On the surjectivity of the Galois representations associated to non-CM elliptic curves

Alina Carmen Cojocaru ^{∗}

The Fields Institute 222 College Street Toronto, Ontario, Canada, M5T 3J1

                                 alina@fields.utoronto.ca

(with an appendix by Ernst Kani, Queen’s University, Kingston, Canada)

January 17, 2004

Abstract

1 Let E be an elliptic curve defined over ℚ, of conductor N and without complex multiplication. For any positive integer k, let ϕ_{k} be the Galois representation associated to the k-division points of E. From a celebrated 1972 result of Serre we know that ϕ_{l} is surjective for any sufficiently large prime l. In this paper we find conditional and unconditional upper bounds in terms of N for the primes l for which ϕ_{l} is not surjective.

1 Introduction

Let E be an elliptic curve defined over ℚ and of conductor N. For primes p of good reduction for E, that is, primes p ∤ N, let E_{p} be the reduction of E modulo p and write

|E_{p}(𝔽_{p})| = p + 1 − a_{p}

for the number of 𝔽_{p}-rational points of E_{p}. We know from Hasse’s inequality that

|a_{p}| ≤ 2\sqrt{p}.  (1)

For a positive integer k, let E[k] denote the group of complex points on E whose order divides k, called the group of k-division points of E, and let L_{k} = ℚ(E[k]) be the field obtained by adjoining to ℚ the x− and y−coordinates of the points in E[k], called the k-division field of E. We know that E[k] is isomorphic to ℤ/kℤ × ℤ/kℤ and that L_{k}/ℚ is a finite Galois extension which is unramified outside kN (see [Si86]).

There has been great interest in determining the size of the degree [L_{k} : ℚ] of the extensions L_{k}/ℚ. For example, this problem has applications in the study of certain diophantine equations (see [Mer]) or in the study of the curve E_{p} as p varies (see [acC]).

∗Research supported in part by an Ontario Graduate Scholarship.

12000 Mathematics Subject Classification. Primary 11G05; Secondary 11N36, 11R45.

A natural way of finding upper bounds for [L_{k} : ℚ] is by observing that the absolute Galois group Gal(\overline{ℚ}/ℚ), where \overline{ℚ} is an algebraic closure of ℚ, acts on the k-division points E[k] for any k. This allows us to define a representation

ϕ_{k} : Gal(L_{k}/ℚ) ⟶ GL_{2}(ℤ/kℤ),

called the Galois representation associated to E[k], which is easily seen to be injective. Therefore for any k we have

[L_{k} : ℚ] ≤ | GL_{2}(ℤ/kℤ)| = k^{4} ∏_{{l|k \atop l prime}} (1 − l^{−2}) (1 − l^{−4}) .

Determining the size of the image of ϕ_{k} is a challenging question.

By classical results in the theory of complex multiplication we know that if E has complex multiplication (denoted CM), then

ϕ(k)^{2} ≪ [L_{k} : ℚ] ≪ k^{2}

for any k > 2, where ϕ(·) denotes the Euler function. Thus ϕ_{k} cannot be surjective in this case. In big contrast, if E does not have complex multiplication (denoted non-CM), a celebrated result of J. -P. Serre [Se72] asserts that there exists a finite set of primes S_{E} such that ϕ_{l} is surjective for any prime l ∉ S_{E}. Moreover, if we define

A(E) := 2 · 3 · 5 · ∏_{l∈S_{E}} l,

we can deduce that ϕ_{k} is surjective for any positive integer k coprime to A(E) (see Appendix 1 of [acC]). We will refer to A(E) as Serre’s constant associated to E.

Now a natural question to ask is how large the prime divisors of A(E) can be. In his 1972 paper, and also later on in a 1981 paper, J.-P. Serre asked if A(E) is an absolute constant for any non-CM elliptic curve E defined over ℚ (see [Se72, p. 299] and [Se81, p. 199]). In 1978 [Maz], B. Mazur showed that A(E) is indeed an absolute constant if E is a semistable elliptic curve, that is, if the conductor N of E is square-free. Later on in 1997 [DaMe], H. Darmon and L. Merel proved that a similar result also holds for non-CM Frey curves, that is, elliptic curves given by equations of the form y^{2} = x(x − a)(x − b) with a, b ∈ ℤ.

In this paper we consider the more modest problem of finding upper bounds for the prime divisors of A(E) in terms of the conductor N of E.

One approach to this problem was initiated by J. -P. Serre in 1981; we recall it in what follows. In [Se81, p. 197] Serre showed that if l is a prime and if we let G_{l} denote the image of ϕ_{l} in GL_{2}(ℤ/lℤ) and P G_{l} the image of G_{l} in the projective group PGL_{2}(ℤ/lℤ) := GL_{2}(ℤ/lℤ)/(ℤ/lℤ)^{∗}, then, in order to prove that G_{l} = GL_{2}(ℤ/lℤ), it suffices to show that:

• P G_{l} is not contained in a Borel subgroup of PGL_{2}(ℤ/lℤ);

• P G_{l} is not contained in a non-split Cartan subgroup of PGL_{2}(ℤ/lℤ);

• P G_{l} is not isomorphic to the permutation groups A_{4}, S_{4} or A_{5};

