{Y_{N} |Y_{N,ε}} : Y_{N,ε} →^{∼} h_{Y′_{N,ε}}.

Note that although Y′_{N,ε} is a scheme over ℚ_{N}, we view it here as a scheme over ℚ; in particular, Y′_{N,ε}/ℚ is not geometrically connected.

4.2 Quotient moduli

Let Z_{N} be the contravariant functor of ℚ-schemes defined by

Z_{N}(S) = {⟨E_{1}/S, E_{2}/S, ψ⟩ :

E_{1}/S, E_{2}/S elliptic curves, ψ : E_{1}[N] ⟶^{∼} E_{2}[N]}.

We have natural maps of functors Φ_{N} : Y_{N} → Z_{N} and Ψ : Z_{N} → Y_{1} defined for each ℚ-scheme S by the rules

Φ_{N,S}(⟨E_{1}, α_{1}; E_{2}, α_{2}⟩) = ⟨E_{1}, E_{2}, α_{2}^{−1} ∘ α_{1}⟩,
Ψ_{S}(⟨E_{1}, E_{2}, ψ⟩) = ⟨E_{1}, E_{2}⟩.

Clearly, Ψ ∘ Φ_{N} = π_{N} × π_{N} while Φ_{N} is G̃_{N}-invariant with respect to the diagonal action on Y_{N}, and so Φ_{N} factors over the quotient functor Π_{N} : Y_{N} → \overline{Y}_{N} := G̃_{N}\Y_{N}, i.e. we have an induced map

\overline{Φ}_{N} : \overline{Y}_{N} := G̃_{N}\Y_{N} → Z_{N}

such that Φ_{N} = \overline{Φ}_{N} ∘ Π_{N}. It is easy to see that \overline{Φ}_{N} is an injection. Unfortunately, \overline{Φ}_{N} is not bijective, so Z_{N} is not quite the quotient functor. (However, both functors have the same sheafifications in the étale topology). Nevertheless, Z_{N} still is coarsely represented by the quotient scheme Z′_{N} := G̃_{N}\Y′_{N}, the non-cuspidal part of Z_{N} := G̃_{N}\Y_{N}. This means that we have a natural map (of functors) P_{Z_{N}} : Z_{N} → h_{Z′_{N}} , where h_{Z′_{N}} denotes the functor associated to the ℚ-scheme Z′_{N} (i.e. h_{Z′_{N}} (S) = Z′_{N}(S) = Hom_{Sch}(S, Z′_{N})), which is bijective on geometric points and which is “best possible” in a certain sense. (In particular, it then follows that the above functor Φ_{N} : Y_{N} → Z_{N} is compatible with the quotient morphism Φ_{N} : Y′_{N} → G̃_{N}\Y′_{N} = Z′_{N}.) Since the proof of this general fact is somewhat technical (cf. [7]), we content ourselves with the following result which suffices for our purposes.

Proposition 22. For every field K ⊃ ℚ there is a unique map

P_{Z_{N} ,K} : Z_{N}(K) → Z′_{N}(K)

which is compatible with base change (of fields) such that the diagram

\begin{matrix}Y_{N}(K) & ⟶^{P_{Y_{N} ,K}} & Y′_{N}(K) \\ ^{Φ}^{N,K} ↓ & & ↓ ^{Φ_{N}} \\ Z_{N}(K) & ⟶^{P_{Z_{N} ,K}} & Z′_{N}(K) \\ ^{Ψ}_{K} ↓ & & ↓ ^{Ψ} \\ Y_{1}(K) & ⟶^{P_{Y_{1},K}} & Y′_{1}(K) \end{matrix}  (9)

commutes. Furthermore, P_{Z_{N} ,K} is a bijection if K is algebraically closed.

Proof. Suppose first that K = \overline{K} is algebraically closed. Then the map \overline{Φ}_{N,K} : \overline{Y}_{N}(K) := G̃_{N}\Y_{N}(K) → Z_{N}(K) is bijective.

