EXPLICIT FAMILIES OF ELLIPTIC CURVES WITH PRESCRIBED MOD N REPRESENTATIONS

A. SILVERBERG

Introduction

In Part 1 we explain how to construct families of elliptic curves with the same mod 3, 4, or 5 representation as that of a given elliptic curve over Q. In §4 we give equations for the families in the mod 4 case. The mod 3 and mod 5 cases were given in [9] (see also [8]). The results remain true (with the same proofs) with the field of rational numbers replaced by any field whose characteristic does not divide the level.

In Part 2 we use the work of Wiles, Taylor-Wiles, and Diamond to give explicit equations for infinite families of modular elliptic curves. In §7 (see Theorem 7.3) we show how to find infinite families of modular elliptic curves with the same mod 4 representation. In §8 we prove that if E is an elliptic curve over Q, and the torsion subgroup of E(Q) is not cyclic of order 1, 2, 3, 6, or 9 (i.e., the torsion subgroup is cyclic of order 4, 5, 7, 8, 10, or 12 or is of the form Z/2Z × Z/2NZ for N = 1, 2, 3 or 4), then E is modular (see Theorem 8.1 and Corollary 8.10).

The proofs of the results in §4 use symbolic computer computations, which were done using the programs Pari and Mathematica. I would like to thank Ken Ribet and Karl Rubin for useful conversations, and the IHES for its hospitality.

Notation. Let Z, Q, and C denote, respectively, the integers, rational numbers, and complex numbers.

We will suppose that N is a positive integer, and N ≥ 3. If E is an elliptic curve over a field k with algebraic closure k̄, let E[N] denote the kernel of multiplication by N on E(k̄), and let j(E) denote the j-invariant of E. If F ⊆ Q̄ is a number field, let G_{F} = Gal(Q̄/F). Let μ_{N} be the G_{Q}-module of N-th roots of unity, and let

e_{N} : E[N] × E[N] → μ_{N}

denote the Weil pairing. Let ℌ denote the complex upper half plane, and let

Γ(N) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(Z) : ( \begin{matrix}a & b \\ c & d \end{matrix} ) ≡ ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) (mod N)\}.

Let I denote the 2 × 2 identity matrix.

Part 1. Elliptic curves with the same mod N representation

1. Modular curves and elliptic modular surfaces of level N

Let

V_{N} = Z/NZ × μ_{N},

a G_{Q}-module, and define a G_{Q}-equivariant pairing

η_{N} : V_{N} × V_{N} → μ_{N}

by

η_{N}((a_{1}, ζ_{1}), (a_{2}, ζ_{2})) = ζ_{2}^{a_{1}}/ζ_{1}^{a_{2}}.

Denote by Y_{N} the (non-compact) modular curve over Q which parametrizes triples (E, P, C) where E is an elliptic curve, P is a point of exact order N on E, C is a cyclic subgroup of order N on E, and C and P generate E[N]. Let Y (N) denote the modular curve which parametrizes elliptic curves with full level N structure (see [12]). If ζ is a fixed primitive N-th root of unity in Q̄, then the map

(E, P, C) ↦ (E, P, Q),

where Q is the unique point in C such that e_{N}(P, Q) = ζ, induces an isomorphism (defined over Q(ζ)) from Y_{N} onto one connected component of Y (N). Thus Y_{N}(C) is isomorphic to ℌ/Γ(N). Let X_{N} denote the compactification of Y_{N}. Then X_{N} has genus 0 if and only if N ≤ 5 (see p. 23 of [12]).

Lemma 1.1. The curve Y_{N} parametrizes isomorphism classes of pairs (E, ϕ), where E is an elliptic curve and

ϕ : V_{N} → E[N]

is a group isomorphism with the property that for all u, v ∈ V_{N},

η_{N}(u, v) = e_{N}(ϕ(u), ϕ(v)).

Proof. Given (E, P, C), define ϕ by ϕ(a, ζ) = aP +Q for the unique Q ∈ C such that e_{N}(P, Q) = ζ. Conversely, given (E, ϕ), let P = ϕ(1, 1) and let C = ϕ(0 × μ_{N}).

There is a quasi-projective surface W_{N} defined over Q, with a projection morphism

π_{N} : W_{N} → Y_{N}

and a zero-section Y_{N} → W_{N}, both defined over Q, with N^{2} sections defined over Q̄ of order dividing N, and such that the fibers of π_{N} correspond to the triples (E, P, C) classified by Y_{N}. (Note that this notation differs from that of [9], where W_{N} denoted a compactification.) The variety W_{N} can be viewed as the universal elliptic curve with level structure as above. See [13] for the theory of elliptic modular surfaces of level N. Models for W_{3} and W_{5} are given in [9]. A model for W_{4} is given in §4 below. Analytically, we have

W_{N}(C) ≅ (ℌ × C)/(Γ(N) ⋉ Z^{2}).