• P G_{l} is not contained in the normalizer N_{l} of a Cartan subgroup C_{l} of PGL_{2}(ℤ/lℤ) such that P G_{l} ⊈ C_{l}; (for the definitions of the Borel and Cartan subgroups of PGL_{2}(ℤ/lℤ) the reader may consult [Se72, Section 2, p. 278] or [La, Sections 1 and 2 of Ch. XI]). Serre showed that the first three situations above hold if l ≥ 19 and l ≠ 37 (see [Se81, Lemmas 16, 17, 18, p. 197]). If there exists a Cartan subgroup C_{l} of PGL_{2}(ℤ/lℤ) such that P G_{l} is contained in the normalizer N_{l} of C_{l} in PGL_{2}(ℤ/lℤ) and is not contained in C_{l}, then we identify \frac{N_{l}}{C_{l}} with the group \{±1\} and we let ϵ_{l} be the composition

Gal(\overline{ℚ}/ℚ) ⟶ G_{l} ⟶ \frac{N_{l}}{C_{l}} ⟶ \{±1\}.

We see that ϵ_{l} is a quadratic character of Gal (\overline{ℚ}/ℚ) , which we can identify with a Dirichlet character. Serre [Se72, p. 317] showed that

ϵ_{l} is unramified at the primes p which do not divide N,  (2)

and

a_{p} ≡ 0(mod l) for any prime p ∤ N such that ϵ_{l}(p) = −1.  (3)

In other words, if l is a prime divisor of A(E), then either l|37 ∏_{{q≤19 \atop q prime}} q or (3) holds.

Now let p_{0} be the smallest prime p of good reduction for E such that ϵ_{l}(p) = −1 and a_{p} ≠ 0. Serre proved that if we assume GRH for the Dedekind zeta functions of the division fields of E, there exists a positive absolute constant c_{0} such that

p_{0} ≤ c_{0}(log N)^{2}(log log(2N))^{6}  (4)

(see [Se81, proof of Lemma 19, p. 198]). By (1) and (4) we then get the estimate

A(E) ≤ 37 ( ∏_{{q≤19 \atop q prime}} q) ∏_{l|a_{p_{0}} ; {l prime \atop (l,37 ∏_{q≤19} q)=1}} l ≤ 37 ( ∏_{{q≤19 \atop q prime}} q) |a_{p_{0}}| ≤ c_{1}(log N)(log log(2N))^{3}

for some positive absolute constant c_{1}. We record this result as:

Theorem 1 Let E be a non-CM elliptic curve defined over ℚ and of conductor N. Let A(E) be Serre’s constant associated to E. If we assume GRH for the Dedekind zeta functions of the division fields of E, there exists a positive absolute constant c_{1} such that

A(E) ≤ c_{1}(log N)(log log(2N))^{3}.

Another approach to the problem of finding upper bounds for the prime divisors of A(E) is due to D. W. Masser and G. Wüstholz (see [MaWü, p. 247]) who showed that there exist absolute positive constants c_{2} and γ, with γ effective, such that ϕ_{l} is surjective for any prime l satisfying

l > c_{2} h(E)^{γ},  (5)

where h(E) denotes the Faltings height of E (for a definition, we refer the reader to [CoSi, p. 248 of Chai’s article]). We will refer to γ as the Masser-Wüstholz constant.

In this paper we will first derive an unconditional upper bound for A(E). After obtaining this estimate, we discovered that A. Kraus had already obtained a similar result a few years earlier (see [Kr]). The proof we give is, conceptually, the same as the one of Kraus, but much simpler, in that it is essentially self-contained; for example, Kraus uses certain theorems of Deligne [De] without proof. We will also derive a conditional upper bound for the prime divisors of A(E) based on a different hypothesis than GRH.

Before stating the precise results that we are proving in this paper, let us point out that one of the key facts used in our proofs is the modularity of elliptic curves defined over ℚ, a long-standing conjecture formulated by Shimura and Taniyama, eventually proven by Wiles, Taylor-Wiles and Breuil-Conrad-Diamond-Taylor. One formulation of this conjecture is that for any elliptic curve E defined over ℚ and of conductor N, there exists a non-trivial surjective morphism

ϕ : X_{0}(N) ⟶ E,

defined over ℚ, called a modular parametrization of E. Here X_{0}(N) is the modular curve obtained by the action of the complex upper-half plane on the matrix subgroup of SL_{2}(ℤ) composed of elements whose left lower entry is 0 modulo N.

The main results of our paper are as follows.

Theorem 2 Let E be a non-CM elliptic curve defined over ℚ and of conductor N. Then, for any prime l satisfying the inequality

l ≥ \frac{4\sqrt{6}}{3}N ∏_{p|N_{p prime}} (1 + \frac{1}{p})^{1/2} + 1,

the Galois representation ϕ_{l} associated to E is surjective. Moreover,

A(E) ≪ N(log log N)^{1/2},

where the implied ≪-constant is absolute.

Theorem 3 Let E be a non-CM elliptic curve defined over ℚ and of conductor N. Let ϕ be a modular parametrization for E of minimal degree deg ϕ. Assuming the degree conjecture for ϕ, that is, assuming that for any ε > 0,

deg ϕ = O_{ε} (N^{2+ε}) ,

there exists an absolute positive constant c_{3} such that, for any prime l satisfying the inequality

l ≥ c_{3}(log N)^{γ},

with γ denoting the Masser-Wüstholz constant, the Galois representation ϕ_{l} associated to E is surjective.