Indeed, since we already know that \overline{Φ}_{N,K} is injective, it is enough to show that Φ_{N,K} (and hence \overline{Φ}_{N,K}) is surjective. For this, let ⟨E_{1}, E_{2}, ψ⟩ ∈ Z_{N}(K). Since K is algebraically closed (and char(K) = 0), there is a K-rational level N structure α_{2} : E_{2}[N] →^{∼} ℤ/Nℤ × ℤ/Nℤ on E_{2}. Put α_{1} := α_{2} ∘ ψ. Then ⟨E_{1}, α_{1}; E_{2}, α_{2}⟩ ∈ Y_{N}(K) and Φ_{N,K}(⟨E_{1}, α_{1}; E_{2}, α_{2}⟩) = ⟨E_{1}, E_{2}, ψ⟩. This means that Φ_{N,K} is surjective and so Z_{N}(K) is the G̃_{N}-quotient of Y_{N}(K) by Φ_{N,K}.

On the other hand, since Z′_{N} = G̃_{N}\Y′_{N} is the geometric quotient of Y′_{N}, we know (since K is algebraically closed) that Z′_{N}(K) is the (set-theoretic) G̃_{N}-quotient of Y′_{N}(K). Thus, since P_{Y_{N} ,K} : Y_{N}(K) → Y′_{N}(K) is bijective, it follows that (Z_{N}(K), Φ_{N,K}) and (Z′_{N}(K), Φ_{N} ∘ P_{Y_{N} ,K}) are both set-theoretic quotients of Y_{N}(K), so there is a unique bijection P_{Z_{N} ,K} : Z_{N}(K) →^{∼} Z′_{N}(K) such that P_{Z_{N} ,K} ∘ Φ_{N,K} = Φ_{N} ∘ P_{Y_{N} ,K}, i.e. such that the top part of diagram (9) commutes.

We observe that this bijection is Aut(K)-equivariant, i.e. we have

P_{Z_{N} ,K}(⟨E_{1}, E_{2}, ψ⟩^{σ}) = P_{Z_{N} ,K}(⟨E_{1}, E_{2}, ψ⟩)^{σ}, for σ ∈ Aut(K).

Here σ acts (on the right) on Z_{N}(K) by functoriality (i.e. by Z_{N}(σ)) and on Z′(K) by the usual action on K-rational points.

To prove that the bottom square of diagram (9) commutes, observe that the outer rectangle of (9) commutes because Ψ ∘ Φ_{N} = π_{N} × π_{N}, Ψ ∘ Φ_{N} = π_{N} × π_{N}, and P_{X_{1},N} ∘ π_{N,K} = π_{N} ∘ P_{X_{N} ,K}, and so the assertion follows by a diagram chase since Φ_{N,K} is surjective (when K is algebraically closed).

Now consider the case of an arbitrary field K with algebraic closure \overline{K} and let G_{K} = Gal(\overline{K}/K) denotes its absolute Galois group. Then we can identify Z′_{N}(K) as the subset of G_{K}-invariant elements of Z′_{N}(\overline{K}), i.e. Z′_{N}(K) = Z′_{N}(\overline{K})^{G_{K}}. The inclusion i^{∗} : K → \overline{K} induces a morphism i : Spec(\overline{K}) ↪ Spec(K) and hence a map Z_{N}(i) : Z_{N}(K) → Z_{N}(\overline{K}). Now the image of Z_{N}(i) is actually G_{K}-invariant because i ∘ σ = i, for all σ ∈ G_{K}, and so we can define P_{Z_{N} ,K} as the composition

\begin{aligned}P_{Z_{N} ,K} & = P_{Z_{N} ,\overline{K}}^{G_{K}} ∘ Z_{N}(i) : \\ & Z_{N}(K) → Z_{N}(\overline{K})^{G_{K}} → Z_{N}(\overline{K})^{G_{K}} = Z_{N}(K). \end{aligned}

It is then clear that the diagram (9) commutes and that the P_{Z_{N} ,K} commute with base change. Furthermore, P_{Z_{N} ,K} is unique, for compatibility with base change (which includes Galois invariance) forces the above definition.