If τ ∈ ℌ, then the equivalence class of τ in ℌ/Γ(N) corresponds to the C-iso-morphism class of the triple (C/Zτ + Z, τ/N, ⟨1/N⟩). Let W_{N}[N] denote the N^{2} sections of π_{N} of order dividing N, viewed as a G_{Q}-module.

2. Twists of Y_{N} and W_{N}

Let Aut(V_{N}, η_{N}) denote the group of automorphisms of V_{N} which preserve η_{N}. Suppose V is a free rank-2 module over Z/NZ with a continuous and linear G_{Q}- action and suppose

η : V × V → μ_{N}

is a non-degenerate alternating G_{Q}-equivariant pairing. Fix a group isomorphism φ : V_{N} → V under which the pairing η_{N} corresponds to the pairing η. Then τ ↦ φ^{−1} ∘ τ(φ) defines a cocycle on G_{Q} with values in Aut(V_{N}, η_{N}). By the universal property of W_{N}, there is a natural injective G_{Q}-equivariant homomorphism

Aut(V_{N}, η_{N}) ↪ Aut(W_{N}).

There is also a natural G_{Q}-equivariant homomorphism

Aut(W_{N}) → Aut(Y_{N}).

Therefore, the above cocycle induces cocycles c and c_{0} on G_{Q} with values in Aut(W_{N}) and in Aut(Y_{N}), respectively. Let W (respectively, Y) denote the twist of W_{N} (respectively, Y_{N}) by the cocycle c (respectively, c_{0}) (see [10]). Then W and Y are quasi-projective varieties defined over Q. Up to isomorphism, W and Y are independent of the choice of φ. We obtain isomorphisms

ψ : W → W_{N}  and  ψ_{0} : Y → Y_{N}

defined over Q̄, and a projection morphism π : W → Y defined over Q, such that the diagram

W ⟶^{ψ} W_{N}
↓ π  ↓ π_{N}
Y ⟶^{ψ_{0}} Y_{N}

commutes, and such that for every τ ∈ G_{Q},

c(τ) = ψ ∘ τ(ψ)^{−1} and c_{0}(τ) = ψ_{0} ∘ τ(ψ_{0})^{−1}.

It follows from the definition of W that if t ∈ Y (C) and E_{t} is the elliptic curve π^{−1}(t), then E_{t}[N] and V are isomorphic as Gal(\overline{Q(t)}/Q(t))-modules.

Theorem 2.1. Suppose N = 3, 4, or 5, and E is an elliptic curve over Q. Then there are infinitely many elliptic curves E′ over Q such that E[N] and E′[N] are isomorphic as G_{Q}-modules.

Proof. Let V = E[N] and let η = e_{N}, the Weil pairing on E[N]. Let W and Y be the varieties constructed as above, for V and η, and let X denote the compactification of Y. Since E is defined over Q, Y (Q) is nonempty. Since N ≤ 5, X has genus 0. Therefore, X is isomorphic to P^{1}, and X(Q) and Y (Q) are infinite. The points of Y (Q) correspond to the desired elliptic curves E′.

3. Model for W when N = 3, 4, or 5

Suppose now that N = 3, 4, or 5 and E is an elliptic curve over Q with Weierstrass model y^{2} = x^{3} + ax + b, with a, b ∈ Q. We will construct a model for W, where Y and W are the twists of Y_{N} and W_{N} as in §2, with V = E[N] and η = e_{N}. For N = 3, 4, and 5, let m = 1, 2, and 5, respectively. Then m = #PSL_{2}(Z/NZ).

We can find a model for W_{N} (see (1) and (3) of [9] for N = 3 and N = 5, respectively, and (5) below for N = 4) such that for u ∈ Y_{N}, the fiber E_{u} = π_{N}^{−1}(u) is of the form

E_{u} : y^{2} = x^{3} + a_{4}(u)x + a_{6}(u) (1)

where a_{4}(u), a_{6}(u) ∈ Q[u] and deg(a_{j}) = jm for j = 4, 6. Let u_{0} be an algebraic number such that E_{u_{0}} is isomorphic (over Q̄) to E. The isomorphism ψ_{0} : Y → Y_{N} extends to an isomorphism ψ_{0} : X → X_{N} on the compactifications. Since X and X_{N} are isomorphic to P^{1}, the isomorphism ψ_{0} can be given by a linear fractional transformation, which can be normalized so that 0 is sent to u_{0}. Let