2 Preliminaries

In this section we review the properties of modular forms that will be needed for proving the two results of the paper. For more details we refer the reader to [La].

Let k, N be positive integers and let

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

Also, let M_{k}(Γ_{0}(N)) and S_{k}(Γ_{0}(N)) be the complex vector spaces of modular forms and cusp forms of weight k on Γ_{0}(N), respectively.

On the space S_{k}(Γ_{0}(N)) we can define an inner product, called the Petersson inner product. We can also define certain linear transformations on M_{k}(Γ_{0}(N)) and S_{k}(Γ_{0}(N)), called Hecke operators. A cusp form f which is an eigenform for all the Hecke operators is called an eigenform; if its first Fourier coefficient a_{1}(f) is 1, then it is called a normalized form. A nice property of an eigenform f is that its Fourier coefficients a_{n}(f) are multiplicative in n. Moreover, for any prime q ∤ N and any positive integer t, the Fourier coefficients a_{q^{t}}(f) satisfy the recurrence relations

a_{q^{t}}(f)a_{q}(f) = a_{q^{t+1}}(f) + q^{k−1}a_{q^{t−1}}.  (6)

A natural question to ask is when two modular forms are equal. We have the following answer:

Proposition 2.1 Let f and f′ be two distinct modular forms of weight k and levels N and N′, respectively. Let M := lcm(N, N′). Then there exists a positive integer n such that

n ≤ \frac{k}{12}M ∏_{p|M_{p prime}} (1 + \frac{1}{p}) + 1

and

a_{n}(f) ≠ a_{n}(f′),

where a_{n}(f) and a_{n}(f′) are the n-th Fourier coefficients of f and f′.

Proof. The following argument was given in [RM97]. We consider the function

ϕ := \frac{(f − f′)^{12}}{Δ^{k}} ,

where Δ is the classical delta function defined by

Δ(z) := e^{2πiz} ∏_{n=1}^{∞} (1 − e^{2πinz})^{24} .

We recall that Δ is a cusp form of weight 12 on SL_{2}(ℤ) = Γ_{0}(1), which is nonzero on the complex upper-half plane and has a simple zero at infinity. Then ϕ is a meromorphic function on X_{0}(M), for which we have

12m_{0} − k ≤ number of zeroes of ϕ
= number of poles of ϕ
≤ k ([SL_{2}(ℤ) : Γ_{0}(M)] − 1)

= k (M ∏_{{p|M \atop p prime}} (1 + \frac{1}{p}) − 1) ,

with m_{0} denoting the order of the zero at infinity of f − f′. Thus

m_{0} ≤ \frac{k}{12}M ∏_{{p|M \atop p prime}} (1 + \frac{1}{p}) .

This completes the proof of the proposition. □

Another natural question to ask is how to generate new modular forms from a given modular form. One way of answering this question is to associate to a given f ∈ M_{k}(Γ_{0}(N)) and a primitive Dirichlet character χ a function f̃ defined by

f̃(z) := ∑_{n=0}^{∞} χ(n)a_{n}(f)e^{2πinz},

called the twist of f by χ, where

f(z) = ∑_{n=0}^{∞} a_{n}(f)e^{2πinz}

is the Fourier expansion of f. We have the following useful properties (for a proof, see [Iw, p. 124]):

Proposition 2.2 The twist f̃ of a modular form f ∈ M_{k}(Γ_{0}(N)) by a quadratic character χ of conductor r is a modular form in M_{k}(Γ_{0}(M)), where M = lcm (N, r^{2}) . Moreover, if f is a cusp form, so is f̃.

We can also generate new modular forms by remarking that if N′|N, then a cusp form f ∈ S_{k}(Γ_{0}(N′)) can be viewed as an element of S_{k}(Γ_{0}(N)). More generally, if δ|\frac{N}{N′} , we have that f(δz) ∈ S_{k}(Γ_{0}(N)). The ℂ−span of

⋃_{{N′|N \atop N′≠N}} ⋃_{δ| \frac{N}{N′}} \{f(δz) : f ∈ S_{k}(Γ_{0}(N′))\}

is called the space of oldforms on Γ_{0}(N). The eigenforms in the orthogonal complement (under the Petersson inner product) of the space of oldforms in S_{k}(Γ_{0}(N)) are called newforms.

We conclude our review with a few words on the modularity of the elliptic curves defined over ℚ. The famous Shimura-Taniyama conjecture, now known to be true (see [BCDT]), gives a one-to-one correspondence between isogeny classes of elliptic curves defined over ℚ and of conductor N, and normalized newforms in S_{2}(Γ_{0}(N)) with integer Fourier coefficients. An equivalent formulation is as follows:

Theorem 2.3 (Shimura-Taniyama conjecture) For every elliptic curve E defined over ℚ and of conductor N there exists a surjective non-trivial morphism ϕ : X_{0}(N) ⟶ E, defined over ℚ.

Regarding the size of the degree of ϕ we have another conjecture, unsolved for the moment, formulated by G. Frey:

Conjecture 2.4 (Degree conjecture) Let E be an elliptic curve defined over ℚ and of conductor N, and let ϕ be a modular parametrization of E of minimal degree. Then, for any ε > 0, we have deg ϕ = O_{ε} (N^{2+ε}) . Here, the O_{ε}-constant depends only on ε.