Remark 23. The above proof shows that we have the following formula for P_{Z_{N} ,K}:

P_{Z_{N} ,K}(⟨E_{1}, E_{2}, ψ⟩) = Φ_{N}(P_{Y_{N} ,\overline{K}}(⟨E_{1} ⊗ \overline{K}, α_{2} ∘ ψ; E_{2} ⊗ \overline{K}, α_{2}⟩)),

in which α_{2} : E_{2}[N]_{/\overline{K}} → (ℤ/Nℤ)^{2} is any level N-structure of E_{2} ⊗ \overline{K}.

Note that for a non-algebraically closed field K the map P_{Z_{N} ,K} is not injective. Indeed, for any ⟨E_{1}, E_{2}, ψ⟩ ∈ Z_{N}(K) and any quadratic character χ : G_{K} → {±1} we have

P_{Z_{N} ,K}(⟨E_{1}, E_{2}, ψ⟩) = P_{Z_{N} ,K}(⟨E_{1,χ}, E_{2,χ}, ψ_{χ}⟩),

where E_{i,χ} = (E_{i})_{χ} denotes the quadratic twist of E_{i}/K, and

ψ_{χ} : E_{1,χ}[N] → E_{2,χ}[N]

is the χ-twist of ψ, i.e. the unique isomorphism such that ψ_{χ} ⊗ \overline{K} = f_{2}^{−1} ∘ (ψ ⊗ \overline{K}) ∘ f_{1}_{|E_{χ}[N]}, where f_{i} : E_{i,χ} ⊗ \overline{K} →^{∼} E_{i} ⊗ \overline{K} is the twist isomorphism associated to χ. (Thus, f_{i} is a \overline{K}-isomorphism such that f_{i}(id_{E_{i,χ}} × σ) = χ(σ)(id_{E_{i}} × σ), for all σ ∈ G_{K}.) Note that it follows from the definition that ψ_{χ} ⊗ \overline{K} ∘ (id_{E_{1,χ}} × σ) = (id_{E_{2,χ}} × σ) ∘ ψ_{χ}, so ψ_{χ} exists by descent theory.

However, we do have that P_{Z_{N} ,K} is always surjective. For simplicity, we prove the following slightly weaker assertion in which we replace Z′_{N} by the open subscheme Z′′_{N} := Φ_{N}(X′′_{N} × X′′_{N}), where X′′_{N} = X_{N} \ π_{N}^{−1}({P̄_{0}, P̄_{1}, P̄_{∞}}).

Proposition 24. For every field K we have Z′′_{N}(K) ⊂ Im(P_{Z_{N} ,K}).

Before proving this, we first establish the following two facts.

Lemma 25. Let z ∈ Z′_{N}(K). Then there exist elliptic curves E_{1}/K and E_{2}/K and level N structures α_{i} : E_{i}[N]_{/\overline{K}} →^{∼} ℤ/Nℤ × ℤ/Nℤ for i = 1, 2 such that

Φ_{N}(P_{Y_{N} ,\overline{K}}(⟨E_{1} ⊗ \overline{K}, α_{1}; E_{2} ⊗ \overline{K}, α_{2}⟩)) = z.

Proof. Since Φ_{N} : Y′_{N} → Z′_{N} is a geometric quotient, there exists y ∈ Y′_{N}(\overline{K}) such that Φ_{N}(y) = z. Then by Proposition 20 there are elliptic curves E_{i}/\overline{K} and level N structures α_{i} : E_{i}[N] → (ℤ/Nℤ)^{2} such that P_{Y_{N}} (⟨E_{1}/\overline{K}, α_{1}; E_{2}/\overline{K}, α_{2}⟩) = y.

Since z ∈ Z′_{N}(K), we have that Ψ(z) ∈ X′_{1}(K)×X′_{1}(K) ≃ 𝔸^{1}(K)×𝔸^{1}(K). But Ψ(z) = ΨΦ_{N}(y) = (j(E_{1}/\overline{K}), j(E_{2}/\overline{K})), so we see that j(E_{i}/\overline{K}) ∈ K. This means that there exist elliptic curves E′_{i}/K such that E′_{i} ⊗ \overline{K} ≃ E_{i}. Taking these as representatives of the isomorphism class (and adapting the α_{i} accordingly) yields the assertion.