A = (\begin{matrix}α & u_{0} \\ γ & 1 \end{matrix}) ∈ GL_{2}(Q̄)

be such a transformation. Since ψ takes a fiber E_{t} = π^{−1}(t) in W isomorphically onto a fiber E_{u} = π_{N}^{−1}(u) in W_{N}, the isomorphism ψ takes a point (t, x, y) ∈ W ⊆ P^{1} × P^{2} to a point of the form (A(t), h(t)^{−2}x, h(t)^{−3}y) ∈ W_{N} ⊆ P^{1} × P^{2}, for appropriate h(t). Therefore, (h(t)^{−2}x, h(t)^{−3}y) lies on E_{A(t)}. Using (1), it follows that (t, x, y) satisfies

y^{2} = x^{3} + h(t)^{4}a_{4}(A(t))x + h(t)^{6}a_{6}(A(t)). (2)

When t = 0, we would like (2) to be an equation for the elliptic curve E. We will solve for h(t)^{2}, α, and γ so that

h(t)^{4}a_{4}(A(t)) and h(t)^{6}a_{6}(A(t)) (3)

are in Q[t] (i.e., so that (2) is a model over Q for W) and take on the values a and b, respectively, when t = 0. From

h(0)^{4}a_{4}(A(0)) = a  and  h(0)^{6}a_{6}(A(0)) = b, (4)

we can solve for h(0)^{2}. Write h(t)^{2} = h(0)^{2}(γt+1)^{2m} and substitute into (3). Then the expressions in (3) become polynomials in Q̄[t], and have constant terms a and b, respectively. In particular, (2) is E when t = 0. Take any ordered pair of rational numbers (r, s) which is not a rational multiple of (4a, 6b), set the coefficients of t in the polynomials in (3) equal to r and s, respectively, and solve for α and γ. With these values, (2) is a model over Q for W. Different choices of the pair (r, s) give rise to Q-isomorphic elliptic surfaces.

Let J = j(E)/1728. The resulting model for W is of the form

y^{2} = x^{3} + af_{4}(J, t)x + bf_{6}(J, t)

where f_{4}, f_{6} ∈ Z[J, t], f_{4} and f_{6} depend only on N, and deg_{t}(f_{j}) = jm for j = 4, 6.

4. Level 4

We write down a model for the elliptic modular surface W_{4}, following [14]. We can view W_{4} as a surface over Q or as an elliptic curve over a function field in one variable. Define

A_{u} : y^{2} = x(x − 1)(x − \frac{(u + u^{−1})^{2}}{4} ).

If u ∈ C and u ∉ {0, 1, −1, i, −i}, then A_{u} is an elliptic curve over Q(u), and a Weierstrass model for A_{u} is given by

E_{u} : y^{2} = x^{3} − 27(u^{8} + 14u^{4} + 1)x − 54(u^{12} − 33u^{8} − 33u^{4} + 1). (5)

We have

j(A_{u}) = j(E_{u}) = \frac{16(u^{8} + 14u^{4} + 1)^{3}}{u^{4}(u^{4} − 1)^{4}} . (6)

Let

P_{u} = (\frac{u^{2} + 1}{2u^{2}}, \frac{1 − u^{4}}{4u^{3}} ) ∈ A_{u}[4].

Let C_{u} be the Gal(\overline{Q(u)}/Q(u))-invariant cyclic subgroup of A_{u}[4] generated by the point (of order 4)

(\frac{u^{2} + 1}{2u}, \frac{i(u^{2} + 1)(u − 1)^{2}}{4u^{2}} ).

The map u ↦ (A_{u}, P_{u}, C_{u}) induces a morphism f : P^{1} → X_{4} defined over Q. The morphism j : X_{4} → P^{1} induced by (E, P, C) ↦ j(E) has degree #PSL_{2}(Z/4Z) = 24. By (6), the degree of the composition j ∘f is 24, which is the same as the degree of j. Therefore, f is an isomorphism. Identify X_{4} with P^{1} via f.

Theorem 4.1. Fix an elliptic curve over Q:

E : y^{2} = x^{3} + ax + b,

with a, b ∈ Q, and let J = j(E)/1728 = 4a^{3}/(4a^{3} + 27b^{2}). Let E_{t} be

E_{t} : y^{2} = x^{3} + a(t)x + b(t), (7)

where

a(t) = ((J − 1)^{4}(144J^{2} − 56J − 7)t^{8} − 48(J − 1)^{4}(4J + 1)t^{7} +
28(J − 1)^{3}(4J + 5)t^{6} + 224(J − 1)^{3}t^{5} + 42(J − 1)^{2}(4J − 5)t^{4} −
112(J − 1)^{2}t^{3} + 28(J − 1)t^{2} + 1)a,
b(t) = ((J − 1)^{6}(1728J^{3} − 144J^{2} + 116J + 1)t^{12} −
12(J − 1)^{5}(288J^{3} − 128J^{2} + 82J + 1)t^{11} +
66(J − 1)^{5}(48J^{2} − 56J − 1)t^{10} − 44(J − 1)^{4}(208J^{2} − 176J − 5)t^{9} −
99(J − 1)^{4}(48J^{2} − 104J − 5)t^{8} + 792(J − 1)^{3}(8J^{2} − 10J − 1)t^{7} −
924(J − 1)^{3}(4J + 1)t^{6} + 792(J − 1)^{2}t^{5} − 99(4J − 5)(J − 1)^{2}t^{4} +
44(J − 1)(6J − 5)t^{3} − 66(J − 1)t^{2} + 12t + 1)b.

