BOUNDS FOR CONGRUENCE PRIMES

M. Ram Murty^{1}

Abstract. Given a normalized newform f of weight 2 and level N and rational integral Fourier coefficients, a_{n}(f), we will be interested in estimating primes ℓ for which there exists a cusp form (not necessarily an eigenform) g of weight 2 and level N with rational integer coefficients a_{n}(g), such that ℓ divides the difference of all the Fourier coefficients of a_{n}(f) − a_{n}(g) with n coprime to N. Such primes are called congruence primes. They are intimately related to the prime divisors of the minimal degree of the modular parametrization of the elliptic curve attached to E via the Eichler-Shimura mapping. We conjecture that this degree is O(N^{2+ϵ}) and show that it is equivalent to the celebrated ABC conjecture. For CM elliptic curves, we prove that this degree is O(N^{3/2+ϵ}). We also study the related question: given two non-isogenous elliptic curves E_{1} and E_{2}, without complex multiplication and conductors N_{1} and N_{2} respectively, how large must ℓ be so that the Galois group of ℚ(E_{1}[ℓ], E_{2}[ℓ]) over ℚ is as large as possible.

Dedicated to Goro Shimura

1. Introduction. Let f be a normalized newform of weight 2 for Γ_{0}(N). Suppose that the Fourier coefficients of f are rational integers. We will be interested in those primes ℓ for which there exists a cusp form g (not necessarily an eigenform) of weight 2 and level N, also with integer coefficients such that f and g are orthogonal with respect to the Petersson inner product and

f ≡ g( mod ℓ).

This means that, if a_{n}(f) and a_{n}(g) denote the n-th Fourier coefficient of f and g respectively, then

a_{n}(f) ≡ a_{n}(g)( mod ℓ)

for all n coprime to N. Such a prime ℓ will be called a congruence prime for f.

More generally, we are interested in natural numbers δ for which there is a congruence

f ≡ g( mod δ)

with g orthogonal to f and a_{n}(g) ∈ ℤ. We will then say δ is a congruence divisor for f.

Let S denote the set of cusp forms of weight 2 on Γ_{0}(N) with rational integer Fourier coefficients. Denote by L the orthogonal complement of f in S. Then, one knows that there are only finitely many congruence divisors for f and the largest such, denote it by D = D_{f}, is the exponent of the finite group S/(ℤf + L) (see Zagier [Z, p. 381]). In [Ma1], Mazur introduces the notion of primes of fusion. That this notion essentially coincides with the concept of a congruence prime has been shown by Ribet [Ri].

1 Research partially supported by NSERC and FCAR grants. AMS Classification: Primary 11F33, 11F30 Secondary 14G20. Key words: modular forms, congruences, Fourier coefficients.

Our aim is to obtain bounds for congruence primes and congruence divisors of f in terms of the level of f. One conjectures that

Conjecture D: D_{f} = O(N^{2+ϵ}) for any ϵ > 0.

A classical theorem of Eichler and Shimura associates an elliptic curve E_{f}/ℚ such that the L-function of E_{f} agrees with the L-function of f. More precisely, consider the map

ϕ(τ) = 2πi ∫_{i∞}^{τ} f(z)dz (τ ∈ H^{∗})

where H^{∗} = H ∪ ℚ ∪ {i∞} and H denotes the upper half plane. It is easily seen that ϕ is independent of the path chosen from i∞ to τ. If γ ∈ Γ_{0}(N), then

\begin{aligned}ω(γ) : & = ϕ(γτ) − ϕ(γ) \\ & = 2πi ∫_{τ}^{γτ} f(z)dz \end{aligned}

is independent of τ. Moreover, ω : Γ_{0}(N)→ℂ is a homomorphism. The image of this map is a lattice of rank 2 in ℂ. If we denote this lattice by Λ_{f}, then we have the Eichler-Shimura map ϕ_{ES} given by

ϕ_{ES} : X_{0}(N) = H^{∗}/Γ_{0}(N)→ℂ/Λ_{f}.

Eichler and Shimura have shown that the torus ℂ/Λ_{f} is the set of complex points of an elliptic curve E_{f} defined over ℚ. An elliptic curve E/ℚ which is isogenous to E_{f} for some f will be called modular.