3 Proof of Theorem 2

As recalled in the Introduction, Serre showed that if l is a prime such that l ≥ 19 and l ≠ 37, then P G_{l} cannot be contained in a Borel subgroup or a Cartan subgroup of PGL_{2}(ℤ/lℤ), nor can it be isomorphic to the permutation groups A_{4}, S_{4} or A_{5}. In what follows we will show that if

l ≥ \frac{4\sqrt{6}}{3}N ∏_{{p|N \atop p prime}} (1 + \frac{1}{p})^{1/2} + 1,

then P G_{l} cannot be contained in the normalizer of a Cartan subgroup C_{l} of PGL_{2}(ℤ/lℤ) such that P G_{l} ⊈ C_{l}. This will imply that ϕ_{l} is surjective.

Let us suppose that there exists a Cartan subgroup C_{l} of PGL_{2}(ℤ/lℤ) such that P G_{l} is contained in the normalizer N_{l} of C_{l} in PGL_{2}(ℤ/lℤ) and is not contained in C_{l}. As explained in the Introduction, this implies the existence of a Dirichlet quadratic character ϵ_{l} satisfying (2) and (3). We also note that since ϵ_{l} is a quadratic character, property (2) implies that the conductor D of ϵ_{l} is of the form D_{0} or 4D_{0} for some square-free integer D_{0} such that D_{0}|N.

We follow Serre’s idea of estimating l, that is, we first look for a prime q of good reduction for E for which a_{q} ≠ 0 and ϵ_{l}(q) = −1, and then we look for a non-trivial upper bound for q. By (3) and Hasse’s bound (1) we obtain that l ≤ 2\sqrt{q}, hence the upper bound for q will give us an upper bound for l.

Let E′ be the elliptic curve defined over ℚ and obtained by taking the twist of E with respect to ϵ_{l}. We denote its conductor by N′. Let f and f′ be the weight 2 normalized newforms with integer Fourier coefficients, of levels N and N′, respectively, associated to E and E′ via the Shimura-Taniyama Conjecture. Proposition 2.2 gives us that N′| lcm (N, D^{2}) . Thus the primes of bad reduction for E′ are divisors of 2N, and so

a′_{p} = ϵ_{l}(p)a_{p} for any prime p ∤ 2N,  (7)

where p + 1 − a′_{p} denotes the number of 𝔽_{p}-rational points of the reduction of E′ modulo p. We want to prove that there exists a prime q which does not divide N and which satisfies

a_{q} ≠ a′_{q},  (8)

ϵ_{l}(q) = −1,  (9)

q ≤ \frac{8}{3}N^{2} ∏_{{p|N \atop p prime}} (1 + \frac{1}{p}) + 1.  (10)

To do this, we first observe that f ≠ f′, since E is a non-CM elliptic curve. Indeed, if we assume that f = f′, then by the definition of f′ we must have a_{p} = 0 for any prime p such that ϵ_{l}(p) = −1. But, on one hand, by Dirichlet’s theorem, the number of primes p ≤ x for which ϵ_{l}(p) = −1 is asymptotically equal to \frac{x}{2 log x}; on the other hand, the number of primes p ≤ x for which a_{p} = 0 is,

in the case of a non-CM elliptic curve, o (\frac{x}{log x}) (see [Se81, p. 189, Corollary 2]); thus we are led to a contradiction.

Now we twist f and f′ by a quadratic character which is zero only at the primes dividing N, and obtain cusp forms

f̃, f̃′ ∈ S_{2}(Γ_{0}(M))

with M|16N^{2} (see Proposition 2.2) and such that f̃, f̃′ vanish at all the primes of bad reduction for E. By the same argument as the one used for showing that f ≠ f′, we have f̃ ≠ f̃′. By Proposition 2.1 this last assertion implies that there exists a positive integer n such that

n ≤ \frac{1}{6}M ∏_{{p|M \atop p prime}} (1 + \frac{1}{p}) + 1  (11)

and

a_{n} \binom{}{f̃} ≠ a_{n} \binom{}{f̃′} ,  (12)

where a_{n} \binom{}{f̃} and a_{n} \binom{}{f̃′} are the n-th Fourier coefficients of f̃ and f̃′, respectively. In particular,

(12) tells us that we must have (n, N) = 1.

Using (11) and that f̃ and f̃′ are twists of f and f′ by a quadratic character vanishing at the primes of bad reduction for E, we obtain that

a_{n}(f) ≠ a_{n}(f′)

for some

n ≤ \frac{1}{6}M ∏_{{p|M \atop p prime}} (1 + \frac{1}{p}) + 1 ≤ \frac{8}{3}N^{2} ∏_{{p|N \atop p prime}} (1 + \frac{1}{p}) + 1

with (n, N) = 1. Here, a_{n}(f) and a_{n}(f′) denote the n-th Fourier coefficients of f and f′.

Since f and f′ are normalized newforms, the coefficients a_{n}(f) and a_{n}(f′) are multiplicative in n, hence there exists a prime q|n such that

a_{q^{α}}(f) ≠ a_{q^{α}}(f′),

with the power α such that q^{α}||n. Moreover, the recurrence relations (6) for a_{q^{t}}(f) and their analogues for a_{q^{t}}(f′), where t is an arbitrary positive integer, give us that

a_{q} = a_{q}(f) ≠ a_{q}(f′) = a′_{q}.  (13)

Now we want to show that ϵ_{l}(q) = −1. Let us suppose that ϵ_{l}(q) ≠ −1. We notice that ϵ_{l}(q) ≠ 0, since ϵ_{l} is unramified at the primes of good reduction for E and q is such a prime. Then we must have ϵ_{l}(q) = 1, which implies

a′_{q} = ϵ_{l}(q)a_{q} = a_{q},

a contradiction with (13).