Then for every rational number t such that E_{t} is nonsingular, E_{t}[4] is isomorphic as a G_{Q}-module to E[4]. If ab ≠ 0, then (7) is a model for W over Q, where W is constructed as in §2 from V = E[4], η = e_{4}.

Proof. If a = 0 then E_{t} is y^{2} = x^{3} +(t+1)^{12}b and if b = 0 then E_{t} is y^{2} = x^{3} +ax. In both cases the elliptic surface is isotrivial, and E_{t}[4] is isomorphic as a G_{Q}-module to E[4]. Now assume ab ≠ 0. Let j = j(E). Using (6), a computation shows that

j(E_{u}) − j(E) =

\frac{16(u^{24} + 1) + (672 − j)(u^{20} + u^{4}) + (9456 + 4j)(u^{16} + u^{8}) + (45248 − 6j)u^{12}}{u^{4}(u^{4} − 1)^{4}} .

Let u_{0} be a root of the numerator. Then j(E_{u_{0}}) = j(E). Following the algorithm in §3, we deduce from (4) that

h(0)^{2} = \frac{a_{4}(u_{0})b}{a_{6}(u_{0})a} = \frac{b(u_{0}^{8} + 14u_{0}^{4} + 1)}{2a(u_{0}^{12} − 33u_{0}^{8} − 33u_{0}^{4} + 1)} ∈ Q̄^{×}.

Now solve for α and γ so that the coefficients of t in the polynomials in (3) are 0 and 12b, respectively. (This choice (r, s) = (0, 12b) leads to the relatively simple polynomials a(t) and b(t) in the statement of the theorem.) We obtain

α = \frac{(7u_{0}^{4} + 1)b}{2^{2}3^{5}u_{0}^{3}(1 − u_{0}^{4})^{3}h(0)^{6}} , γ = \frac{(u_{0}^{4} + 7)b}{2^{2}3^{5}(u_{0}^{4} − 1)^{3}h(0)^{6}} .

With these values, (2) is a model over Q for W, and (2) is (7) with the stated a(t) and b(t). This elliptic surface is not isotrivial.

Theorem 4.2. Fix a nonzero integer D and define E_{t} by

y^{2} = x^{3} + a(t)x + b(t)

where

a(t) = D(81D^{2}t^{4} + 6Dt^{2} + 1)(81D^{2}t^{4} − 90Dt^{2} + 1),

b(t) = 8D^{2}t(9Dt^{2} + 1)(9D^{2}t^{4} − 2Dt^{2} + 1)(729D^{2}t^{4} − 18Dt^{2} + 1).

If t ∈ Q and 9Dt^{2} ≠ 1, then E_{t} is an elliptic curve over Q and E_{t}[4] is isomorphic as a G_{Q}-module to E[4], where E is the elliptic curve

y^{2} = x^{3} + Dx.

Proof. Using (6), a computation shows that

j(E_{u}) − j(E) = \frac{2^{4}(u^{2} − 2u − 1)^{2}(u^{2} + 2u − 1)^{2}(u^{4} + 1)^{2}(u^{4} + 6u^{2} + 1)^{2}}{u^{4}(u^{4} − 1)^{4}} .

Let u_{0} = 1 + \sqrt{2}, a root of u^{2} − 2u − 1. Now follow the algorithm in §3. We obtain

h(0)^{4} = \frac{D}{a_{4}(u_{0})} = \frac{(12\sqrt{2} − 17)D}{2^{4}3^{4}} .

Let

h(0)^{2} = \frac{(2\sqrt{2} − 3)\sqrt{−D}}{36} , r = 0, and s = 8D^{2}.

Then

α = 3\sqrt{−D}, γ = −3(1 + \sqrt{2})\sqrt{−D},

and we obtain E_{t} as in the statement of the theorem. The discriminant of E_{t} is

Δ(E_{t}) = −2^{6}D^{3}(9Dt^{2} − 1)^{4}(81D^{2}t^{4} + 54Dt^{2} + 1)^{4}.

Thus if t ∈ P^{1}(Q) and 9Dt^{2} ≠ 1, then E_{t} is an elliptic curve. The j-invariant of E_{t} is

j(E_{t}) = \frac{1728(81D^{2}t^{4} + 6Dt^{2} + 1)^{3}(81D^{2}t^{4} − 90Dt^{2} + 1)^{3}}{(9Dt^{2} − 1)^{4}(81D^{2}t^{4} + 54Dt^{2} + 1)^{4}} .