The map ϕ = ϕ_{ES} is a branched covering between the two Riemann surfaces X_{0}(N) and E_{f}. From the work of Doi, Hida, Ohta, Mazur, Ribet and others, we know that the largest congruence divisor D for f and the topological degree of ϕ are related (see Zagier[Z, p. 381]). The precise relationship has not been spelled out in the literature. For a given modular elliptic curve E/ℚ, of conductor N, we choose a modular parametrization

ϕ_{E} : X_{0}(N)→E

of minimal degree. In any ℚ isogeny class of modular elliptic curves, there is a curve E with deg ϕ minimal. We call such an E a strong modular elliptic curve. Mazur has shown that if E and E′ are isogenous over ℚ, then there is some isogeny between them of degree ≤ 163. Thus, deg ϕ_{E_{f}} ≍ deg ϕ_{E} for the strong modular elliptic curve in the isogeny class of E_{f}. We have the following conjecture of Frey [Fr]:

Degree conjecture: For any modular elliptic curve E/ℚ of conductor N,

deg ϕ_{E} = O(N^{2+ϵ}).

The precise relationship between the minimal degree of a modular parametrization of E_{f} and the congruence divisor D = D_{f} is given by:

Proposition 0. (Mazur - Ribet) For a strong modular elliptic curve E with corresponding newform f, we have deg ϕ_{E}|D_{f}. The only primes dividing D_{f}/deg ϕ_{E} are primes dividing the conductor of E.

Proof. The proof of this theorem will appear in a forthcoming paper by Mazur and Ribet [MR].

Following Cremona [Cr], the topological degree can be computed by observing that

4π^{2}(f, f) = 2π^{2}i ∫_{X_{0}(N)(ℂ)} f(τ)dτ ∧ \overline{f(τ)dτ}

= \frac{i}{2} ∫_{X_{0}(N)(ℂ)}(2πif(τ)dτ) ∧ \overline{2πif(τ)dτ}

= \frac{i}{2} ∫_{X_{0}(N)(ℂ)} ϕ^{∗}(dz) ∧ \overline{ϕ^{∗}(dz)},

since ϕ^{∗}(dz) = 2πif(τ)dτ, where z = x + iy is the co-ordinate on E_{f}. Whence,

4π^{2}(f, f) = \frac{i}{2}(deg ϕ) ∫_{E_{f} (ℂ)} dz ∧ \overline{dz}
= (deg ϕ) ∫_{E_{f} (ℂ)} dxdy
= (deg ϕ)V ol(E_{f})

where V ol(E_{f}) is the area of the fundamental parallelogram of the lattice of periods of E_{f}. As in Cremona [Cr], the Manin constant does not enter into this calculation.

Conjecture D is not unrelated to the celebrated ABC conjecture. In fact, as we shall show below, it is essentially equivalent to it. This relationship is similar in spirit to Szpiro’s conjecture [Sz] and its relationship to ABC as exposed in Oesterlé [Oe] or Hindry [H]. In fact, as we shall see below,

log D_{f} ∼ log N + \frac{1}{6} log max\{|c_{4}|^{3}, |c_{6}|^{2}\}

so that Conjecture D is easily seen to be equivalent to Conjecture 4’ in [Oe]. The ABC conjecture predicts that if

A + B = C with A, B, C ∈ ℤ

and mutually coprime, then

max(|A|, |B|, |C|) ≤ κ(ϵ)( ∏_{p|ABC} p)^{1+ϵ}

for any ϵ > 0 and κ(ϵ) is a constant depending only on ϵ.

To understand the ABC conjecture, Frey [Fr] had the idea of studying the associated elliptic curve

E_{A,B,C} :  y^{2} = x(x − A)(x + B).

We will call such curves Frey curves. They have conductor

N = 2^{δ} ∏_{p|ABC} p

with δ bounded (see [DK]).

Rubin and Silverberg observed that the modularity of Frey curves follows from Diamond [D] which is an extension of the work of Wiles [W]. Later, Diamond and Kramer [DK] showed that it follows directly from Wiles [W] and Taylor-Wiles [TW]. That is, such a curve is isogenous to a curve of the form E_{f} for some newform f with a_{n}(f) ∈ ℤ. We have:

Theorem 1.

(i) The degree conjecture ⇒ABC; (ii) ABC⇒ the degree conjecture for all Frey curves.

Remark. A weaker version of (i) was first proved by Frey [Fr]. Namely, he established that some bound for the degree gives some version of ABC (that is, with 1 + ϵ replaced by some larger number). Mai and Murty established (ii) in [MM]. As we shall see below, the precise relationship between the exponents in these two conjectures is determined by invoking a Siegel-type theorem for the lower bound of the symmetric square L-function at s = 2 due to Hoffstein and Lockhart [HL].