Lemma 26. Suppose that E_{1} and E_{2} are elliptic curves over a field K ⊃ ℚ and ψ : E_{1}[N]_{/\overline{K}} →^{∼} E_{2}[N]_{/\overline{K}} is an isomorphism such that

P_{Z_{N}} (⟨E_{1} ⊗ \overline{K}, E_{2} ⊗ \overline{K}, ψ⟩) ∈ Z′′_{N}(K).

Then there is a character χ : G_{K} → {±1} such that

ψσ = σψχ(σ),  for all σ ∈ G_{K}.  (10)

Proof. Let α_{2} : E_{2}[N]_{/\overline{K}} → ℤ/Nℤ × ℤ/Nℤ be a level N-structure and put α_{1} = α_{2} ∘ ψ. Then, writing E_{i}/\overline{K} in place of E_{i} ⊗ \overline{K}/\overline{K}, we have Φ_{N}(⟨E_{1}/\overline{K},α_{1}; E_{2}/\overline{K},α_{2}⟩) = ⟨E_{1}/\overline{K}, E_{2}/\overline{K}, ψ⟩ and thus, if we let

z = P_{Z_{N}} (⟨E_{1}/\overline{K}, E_{2}/\overline{K}, ψ⟩) and y = P_{Y_{N}} (⟨E_{1}/\overline{K}, α_{1}; E_{2}/\overline{K}, α_{2}⟩),

then Φ_{N}(y) = z. Since z is G_{K}-invariant, we have for any σ ∈ G_{K} that y^{σ} ∈ Φ_{N}^{−1}(z), so y^{σ} = g_{σ} · y, for some g_{σ} ∈ GL_{2}(ℤ/Nℤ). Note that g_{σ} is unique up to ±1 because G̃_{N} = GL_{2}(ℤ/Nℤ)/ ± 1 acts freely on Φ_{N}^{−1}(z); here we use the fact that z ∈ Z′′_{N}(K).

On the moduli problem, this means that

⟨E_{1}/\overline{K}, α_{1}; E_{2}/\overline{K}, α_{2}⟩^{σ} = g_{σ} · ⟨E_{1}/\overline{K}, α_{1}; E_{2}/\overline{K}, α_{2}⟩.

Hence, since E_{1}/\overline{K} and E_{2}/\overline{K} are G_{K}-invariant,

⟨E_{1}/\overline{K}, α_{1}σ; E_{2}/\overline{K}, α_{2}σ⟩ = ⟨E_{1}/\overline{K}, g_{σ}α_{1}; E_{2}/\overline{K}, g_{σ}α_{2}⟩.

In other words, there are automorphisms χ_{i,σ} of E_{i}/\overline{K} (which depend on the choice of g_{σ}), such that g_{σ}α_{i}χ_{i,σ} = α_{i}σ, for i = 1, 2. Now since z ∈ Z′′_{N}(K), we have j(E_{i}/\overline{K}) ≠ 0, 1728, so Aut(E_{i}/\overline{K}) = \{id_{E_{i}}\} ≃ \{±1\}, and hence χ_{i,σ} ∈ {±1}. By replacing g_{σ} with −g_{σ} if necessary (and hence replacing χ_{i,σ} by −χ_{i,σ}), we may suppose that χ_{2,σ} = 1, for all σ ∈ G_{K}. Write χ(σ) = χ_{1,σ}. We now have, for all σ ∈ G_{K},

ψσ = α_{2}^{−1}α_{1}σ = α_{2}^{−1}g_{σ}α_{1}χ(σ) = σα_{2}^{−1}α_{1}χ(σ) = σψχ(σ),

which is (10). Note that it follows from this equation that χ is a character.