Theorem 4.3. Fix a nonzero integer D and define E_{t} by

y^{2} = x^{3} − 12Dt(8Dt^{3} − 1)(Dt^{3} + 1)x − D(8D^{2}t^{6} + 88Dt^{3} − 1)(8D^{2}t^{6} + 1).

For every rational number t, E_{t} is an elliptic curve over Q and E_{t}[4] is isomorphic as a G_{Q}-module to E[4], where E is the elliptic curve

y^{2} = x^{3} + D.

Proof. We have j(E) = 0 and

j(E_{u}) = \frac{2^{4}(u^{4} − 2u^{3} + 2u^{2} + 2u + 1)^{3}(u^{4} + 2u^{3} + 2u^{2} − 2u + 1)^{3}}{u^{4}(u^{4} − 1)^{4}} .

Let u_{0} be a root of u^{4} − 2u^{3} + 2u^{2} + 2u + 1. Let

β = u_{0}(1 + u_{0} − u_{0}^{2})  and  λ = \frac{(39β + 18)^{2/3}D^{1/3}}{6} .

Applying the algorithm of §3, we have

h(0)^{6} = \frac{D(13β − 84)}{2^{6}3^{5}}.

Let

h(0)^{2} = \frac{[3D(13β − 84)]^{1/3}}{36} , r = 12D, and s = 0.

Then

α = (11u_{0}^{3} − 33u_{0}^{2} + 49u_{0} − 11)λ, γ = (3β − 19)λ,

and we obtain E_{t} as in the statement of the theorem. The discriminant and jinvariant of E_{t} are given by

Δ(E_{t}) = −2^{4}3^{3}D^{2}(8D^{2}t^{6} − 20Dt^{3} − 1)^{4},

j(E_{t}) = \frac{−2^{14}3^{3}Dt^{3}(8Dt^{3} − 1)^{3}(Dt^{3} + 1)^{3}}{(8D^{2}t^{6} − 20Dt^{3} − 1)^{4}} .

Since Δ(E_{t}) has no rational roots, the theorem follows.

Part 2. Explicit families of modular elliptic curves

5. Modular j-invariants

If E and E′ are elliptic curves over Q, and E and E′ are isomorphic over C, then E is modular if and only if E′ is modular. It therefore makes sense to talk about modular j-invariants, i.e., the rational numbers which are j-invariants of modular elliptic curves. Before the work of Wiles, it was not known that there are infinitely many modular j-invariants. Using the results of Wiles [19], Taylor-Wiles [18], and Diamond [2], it is now very easy to write down infinite families of modular j-invariants.

We begin by stating Diamond’s improvement of the results of Wiles and Taylor-Wiles. While Theorems 7.3 and 8.1 below follow easily from this statement, in fact such results generally follow from the theorems stated in [19], with some additional work.

Theorem 5.1. If E is an elliptic curve over Q which has semistable reduction at 3 and at 5, then E is modular.

Proof. See Theorem 1.2 of [2].

6. Semistable reduction

We next state some results which will be used in the proofs of Propositions 7.1 and 8.4. If F is a number field, and v is a prime ideal of F, let I_{v} denote the inertia subgroup of G_{F} corresponding to an extension to Q̄ of the v-adic valuation on F.

Theorem 6.1. If E is an elliptic curve over Q, then there is a number field over which E has everywhere semistable reduction.

Proof. See Proposition 3.6 of [4]. See also Proposition 5.4 on p. 181 of [17].

Theorem 6.2. If E is an elliptic curve over Q, p and ℓ are distinct prime numbers, ρ_{p} : G_{Q} → GL_{2}(Z_{p}) is the p-adic representation associated to E, and τ ∈ I_{ℓ}, then the characteristic polynomial of ρ_{p}(τ) has integer coefficients which are independent of p.

Proof. See Theorem 4.3 of [4].

Theorem 6.3. Suppose E is an elliptic curve over a number field F, p and ℓ are distinct prime numbers, v is a prime ideal of F dividing ℓ, and ρ_{p} : G_{F} → GL_{2}(Z_{p}) is the p-adic representation associated to E. Then E has good reduction at v if and only if ρ_{p}(I_{v}) is trivial.

Proof. See Theorem 1 of [11].

Theorem 6.4. Suppose E is an elliptic curve over a number field F, p and ℓ are distinct prime numbers, v is a prime ideal of F dividing ℓ, and ρ_{p} : G_{F} → GL_{2}(Z_{p}) is the p-adic representation associated to E. Then the following are equivalent:

(i) E has semistable reduction at v, (ii) for every τ ∈ I_{v}, all the eigenvalues of ρ_{p}(τ) are 1 (i.e., I_{v} acts unipotently

on the p-adic Tate module of E), (iii) for every τ ∈ I_{v}, (ρ_{p}(τ) − I)^{2} = 0.