We have therefore proven that there exists a prime q of good reduction for E satisfying conditions (8), (9), (10). By (8) we deduce that a_{q} ≠ 0, and by (10) and (3) we deduce that l|a_{q}. Using Hasse’s bound and (10) we obtain

l < \frac{4\sqrt{6}}{3}N ∏_{{p|N \atop p prime}} (1 + \frac{1}{p})^{1/2} + 1,

which completes the proof of the first part of Theorem 2.

Let us remark that

A(E) ≤ 37 ( ∏_{{p≤19 \atop p prime}} p) |a_{q}| ≪ q^{1/2},  (14)

where q is the prime satisfying (8)-(10). We also recall from elementary number theory that

∏_{{p|N \atop p prime}} (1 + \frac{1}{p}) ≪ log log N.  (15)

Combining (10), (14) and (15) gives the second assertion of Theorem 2. This completes the proof.

4 Proof of Theorem 3

We recalled in the Introduction that for any prime l such that l > c_{2} h(E)^{γ}, with h(E) denoting the Faltings height of E and with c_{2} and γ as in (5), the representation ϕ_{l} is surjective.

Now we want to relate h(E) to the conductor N of the elliptic curve E. To do so, we let f ∈ S_{2}(Γ_{0}(N)) be the normalized newform associated to E and ϕ : X_{0}(N) ⟶ E be a minimal modular parametrization of E given by Theorem 2.3. As shown in [RM99, pp. 181-182], we have that

2 h(E) + log⟨f, f⟩ = log deg ϕ + O(1),

and that for any ε > 0 and N sufficiently large,

(1 − ε) log N < log⟨f, f⟩ < (1 + ε) log N.

Here, ⟨·, ·⟩ denotes the Petersson inner product and deg ϕ denotes the degree of ϕ. We obtain

(1 − ε) log N + 2 h(E) < log deg ϕ + O(1) < (1 + ε) log N + 2 h(E).

By the degree conjecture for E, deg ϕ = O_{ε} (N^{2+ε}) , hence

h(E) ≪ log N,  (16)

where the implied constant is absolute. We use (16) in (5) and deduce that there exists an absolute positive constant c_{3} such that, for any prime l satisfying

l > c_{3}(log N)^{γ} (≥ c_{2} h(E)^{γ}) ,

the representation ϕ_{l} is surjective. This completes the proof of Theorem 3.

5 Final remarks

The bound provided by Theorem 3 is weaker than the bound (4) obtained by Serre by assuming GRH, since the Masser-Wüstholz constant γ, even though effective, is not yet determined (see [MaWü, p. 248]). On the other hand, the hypothesis of Theorem 3 is different, as illustrated in [RM99], where it is shown that the ABC conjecture is equivalent to the degree conjecture for Frey curves. ^{2} It will be interesting to find the value of the Masser-Wüstholz constant γ, and, consequently, to find the precise estimate A(E) ≪ N^{b} provided by Theorem 3. As shown in Theorem 2, we have A(E) ≪_{ε} N^{1+ε} for any small positive ε. Currently, this is the best (unconditional) estimate for A(E). We hope to address the problem of showing that A(E) ≪ N^{b} for some b < 1 in future research.

Acknowledgements The results of this paper were obtained in the fall of 2001, while I was a graduate student at Queen’s University at Kingston. I wish to thank Prof. M. Ram Murty for valuable discussions on these results and for encouraging me to write the paper. Also, I wish to thank Prof. Ernst Kani for useful discussions on the constant A(E).

A Appendix: The Surjectivity of mod m Galois Representations

Ernst Kani

Queen’s University Department of Mathematics and Statistics Kingston, Ontario, Canada, K7L 3N6

                                   kani@mast.queensu.ca

The aim of this appendix is to prove the following result.

Theorem A.1 If E is an elliptic curve defined over ℚ and m is any integer with (m, 30) = 1, then the Galois representation

\overline{ρ}_{E,m} : Gal(\overline{ℚ}/ℚ) → Aut(E[m]) ≃ GL_{2}(ℤ/mℤ)

is surjective if and only if the Galois representations \overline{ρ}_{E,p} are surjective for every prime p|m. In particular, if E is a non-CM elliptic curve and A(E) is Serre’s constant associated to E (see the definition in Section 1 of the paper), then \overline{ρ}_{E,m} is surjective for every integer m with (m, A(E)) = 1.

This theorem is essentially due to Serre and can be deduced from the Main Lemma in his book [Se68] on p. IV-19. However, since this fact does not seem to be widely known and since there is a small gap in his argument (similar to the gap which is mentioned on p. 715 of his Collected Papers [Se85]), it seems useful to present the following (modified) proof.

As in Serre’s book, the above assertion can be deduced from a purely group-theoretical statement about subgroups of GL_{2}(ℤ/mℤ), which is perhaps also of independent interest. To state this result, we use the following notation introduced in Serre[Se68], p. IV-25.