of Proposition 24. Let z ∈ Z′′_{N}(K). Then by Lemma 25 there are elliptic curves E_{i}/K and level N structures α_{i} on E_{i} ⊗ \overline{K} for i = 1, 2 such that P_{Z_{N} ,\overline{K}}(⟨E_{1}/\overline{K}, E_{2}/\overline{K}, ψ⟩) = z, where ψ = α_{2}^{−1} ∘ α_{1}. Let χ be the corresponding quadratic character given by Lemma 26. If E_{2,χ}/K is the twist of E_{2}/K by χ and τ : E_{2} ⊗ \overline{K} →^{∼} E_{2,χ} ⊗ \overline{K} the corresponding isomorphism such that τχ(σ)σ = στ, then we have ⟨E_{1}/\overline{K}, E_{2,χ}/\overline{K}, τψ⟩ = ⟨E_{1}/\overline{K}, E_{2}/\overline{K}, ψ⟩. By equation (10) we have τψσ = τχ(σ)σψ = στψ, for every σ ∈ G_{K}. Thus, τψ descends to K and so ⟨E_{1}/K, E_{2,χ}/K, τψ⟩ ∈ Z_{N}(K). We thus have (cf. Remark 23)

P_{Z_{N} ,K}(⟨E_{1}/K, E_{2,χ}/K, τψ⟩) = P_{Z_{N} ,\overline{K}}(⟨E_{1}/\overline{K}, E_{2,χ}/\overline{K}, τψ⟩)
= P_{Z_{N} ,\overline{K}}(⟨E_{1}/\overline{K}, E_{2}/\overline{K}, ψ⟩) = z,

which shows that z ∈ Im(P_{Z_{N} ,K}), as claimed.

For each ε ∈ (ℤ/Nℤ)^{×}, we also have the following subfunctor Z_{N,ε} of Z_{N} which is defined by

Z_{N,ε}(S) = {⟨E_{1}/S, E_{2}/S, ψ⟩ ∈ Z_{N}(S) : e_{N} ∘ ψ = e_{N}^{ε}}.

It is then clear that Z(S) = ∐ Z_{N,ε}(S).

Restricting the above (functor) morphisms Φ_{N} and P_{Y_{N}} (cf. Remark 21 and Prop. 22) to the component Y_{N,ε} of Y_{N} yields morphisms

Φ_{N,ε} = Φ_{N}_{|Y_{N,ε}} : Y_{N,ε} → Z_{N,ε} and P_{Y_{N,ε}} = P_{Y_{N} |Y_{N,ε}} : Y_{N,ε} →^{∼} h_{Y′_{N,ε}}.

On the other hand, we also have the quotient morphism

Φ_{N,ε} : Y′_{N,ε} → Z′_{N,ε/ℚ} = Δ̃_{ε}\Y′_{N,ε}

where Z′_{N,ε/ℚ} denotes the non-cuspidal part of the surface Z_{N,ε/ℚ} defined in section 3.2 (cf. Theorem 14 and Remark 15). These morphisms are connected as follows:

Theorem 27. For every field K ⊃ ℚ there is a unique map

P_{Z_{N,ε},K} : Z_{N,ε}(K) → Z′_{N,ε/ℚ}(K)

which is compatible with base change (of fields) such that

Φ_{N,ε} ∘ P_{Y_{N,ε},K} = P_{Z_{N,ε},K} ∘ Φ_{N,ε}.  (11)

In addition, P_{Z_{N,ε},K} is a bijection if K is algebraically closed, whereas for an arbitrary field K we have

Z′′_{N,ε}(K) ⊂ Im(P_{Z_{N,ε},K}).

Proof. Define P_{Z_{N,ε},K} = P_{Z_{N} ,K}_{|Z_{N,ε}(K)} : Z_{N,ε}(K) → Z′_{N}(K). Then it is clear from Proposition 22 that P_{Z_{N,ε},K} is compatible with base change and that equation (11) holds. In addition, it follows that if K is algebraically closed, then P_{Z_{N,ε},K} is a bijection onto its image. Now by (11), this image has to be Z_{N,ε/ℚ}(K) because it is the image of Φ_{N,ε} (and because Φ_{N,ε,K} is surjective). From this we see in particular that for any K, the image of P_{Z_{N,ε},K} is contained in Z_{N}(K) ∩ Z_{N,ε/ℚ}(\overline{K}) = Z_{N,ε/ℚ}(K), and so P_{Z_{N,ε},K} satisfies the requirements of the theorem.