Proof. See Proposition 3.5 and Corollaire 3.8 of [4].

Let Z̄ denote the ring of algebraic integers.

Theorem 6.5. If α is a root of unity in Z̄, and either

(a) 5 ≤ N ∈ Z and (α − 1)^{2} ∈ NZ̄, or

(b) 3 ≤ N ∈ Z and α − 1 ∈ NZ̄, then α = 1.

Proof. Part (b) is well-known. See Theorem 3.1 of [16] for proofs of (a) and (b).

Lemma 6.6. Suppose that N is a positive integer, and for each prime divisor p of N we have a matrix A_{p} ∈ M_{2}(Z_{p}) such that the characteristic polynomials of the A_{p} have integral coefficients independent of p, and such that (A_{p} −I)^{2} ∈ NM_{2}(Z_{p}). Then for every eigenvalue α of A_{p}, (α − 1)/\sqrt{N} satisfies a monic polynomial with integer coefficients.

Proof. See Lemma 5.2 of [15].

7. Mod 4 representations

Proposition 7.1. Suppose E and E′ are elliptic curves over Q, N is a positive integer, E[N] and E′[N] are isomorphic as G_{Q}-modules, and ℓ is a prime number which does not divide N. If

(a) N ≥ 5 and E has semistable reduction at ℓ, or

(b) N = 3 or 4 and E has good reduction at ℓ, then E′ has semistable reduction at ℓ.

Proof. We give a proof in the spirit of [15]. Let I_{ℓ} denote the inertia subgroup of G_{Q} corresponding to an extension λ̄ to Q̄ of the ℓ-adic valuation on Q. Suppose τ ∈ I_{ℓ}, p is a prime divisor of N, and ρ_{E,p} and ρ_{E′,p} are the p-adic representations of G_{Q} associated to E and E′, respectively. Suppose α is an eigenvalue of ρ_{E′,p}(τ). There is a number field F such that E has semistable reduction at the restriction λ of λ̄ to F (by Theorem 6.1). Therefore, τ^{m} ∈ I_{λ} for some positive integer m. By Theorem 6.4,

(ρ_{E′,p}(τ)^{m} − I)^{2} = 0.

Thus, (α^{m} − 1)^{2} = 0, so α^{m} = 1.

Since E has semistable reduction at ℓ,

(ρ_{E,p}(τ) − I)^{2} = 0

by Theorem 6.4. Since E[N] ≅ E′[N], we have

ρ_{E,p}(τ) − ρ_{E′,p}(τ) ∈ NM_{2}(Z_{p})

for appropriate choices of bases for the p-adic Tate modules of E and of E′. Therefore,

(ρ_{E′,p}(τ) − I)^{2} ∈ NM_{2}(Z_{p}).

The characteristic polynomial of ρ_{E′,p}(τ) is independent of the choice of prime divisor p of N (by Theorem 6.2). By Lemma 6.6, (α − 1)^{2} ∈ NZ̄. Suppose N ≥ 5. By Theorem 6.5a, α = 1. Therefore, I_{ℓ} acts on the Tate module of E′ by unipotent operators. By Theorem 6.4, E′ has semistable reduction at ℓ. Now suppose N = 3 or 4 and E has good reduction at ℓ. By Theorem 6.3, ρ_{E,p}(τ) = I. Therefore, ρ_{E′,p}(τ) − I ∈ NM_{2}(Z_{p}), α − 1 ∈ NZ̄, and α = 1 (using Lemma 6.6 and Theorem 6.5b). By Theorem 6.4, E′ has semistable reduction at ℓ.

Examples 7.2. To see that Proposition 7.1a fails for N = 3 or 4, let E be the elliptic curve y^{2} = x^{3} + x + 1. The conductor of E is 2^{4}31, so E has multiplicative reduction at 31. Consider Theorem 4.1 with a = b = 1 and let E′ be the elliptic curve obtained by letting t = 1 (letting t = 0 gives E). Then E[4] ≅ E′[4], and it is easy to check that E′ has additive reduction at 31. Theorem 4.1 of [9] is the analogue of Theorem 4.1 of this paper, with N = 3 instead of N = 4. Consider Theorem 4.1 of [9] with a = b = 1, and let E′′ be the elliptic curve obtained by letting t = 1 (again, t = 0 gives E). Then E[3] ≅ E′′[3], and it is easy to check that E′′ has additive reduction at 31.