By Proposition 0, we infer that bounding D_{f} would give us bounds for deg ϕ_{E_{f}}. In addition, bounding the prime divisors of D_{f} gives us bounds for the prime divisors of deg ϕ_{E_{f}}. It is this thme that we shall explore in this paper.

For CM elliptic curves, we establish

Theorem 2. Let E/ℚ be a CM elliptic curve with conductor N. Then E is modular and the minimal degree of a modular parametrization satisfies N^{3/2−ϵ} ≪ deg ϕ_{E} ≪ N^{3/2+ϵ} for any ϵ > 0. (The implied constant may depend on ϵ.)

For any Frey curve, we have:

Theorem 3. log D = O(N^{1/3+ϵ}).

Corollary 4. If ℓ is a congruence prime for f, then

ℓ ≪ exp(N^{\frac{1}{3} +ϵ}).

Remark. This result also implies that the number of congruence primes for f is

≪ N^{\frac{1}{3} +ϵ}.

The degree can be estimated in several ways. We obtain the following curious properties of various determinants. Let f_{1}, ..., f_{r} be a basis of normalized eigenforms for S_{2}(N), the space of cusp forms of weight 2 and level N. Let S be any set of r distinct natural numbers coprime to N. Consider the r × r matrix

A_{S} = (a_{n}(f_{i}))_{{n∈S \atop i≤r}} .

Theorem 5. (det A_{S})^{2} is a rational integer and

D_{f}^{2}|(det A_{S})^{2}.

Using this theorem, we will deduce:

Corollary 6.

D_{f} = O((2N^{2}r)!^{3/2}(log 2N^{2}r)^{r})

where r = dim S_{2}(N).

Mazur [Ma] asked the question: given f as above, let ℓ be a prime such that there is an eigenform g with integer coefficients such that f ≡ g( mod ℓ). Is ℓ bounded ? We will prove:

Theorem 7. Given a newform f ∈ S_{2}(N) with a_{n}(f) ∈ ℤ, suppose that there is an eigenform g ∈ S_{2}(N) orthogonal to f and with a_{n}(g) ∈ ℤ, and a prime ℓ such that f ≡ g( mod ℓ). Then ℓ ≤ N^{\frac{1}{2} +ϵ}.

We indicate in the last section how one can replace 1/2 by 3/2 using an easier argument. Recently, Frey has given examples of non-isogenous E and E′ over ℚ, such that E[11] ≃ E′[11] so that ℓ ≥ 13 in Theorem 7.

Theorem 7 has an application to the computation of Galois groups. Recall the well-known theorem of Serre [Ser]. Let E be an elliptic curve /ℚ without complex multiplication. Then, the field ℚ(E[ℓ]) obtained by adjoining the ℓ-division points of E has Galois group over ℚ isomorphic to GL_{2}(𝔽_{ℓ}) for ℓ sufficiently large. How large ℓ should be is answered by means of the effective Chebotarev density theorem. Serre [Ser2] showed that assuming the generalized Riemann hypothesis (GRH) for the Dedekind zeta functions of ℚ(E[ℓ]), one can prove that ℓ ≫ (log N)^{2}. It is also possible to show that unconditionally, there is a constant c > 0 so that the result holds for ℓ ≥ N^{c}. (See [Ser3, p. 716].)

Given two distinct non-isogenous elliptic curves E_{1} and E_{2} over ℚ, without complex multiplication, how large is

Gal(ℚ(E_{1}[ℓ], E_{2}[ℓ])/ℚ)?

The largest it can be is the subgroup

H_{ℓ} = {(a, b) ∈ GL_{2}(𝔽_{ℓ}) × GL_{2}(𝔽_{ℓ}) : det a = det b}

of

GL_{2}(𝔽_{ℓ}) × GL_{2}(𝔽_{ℓ}).

Theorem 8. Suppose E_{1} and E_{2} are non-isogenous modular elliptic curves /ℚ without complex multiplication and having conductors N_{1} and N_{2} respectively. Let N = max(N_{1}, N_{2}). There is an effective constant c > 0, such that if ℓ ≥ N^{c}, then

Gal(ℚ(E_{1}[ℓ], E_{2}[ℓ])/ℚ) = H_{ℓ}.

If we assume the GRH, we can improve this to ℓ ≫ log N.

Remark. Mazur’s conjecture would imply the result for all ℓ ≥ c_{1} for some constant c_{1}.