4.3 Proof of Theorem 2

Lemma 28. Let ⟨E_{1}/ℚ, E_{2}/ℚ, ψ⟩ ∈ Z_{N,1}(ℚ). If E_{1} and E_{2} are ℚ-isoge-neous, then P_{Z_{N,1},ℚ}(⟨E_{1}, E_{2}, ψ⟩_{ℚ}) ∈ M(Z_{N,1}), i.e., it is a Mazur trivial point.

Proof. Since E_{1} and E_{2} are ℚ-isogeneous, there is a cyclic ℚ-isogeny between them. If n is its degree, then by the modular description of X′_{0}(n) we have P_{Y_{1},ℚ}(⟨E_{1}, E_{2}⟩_{ℚ}) ∈ ξ_{n}(X′_{0}(n)(ℚ)), where ξ_{n} is the normalization map as in section 3.3. By Proposition 22, this implies that Ψ ∘ P_{Z_{N,1},ℚ}(⟨E_{1}, E_{2}, ψ⟩_{ℚ}) ∈ ξ_{n}(X′_{0}(n)(ℚ)) and so the assertion follows from the definition of M(Z_{N,1}) (cf. section 3.3).

of Theorem 2. Consider the infinite number of sections of S̄_{i} : ℙ_{ℚ}^{1} → Z̄ given by Theorem 19. By Propositions 4 and 7, the proper transform S_{i} in Z/ℚ of each S̄_{i} is a curve S_{i} ≃ ℙ_{ℚ}^{1} on Z; by construction, |S_{i}(ℚ) \ M| = ∞, where M = M(Z_{11,1}). By Theorem 27 and Lemma 28, the points of S_{i}(ℚ) ∩ Z′(ℚ) \ M correspond to a one-parameter family of isomorphism classes of pairs of elliptic curves over ℚ with symplectically isomorphic 11-structure which are not ℚ-isogeneous, and so we have proved the theorem.

The authors are very thankful for helpful and stimulating discussions with G. Frey. The second author wants to thank Queen’s University and the first author for their hospitality and kindness.

This research was supported in part by a grant from the Natural Science and Engineering Research Council of Canada (NSERC).

References

[1] W. Barth, C. Peters, and A. van de Ven, Compact Complex Surfaces, Springer-Verlag, Berlin, 1984.

[2] P. Deligne, Formes modulaires et représentations ℓ-adiques, Sem. Bourbaki 1968/69, No. 355. Springer Lecture Notes 179 (1971), 139–172.

[3] P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques, Modular functions of one variable, II, Lecture Notes in Math. 349, Springer-Verlag, Berlin, 1973, pp. 143–316.

[4] G. Frey and M. Müller, Arithmetic of modular curves and applications, Algorithmic Algebra and Number Theory, Heidelberg 1997 (B. H. Matzat, G.-M. Greuel and G. Hiss, eds.), Springer-Verlag, Berlin, 1999, pp. 11–48.

[5] E. Halberstadt and A. Kraus, On the modular curves Y_{E}(7), To appear in Math. Comp.

[6] C. F. Hermann, Modulflächen quadratischer Diskriminante, Manuscr. Math. 72 (1991), 95–10.

[7] E. Kani, On the moduli functor Z_{N,ε}, in preparation.

[8] E. Kani and W. Schanz, Diagonal quotient surfaces, Manuscr. Math. 93 (1997), 67–108.

[9] E. Kani and W. Schanz, Modular diagonal quotient surfaces, Math. Z. 227 (1998), 337–366.

[10] N. M. Katz and B. Mazur, Arithmetic Moduli of Elliptic Curves, Princeton University Press, Princeton, NJ, 1985.

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

[12] A. Kraus and J. Oesterlé, Sur une question de B. Mazur, Math. Ann. 293 (1992), no. 2, 259–275.

[13] S. Lang, Elliptic Functions, Addison-Wesley, Reading, MA, 1973.

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

[15] D. Mumford, Abelian Varieties, Oxford University Press, Oxford, 1970.

[16] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton University Press, Princeton, NJ, 1971.

[17] J. H. Silverman, The Arithmetic of Elliptic Curves, GTM 106, Springer-Verlag, New York, 1986.

[18] J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, GTM 151, Springer-Verlag, New York, 1994.