Notation. For any finite group G, let Occ(G) denote the set of isomorphism classes of non-abelian finite simple groups which occur as a composition factor of some subgroup H ≤ G. Let p and q denote rational primes.

2Using the ‘unconditional ABC’ proven in [StYu], we obtain that for a Frey curve E of conductor N, ϕ_{l} is surjective for any prime l ≫ N^{γ/3}(log N)^{3}.

Theorem A.2 Let G be a subgroup of GL_{2}(ℤ/mℤ) where (m, 30) = 1. Then:

(a) G = GL_{2}(ℤ/mℤ) if and only if G ≥ SL_{2}(ℤ/mℤ) and ϕ(m)|[G : G′].

(b) G ≥ SL_{2}(ℤ/mℤ) if and only PSL_{2}(p) ∈ Occ(G) for all primes p|m (here, PSL_{2}(p) := SL_{2}(ℤ/pℤ)/(±1)).

In order to prove this theorem, we first recall some basic facts about the matrix groups SL_{2}(ℤ/mℤ).

The following (well-known) facts can be found in [Hu], Theorems II.6.13 and II.8.14:

Lemma A.3 Let p and q be prime numbers. Then:

(a) PSL_{2}(p) is a simple group for any p ≥ 5;

(b) PSL_{2}(p) ≃ PSL_{2}(q) if and only if p = q;

(c) If H is a proper subgroup of PSL_{2}(p), then H is solvable or H ≃ A_{5};

(d) PSL_{2}(p) ≃ A_{5} if and only if p = 5.

Moreover, by Lemma 2 of [Se68], p. IV-23, we have

Lemma A.4 Let H be a subgroup of SL_{2}(ℤ/pℤ) such that ±H = SL_{2}(ℤ/pℤ). Then H = SL_{2}(ℤ/pℤ).

Corollary A.5 If p ≥ 5 is a prime, then the commutator subgroup SL_{2}(ℤ/pℤ)′ of SL_{2}(ℤ/pℤ) is SL_{2}(ℤ/pℤ) itself, i.e. SL_{2}(ℤ/pℤ)′ = SL_{2}(ℤ/pℤ).

Proof. Since PSL_{2}(p) is non-abelian and simple (cf. Lemma A.3(a)), we have PSL_{2}(p)′ = PSL_{2}(p) and so ± SL_{2}(ℤ/pℤ)′ = SL_{2}(ℤ/pℤ). Thus, the assertion follows from Lemma A.4. □

To extend the above result to SL_{2}(ℤ/mℤ) for an arbitrary integer m, let d|m be a divisor of m and consider the (surjective) group homomorphism

pr_{d} = pr_{d}^{(m)} : GL_{2}(ℤ/mℤ) ⟶ GL_{2}(ℤ/dℤ),

which is induced by reduction modulo d. We then have:

Lemma A.6 Let H be a subgroup of SL_{2}(ℤ/p^{r}ℤ), where p ≥ 5 is a prime and r is a positive integer. If pr_{p}(H) = SL_{2}(ℤ/pℤ), then H = SL_{2}(ℤ/p^{r}ℤ).

This result can be deduced from the proof of Lemma 3 of [Se68]. As pointed out in [Se68], the result is false for p = 2 and p = 3. The above lemma has the following consequences:

Corollary A.7 For any integer m with (m, 6) = 1 we have that the commutator subgroup SL_{2}(ℤ/mℤ)′ is SL_{2}(ℤ/mℤ).

Proof. Since SL_{2}(ℤ/mℤ) = ∏_{p^{r}||m} SL_{2}(ℤ/p^{r}ℤ), it is enough to verify this for a prime power m = p^{r} with p ≥ 5. In that case we can apply Lemma A.6 to H = SL_{2}(ℤ/p^{r}ℤ)′ because by Corollary A.5 we have pr_{p}(H) = pr_{p}(SL_{2}(ℤ/p^{r}ℤ))′ = SL_{2}(ℤ/pℤ)′ = SL_{2}(ℤ/pℤ). Thus, by Lemma A.6 we obtain that H = SL_{2}(ℤ/p^{r}ℤ), as desired. □

Corollary A.8 Let m be a positive integer coprime to 6. If

SL_{2}(ℤ/mℤ) ≤ H ≤ GL_{2}(ℤ/mℤ),

then H′ = SL_{2}(ℤ/mℤ). In particular, GL_{2}(ℤ/mℤ)′ = SL_{2}(ℤ/mℤ).

Proof. Since H/ SL_{2}(ℤ/mℤ) ≤ GL_{2}(ℤ/mℤ)/ SL_{2}(ℤ/mℤ) is abelian, we see that H′ ≤ SL_{2}(ℤ/mℤ). On the other hand, by Corollary A.7 we also have that SL_{2}(ℤ/mℤ) = SL_{2}(ℤ/mℤ)′ ≤ H′, and so H′ = SL_{2}(ℤ/mℤ). □

As we shall see below, part (a) of Theorem A.2 follows readily from Corollary A.8. We therefore turn now to preliminary preparations for part (b). Here we first note:

Lemma A.9 Let G be a finite group. Then:

(a) Occ(G) = ∅ if and only if G is solvable.

(b) If H is a normal subgroup of G, then Occ(G) = Occ(H) ∪ Occ(G/H).

Proof. Let N (G) denote the set of isomorphism classes of non-abelian composition factors of G, so that Occ(G) = ⋃_{H≤G} N (H).