2. Proof of Theorem 1.

From the formula already stated:

4π^{2}(f, f) = (deg ϕ)V ol(E_{f})

we find by Deligne [De, p. 4] and as in Frey [Fr, p. 131] that

c^{2}4π^{2}(f, f) = (deg ϕ)e^{−2h(E)}

where h(E) is the Faltings height of E and c is Manin’s constant. By recent work of Edixhoven [E], we know that c is bounded. Therefore

2h(E) + log(f, f) = log deg ϕ + O(1).

By Silverman [Si, p. 258]

h(E) = \frac{1}{12}h(j_{E}) + +\frac{1}{12} log γ + O(log h(j_{E})),

where h(j_{E}) is the naive (Weil) height of j_{E} and γ is the unstable minimal discriminant (see [Si, p. 257]). Writing j_{E} = a/b with (a, b) = 1 and a, b ∈ ℤ, we see that the unstable minimal discriminant is Δ/b where Δ is the discriminant of E. It is easily verified that in the case of Frey curves, the unstable minimal discriminant is bounded as the conductor tends to infinity, since

Δ = (ABC)^{2}/2^{8}

and

j_{E} = \frac{2^{8}(A^{2} + B^{2} + AB)^{3}}{(ABC)^{2}} > 1.

Now A^{2} + B^{2} + AB is coprime to ABC so that

h(j_{E}) = 3 log(A^{2} + B^{2} + AB).

A simple calculation shows that

log(A^{2} + B^{2} + AB) = 2 log max(|A|, |B|, |C|) + O(1).

We will now prove that for any ϵ > 0 and N greater than some constant depending only on ϵ, we have the inequality

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

The upper bound is an easy consequence of the Phragmén-Lindelöf theorem applied to the L-function of the symmetric square, as in Mai - Murty [MM]. The L-function attached to the symmetric square of f has the special value

L(2, sym^{2}(f)) = 288π^{3} ∏_{p^{2}|N} (1 − \frac{1}{p^{2}} )\frac{(f, f)}{N}.

By the Phragmén - Lindelöf theorem, we have

L(2, sym^{2}(f)) = O(log N).

Hence,

(f, f) ≤ c_{2}N log N

for some constant c_{2}. By a result of Hoffstein and Lockhart [HL], we have

log(f, f) > (1 − ϵ) log N.

Hence

log max(|A|, |B|, |C|) + O(log log max(|A|, |B|, |C|)) = log deg ϕ − log(f, f) + O(1).

Therefore, if deg ϕ ≤ N^{2+ϵ}, we get the ABC conjecture.

For the converse,

log max(|A|, |B|, |C|) + O(log log max(|A|, |B|, |C|))

≥ log deg ϕ − log N − log log N + O(1).

Therefore, assuming the ABC conjecture gives

deg ϕ = O(N^{2+ϵ}).

This proves (ii).

3. Proof of Theorem 2. Recall that if E/ℚ has CM, and j_{E} ≠ 0, 1728, then E is a quadratic twist of one of a finite set of elliptic curves E_{i} of conductor N_{i} and discriminant Δ_{i} (say). If j_{E} = 0, or 1728, then it has a Weierstrass equation y^{2} = x^{3} + D or y^{2} = x^{3} + Dx respectively (see [Si2, p. 308 and p. 340]). We will discuss the case the curve is a quadratic twist by D (the other cases being handled similarly). In this case, the conductor divides N_{i}D^{2} for some i. An easy calculation shows that the discriminant of E is O(Δ_{i}D^{6}). We therefore obtain

log deg ϕ = log(f, f) + O(1) which implies by the proof of Theorem 1,

N_{E}^{3/2−ϵ} ≪ deg ϕ ≪ N_{E}^{3/2+ϵ}.

This completes the proof of Theorem 2.

4. Proof of Theorem 3.

Stewart and Yu [SY] have proved that for the ABC conjecture, for any ϵ > 0, there is an effectively computable constant κ(ϵ) so that

log max(|A|, |B|, |C|) ≤ κ(ϵ)N^{2/3+ϵ}.

The exponent has recently been improved by Stewart [S] to 1/3 + ϵ. Hence, by the above analysis, we deduce that for some constant κ_{1}(ϵ),

deg ϕ < exp(κ_{1}(ϵ)N^{1/3+ϵ}).

This completes the proof of Theorem 3.

To deduce Corollary 4, we invoke Proposition 0 to deduce that the primes dividing the degree are precisely the congruence primes apart from primes dividing the level N. Therefore, each prime divisor ℓ of deg ϕ satisfies

ℓ ≤ exp(κ_{1}(ϵ)N^{1/3+ϵ}).