To see that Proposition 7.1b fails for N = 2, let ℓ be an odd prime, let E be the elliptic curve y^{2} = x^{3} − x, and let E′ be the elliptic curve y^{2} = x^{3} − ℓ^{2}x. Then E has good reduction at ℓ (the conductor of E is 2^{5}), E′ has additive reduction at ℓ (the conductor of E′ is 2^{5}ℓ^{2}), and E[2] ≅ E′[2] ≅ Z/2Z × Z/2Z.

Theorem 7.3. If E is an elliptic curve over Q which has good reduction at 3 and at 5, and E′ is an elliptic curve over Q such that E[4] and E′[4] are isomorphic as G_{Q}-modules, then E′ is modular.

Proof. By Proposition 7.1b with N = 4, E′ has semistable reduction at 3 and at 5. Therefore E′ is modular by Theorem 5.1.

Therefore, the explicit families of §4 give infinite families of modular elliptic curves, as long as one of the elliptic curves in the family has good reduction at 3 and at 5.

8. Torsion subgroups

Theorem 8.1. If E is an elliptic curve over Q which has:

(1) all its points of order 2,

(2) a cyclic subgroup of order 4,

(3) a point of order 5, or

(4) a point of order 7, defined over Q, then E is modular.

We prove Theorem 8.1 in a series of lemmas.

Lemma 8.2. If E is an elliptic curve over Q, and E(Q) ⊇ Z/2Z × Z/2Z, then E is modular.

Proof. It is easy to see that E is isomorphic over C to an elliptic curve E′ of the form

y^{2} = x(x − A)(x − B)

where A and B are relatively prime integers. Suppose p is an odd prime. Since the right hand side does not have a triple root modulo p, E′ has semistable reduction at p. In other words, one can twist away any additive reduction on E at odd primes. The lemma now follows from Theorem 5.1.

Lemma 8.2 was an observation made with K. Rubin. The main theorem of [3] shows that Lemma 8.2 follows from the results of [19] and [18], without using [2].

Lemma 8.3. If E is an elliptic curve over Q, and E has a cyclic subgroup of order 4 defined over Q, then E is modular.

Proof. Suppose C is a rational cyclic subgroup of E of order 4. Let D = C ∩ E[2]. Then D is a subgroup of E of order 2, and D is defined over Q. Let E′ = E/D, an elliptic curve over Q. The quotient map φ : E → E′ is an isogeny defined over Q. Therefore to show that E is modular, it suffices to show that E′ is modular. Fix a generator x of C and fix y ∈ E[2] − D. Then φ(x) generates C/D, which is a rational subgroup of E′ of order 2. Therefore, φ(x) is defined over Q. Similarly, φ(y) generates E[2]/D, a rational subgroup of E′ of order 2, so φ(y) is defined over Q. Since x − y ∉ C, we have x − y ∉ D, so φ(x) ≠ φ(y). Therefore, E′ has all its points of order 2 defined over Q. By Lemma 8.2, E′ is modular.

Proposition 8.4. If E is an elliptic curve over Q, 5 ≤ N ∈ Z, and E(Q) ⊇ Z/NZ, then E has semistable reduction at every prime which does not divide N.

Proof. We give a proof from [15] (see Theorem 6.2). Suppose ℓ is a prime which does not divide N. Let I_{ℓ} denote the inertia subgroup of G_{Q} corresponding to an extension λ̄ to Q̄ of the ℓ-adic valuation on Q. Suppose τ ∈ I_{ℓ}. Since ℓ does not divide N, Q(ζ_{N}) is unramified at ℓ, so I_{ℓ} acts as the identity on the N-th roots of unity. Suppose p is a prime divisor of N, and let ρ_{p} : G_{Q} → GL_{2}(Z_{p}) denote the p-adic representation associated to E. Since E(Q) has a point of order N,

ρ_{p}(τ) ≡ ( \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} ) (modNM_{2}(Z_{p})),

for an appropriate choice of basis for the p-adic Tate module of E. Therefore, (ρ_{p}(τ) − I)^{2} ∈ NM_{2}(Z_{p}). There is a number field F such that E has semistable reduction at the restriction λ of λ̄ to F (by Theorem 6.1). Then τ^{m} ∈ I_{λ} for some positive integer m. By Theorem 6.4, (ρ_{p}(τ)^{m} − I)^{2} = 0. Let α be an eigenvalue of ρ_{p}(τ). Then (α^{m} − 1)^{2} = 0, so α^{m} = 1. By Lemma 6.6 and Theorem 6.2, (α − 1)^{2} ∈ NM_{2}(Z̄). By Theorem 6.5a, α = 1. Therefore, I_{ℓ} acts on the Tate module of E by unipotent operators. By Theorem 6.4, E has semistable reduction at ℓ.