(a) If Occ(G) = ∅, then also N (G) = ∅, and so by [Hu], I.11.9, G is solvable. Conversely, if G is solvable, then so is any of its subgroups H ≤ G, and hence by [Hu], I.11.9 again, we obtain that Occ(G) = ∅.

(b) From the definition it follows that N (G) = N (H) ∪ N (G/H), and so we see that Occ(G) ⊇ Occ(H) ∪ Occ(G/H). For the reverse inclusion we let K ≤ G. Then H ∩ K is a normal subgroup of K, and so N (K) = N (H ∩K)∪N (K/(H ∩K)) = N (H ∩K)∪N ((HK)/K) ⊆ Occ(H)∪Occ(G/H), which yields the other inclusion. □

We now use the above result to determine Occ(G) for our matrix groups.

Lemma A.10 If m is a positive integer, then

Occ(GL_{2}(ℤ/mℤ)) = Occ(SL_{2}(ℤ/mℤ)) = ⋃_{p|m} Occ(PSL_{2}(p))  (17)

Moreover, if p ≥ 5 is a prime, then

{PSL_{2}(p)} ⊂ Occ(PSL_{2}(p)) ⊂ {PSL_{2}(p), A_{5}} .  (18)

Proof. Since (18) is just a restatement of parts (a) and (c) of Lemma A.3, it is enough to verify (17).

The first equality of (17) follows from Lemma A.10 and from the fact that GL_{2}(ℤ/mℤ)/SL_{2}(ℤ/mℤ) is abelian. For the second equality we first note that Lemma A.9(b) implies that

Occ(SL_{2}(ℤ/mℤ)) = ⋃_{p^{r}||m} Occ(SL_{2}(ℤ/p^{r}ℤ))

because SL_{2}(ℤ/mℤ) = ∏_{p^{r}||m} SL_{2}(ℤ/p^{r}ℤ). Next we observe that

Occ(SL_{2}(ℤ/p^{r}ℤ)) = Occ(SL_{2}(ℤ/pℤ))

because Ker(pr_{p}) is a p-group (and hence is solvable). Since clearly

Occ(SL_{2}(ℤ/pℤ)) = Occ(SL_{2}(ℤ/pℤ)/(±1)),

this proves the second equality of (17). □

Corollary A.11 If p > 5 is a prime and m is an integer such that p ∤ m, then PSL_{2}(p) ∉ Occ(GL_{2}(ℤ/mℤ)).

Proof. Since p > 5 we have by Lemma A.3 that PSL_{2}(p) ≄ A_{5} and that PSL_{2}(p) ≄ PSL_{2}(q) for any prime q|m. Thus, the assertion follows from Lemma A.10. □

Lemma A.12 Let G ≤ GL_{2}(ℤ/p^{r}ℤ), where p ≥ 5 is a prime and r is a positive integer. Then the following statements are equivalent:

(i) PSL_{2}(p) ∈ Occ(G).

(ii) SL_{2}(ℤ/p^{r}ℤ) ≤ G.

(iii) SL_{2}(ℤ/pℤ) ≤ pr_{p}(G).

Proof. (ii) ⇒ (i): This is trivial, since PSL_{2}(p) is a composition factor of SL_{2}(ℤ/p^{r}ℤ).

(i) ⇒ (iii): Since pr_{p}(G) ≃ G/H where H ≤ Ker(pr_{p}) is a p-group, we see by Lemma A.9 that Occ(G) = Occ(pr_{p}(G)).

Now for any subgroup K ≤ GL_{2}(ℤ/pℤ) we know by [La], Theorem XI.2.2 (p. 183), that

K ≥ SL_{2}(ℤ/pℤ) ⇔ p | |K| and K is not contained in a Borel subgroup.

Since the Borel subgroups of GL_{2}(ℤ/pℤ) are solvable, this therefore shows that the condition PSL_{2}(p) ∈ Occ(pr_{p}(G)) implies that pr_{p}(G) ≥ SL_{2}(ℤ/pℤ).

(iii) ⇒ (ii): Consider H := G′ ≤ GL_{2}(ℤ/p^{r}ℤ)′ = SL_{2}(ℤ/p^{r}ℤ) by Corollary A.8. Now since SL_{2}(ℤ/pℤ) ≤ pr_{p}(G) by hypothesis, we have SL_{2}(ℤ/pℤ)′ ≤ pr_{p}(G)′ = pr_{p}(G′) = pr_{p}(H) ≤ SL_{2}(ℤ/pℤ). But SL_{2}(ℤ/pℤ)′ = SL_{2}(ℤ/pℤ) by Corollary A.5, and so SL_{2}(ℤ/pℤ) = pr_{p}(H). It therefore follows from Lemma A.6 that SL_{2}(ℤ/p^{r}ℤ) = H = G′ ≤ G, and hence condition (ii) holds. □

We are now ready to prove Theorem A.2.

Proof of Theorem A.2.

(a) If G = GL_{2}(ℤ/mℤ), then clearly G ≥ SL_{2}(ℤ/mℤ). Moreover, by Corollary A.8 we have ϕ(m) = [GL_{2}(ℤ/mℤ) : SL_{2}(ℤ/mℤ)] = [GL_{2}(ℤ/mℤ) : GL_{2}(ℤ/kℤ)′] = [G : G′], so in particular ϕ(m)|[G : G′].