Examples 8.5. Proposition 8.4 fails for N < 5. The point (48, −15) is a point of order 4 on the elliptic curve y^{2} + xy = x^{3} − x^{2} − 7056x + 229905, and this curve has additive reduction at 3 (it is the curve 63A5 in Cremona’s tables [1]). The point (1, 1) is a point of order 3 on the elliptic curve y^{2} = x^{3} + x^{2} − x, and this curve has additive reduction at 2 (it is 20A2 in [1]).

Lemma 8.6. If E is an elliptic curve over Q, and E(Q) ⊇ Z/7Z, then E is modular.

Proof. By Proposition 8.4 with N = 7, E has semistable reduction at 3 and at 5. By Theorem 5.1, E is modular.

Theorem 8.7. If E is an elliptic curve over Q, p is a prime number, E has semistable reduction at p, the mod p representation ρ̄_{E,p} for E is modular, and the restriction of ρ̄_{E,p} to G_{Q(\sqrt{−1^{(p−1)/2}p})} is absolutely irreducible, then E is modular.

Proof. This is Theorem 5.3 of [2], applied to elliptic curves.

Proposition 8.8. If E is an elliptic curve over Q, the mod 5 representation for E is reducible, and the restriction to G_{Q(\sqrt{−3})} of the mod 3 representation for E is not absolutely irreducible, then E is modular.

Proof. See Proposition 13 of [7] (by work of J. E. Cremona, the elliptic curves over Q whose j-invariants are given in Proposition 13 of [7] are all modular), p. 544 of [19], or the proof of Theorem 5.4 of [2].

Lemma 8.9. If E is an elliptic curve over Q, and E(Q) ⊇ Z/5Z, then E is modular.

Proof. Since E(Q) ⊇ Z/5Z, the mod 5 representation for E is reducible. By Proposition 8.8, we may assume that the restriction to G_{Q(\sqrt{−3})} of the mod 3 representation for E is absolutely irreducible. The mod 3 representation for E is modular by the Langlands-Tunnell Theorem. By Proposition 8.4, E has semistable reduction at 3. By Theorem 8.7, E is modular.

Theorem 8.1 follows from Lemmas 8.2, 8.3, 8.6, and 8.9.

By [6], if E is an elliptic curve over Q, then the torsion subgroup of E(Q) is isomorphic to one of the following 15 groups:

Z/NZ  for N = 1, …, 10 or 12,
Z/2Z × Z/2NZ for N = 1, 2, 3, or 4.

This and Theorem 8.1 immediately imply the following result.

Corollary 8.10. If E is an elliptic curve over Q, and the torsion subgroup of E(Q) is not a cyclic group of order 1, 2, 3, 6, or 9, then E is modular.

Given a model for an elliptic curve E over Q, the Nagell-Lutz Theorem (see Corollary 7.2 of Chapter VIII of [17]) provides an algorithm for computing the torsion subgroup of E(Q).

9. Explicit families of modular elliptic curves

By Theorem 8.1, the elliptic curves below are modular. See Table 3 on p. 217 of [5] for such parametrizations of elliptic curves over Q with points of finite order.

Example 9.1 (rational 2-torsion). Elliptic curves over Q with all points of order 2 defined over Q are given by

Dy^{2} = x(x − 1)(x − λ)

with D, λ ∈ Q^{×}, λ ≠ 1. The corresponding (modular) j-invariants are all the numbers of the form

\frac{2^{8}(λ^{2} − λ + 1)^{3}}{λ^{2}(λ − 1)^{2}} with λ ∈ Q − \{0, 1\}.

Example 9.2 (rational cyclic subgroup of order 4). The family of elliptic curves with a rational cyclic subgroup of order 4 is given by

y^{2} = x^{3} + D(1 − 4b)x^{2} − 8D^{2}bx + 16D^{3}b^{2}

with b, D ∈ Q^{×}, b ≠ \frac{1}{16}. The rational cyclic subgroup of order 4 is

\{0, (4bD, 0), (0, ±4bD\sqrt{D})\}

and the j-invariant is

\frac{(16b^{2} + 16b + 1)^{3}}{(16b + 1)b^{4}} .

Example 9.3 (rational points of order 5). The family of elliptic curves over Q with a rational point of order 5 is given by

y^{2} + (1 − b)xy − by = x^{3} − bx^{2}

with b ∈ Q^{×}. The point (0, 0) has order 5.

Example 9.4 (rational points of order 7). The family of elliptic curves over Q with a rational point of order 7 is given by

y^{2} + (1 − d(d − 1))xy − d^{2}(d − 1)y = x^{3} − d^{2}(d − 1)x^{2}

with d ∈ Q − {0, 1}. The point (0, 0) has order 7.

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Department of Mathematics, Ohio State University, 231 W. 18 Avenue, Columbus, Ohio 43210–1174, USA

   E-mail address: silver@math.ohio-state.edu