Conversely, if G ≥ SL_{2}(ℤ/mℤ) and ϕ(m)|[G : G′], then |GL_{2}(ℤ/mℤ)| = ϕ(m) |SL_{2}(ℤ/mℤ)| is a divisor of [G : G′] |SL_{2}(ℤ/mℤ)| = |G|, where the last equality follows from the fact that G′ = SL_{2}(ℤ/mℤ) by Corollary A.8. Thus |GL_{2}(ℤ/mℤ)| divides |G|, and hence G = GL_{2}(ℤ/mℤ).

(b) If G ≥ SL_{2}(ℤ/mℤ), then G/ SL_{2}(ℤ/mℤ) is abelian and so Occ(G) = Occ(SL_{2}(ℤ/mℤ)) ⊃ {PSL_{2}(p) : p|m} by Lemma A.10.

Conversely, suppose that Occ(G) ⊃ {PSL_{2}(p) : p|m}. For any prime p|m and any subgroup H ≤ GL_{2}(ℤ/mℤ) let H^{(p)} = H ∩ Ker(pr_{m/p^{r}}), where p^{r}||m. Note that H^{(p)} ⊴ H is a normal subgroup of H and that

H^{(p)} ≤ GL_{2}(ℤ/mℤ)^{(p)} ≃ GL_{2}(ℤ/p^{r}ℤ).

We claim that PSL_{2}(p) ∈ Occ(H^{(p)}). Indeed, since H/H^{(p)} ≃ pr_{m/p^{r}}(H), and since PSL_{2}(p) ∉ Occ(GL_{2}(ℤ/\frac{m}{p^{r}} ℤ)) by Corollary A.11 (using the fact that p > 5 by our hypothesis), we have that PSL_{2}(p) ∈ Occ(H^{(p)}) by Lemma A.9(b). Thus, by Lemma A.12 (applied to GL_{2}(ℤ/mℤ)^{(p)} ≃ GL_{2}(ℤ/p^{r}ℤ)) we see that SL_{2}(ℤ/mℤ)^{(p)} ≤ H^{(p)} because SL_{2}(ℤ/mℤ)^{(p)} ≃ SL_{2}(ℤ/p^{r}ℤ) (via the above isomorphism). Since this is true for all p|m, we obtain

∏_{p|m} SL_{2}(ℤ/mℤ)^{(p)} ≤ ∏_{p|m} H^{(p)} ≤ H.

But by the Chinese remainder theorem we have

GL_{2}(ℤ/mℤ) = ∏_{p|m} GL_{2}(ℤ/mℤ)^{(p)} and SL_{2}(ℤ/mℤ) = ∏_{p|m} SL_{2}(ℤ/mℤ)^{(p)}

(as subgroups of GL_{2}(ℤ/mℤ)) and so we obtain SL_{2}(ℤ/mℤ) ≤ H, as desired. This completes the proof of Theorem A.2. □

Remark. Part (b) of this theorem is closely related to Lemma 5 of [Se68], p. IV-26. However, the proof of that lemma has a small gap, for it is not true (or clear) that the subgroup H_{ℓ} (as defined on line 6) actually maps into SL_{2}(𝔽_{ℓ}) (as is asserted on line 13). Nevertheless, the argument is easily repaired by the same argument as in the implication (iii) ⇒ (ii) of Lemma A.12.

Proof of Theorem A.1.

If \overline{ρ}_{E,m} is surjective, then clearly so is \overline{ρ}_{E,p} = pr_{p}^{(m)} ∘ \overline{ρ}_{E,m} for every p|m.

Conversely, suppose \overline{ρ}_{E,p} is surjective for all p|m. Then GL_{2}(ℤ/pℤ) is a quotient of G := Im(\overline{ρ}_{E,m}) ≤ GL_{m}(ℤ/mℤ), and so PSL_{2}(p) ∈ Occ(G), for every p|m and hence G ≥ SL_{2}(ℤ/mℤ) by Theorem A.2(b).

Moreover, since G ≃ Gal(ℚ(E[m])/ℚ) and since ℚ(ζ_{m}) ⊂ ℚ(E[m]), we see that ϕ(m)|[G : G′]. Thus, by Theorem A.2(a) we have that G = GL_{2}(ℤ/mℤ), i.e. that \overline{ρ}_{E,m} is surjective.

The last assertion is clear, for by definition A(E) = 30 ∏_{p∈S_{E}} p, where S_{E} is the set of primes p such that \overline{ρ}_{E,p} is not surjective. □

Corollary A.13 Let E/ℚ be a non-CM elliptic curve and let m be a positive integer coprime to A(E). Then ℚ(ζ_{m}) is the maximal abelian extension of ℚ in ℚ(E[m]) and Gal(ℚ(E[m])/ℚ) ≃ GL_{2}(ℤ/mℤ).

Proof. Let G := Gal(ℚ(E[m])/ℚ) ≃ Im(\overline{ρ}_{E,m}). By Theorem A.1 we know that \overline{ρ}_{E,m} is surjective, so G ≃ Im(\overline{ρ}_{E,m}) = GL_{2}(ℤ/mℤ). Thus, by Corollary A.8 we have [G : G′] = ϕ(m) and so the maximal abelian extension in ℚ(E[m]) has degree ϕ(m). Since ℚ(ζ_{m}) ⊂ ℚ(E[m]), the assertion follows. □

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