5. Proof of Theorem 5.

Let f_{1}, ..., f_{r} be a normalized (i.e. a_{1}(f_{i}) = 1) orthogonal basis for S_{2}(N) which are eigenforms for all the Hecke operators T_{p} for (p, N) = 1. (See Lang [L].) Consider (det A_{S})^{2}. Since A_{S}A_{S}^{(t)} is invariant under Galois conjugation, we find that (det A_{S})^{2} is a rational integer. Suppose it is not zero. By going to a finite algebraic extension, we can solve the system

∑_{j∈S} a_{j}(f_{i})m_{j} = \{ \begin{matrix}det A_{S} & if i = 1 \\ 0 & if i ≠ 1 , \end{matrix}

with the m_{j} algebraic integers. Let g be orthogonal to f = f_{1} with a_{n}(g) ∈ ℤ. Suppose that

f ≡ g( mod D).

Let us write

g = c_{1}f_{1} + c_{2}f_{2} + ⋯ c_{r}f_{r},

so that c_{1} = 0. Now consider

h(z) = ∑_{j∈S} m_{j}T_{j}(g)

where T_{i} denotes the i-th Hecke operator. This is easily seen to be

∑_{i=1}^{r} c_{i}( ∑_{j∈S} a_{j}(f_{i})m_{j})f_{i}

which is equal to 0 since c_{1} = 0. Recall that S consists of numbers coprime to N. Then the n-th Fourier coefficient of h(z) is

0 = ∑_{j∈S} m_{j} ∑_{d|n,j} a_{nj/d^{2}}(g)d

≡ ∑_{j∈S} m_{j} ∑_{d|n,j} a_{nj/d^{2}}(f_{1})d( mod D)

≡ ∑_{j∈S} m_{j}a_{n}(f_{1})a_{j}(f_{1})( mod D)

≡ a_{n}(f_{1}) det A_{S}( mod D)

so that choosing n = 1 gives that D^{2}|(det A_{S})^{2}.

We now prove Corollary 6. By Proposition 2.16 of Shimura [Sh, p. 39], we deduce that the order of vanishing at i∞ of any non-zero cusp form f ∈ S_{2}(N) is

< \frac{N}{6} ∏_{p|N} (1 + \frac{1}{p}) = h (say).

Note that h > 2 dim S_{2}(N) = 2r. Let χ_{N} be a primitive quadratic character with conductor supported at all the primes dividing N. The r vectors

v_{i} ⊗ χ_{N}(n) = (a_{1}(f_{i})χ_{N}(1), ..., a_{h}(f_{i})χ_{N}(h)) 1 ≤ i ≤ r

corresponding to f_{i} ⊗ χ_{N} ∈ S_{2}(M) for some M|N^{3} must be linearly independent for h = 2 dim S_{2}(M). Otherwise, we would have a non-zero element in S_{2}(M) with a zero of order exceeding h contrary to the above remark. Thus, the matrix whose rows are the vectors v_{i} has rank r. Hence, there are r independent columns. Thus, there is a subset

S ⊆ {1, 2, ..., h}

consisting of elements coprime to N, so that |S| = r and

det A_{S} ≠ 0.

This particular choice allows us to estimate D. Since h ≤ 2 dim S_{2}(N^{3}) = 2N^{2}r, we have by Deligne’s estimate for the Fourier coefficients:

D ≤ (2N^{2}r)!^{3/2} ∏_{i=1}^{2N^{2}r} d(i)^{1/2}.

where d(i) denotes the divisor function. By the arithmetic - geometric mean inequality, we deduce

\{ ∏_{i=1}^{2N^{2}r} d(i)\}^{1/2N^{2}r} ≤ \frac{1}{2N^{2}r} ∑_{i=1}^{2N^{2}r} d(i) ≤ log 2N^{2}r,

by well-known estimates of analytic number theory.

Remark. If N is prime, we need not take S consisting of elements coprime to N since all eigenforms are newforms and thus eigenfunctions for all Hecke operators (including T_{p} for p|N). In this case, we can take S to be

{1, 2, ..., r}

then we would get the sharper estimate

det A ≤ r!^{3/2} ∏_{i=1}^{r} d(i)^{1/2} ≤ r!^{3/2}(log r)^{r/2}.

This is related to the question of whether i∞ is a Weierstrass point for Γ_{0}(N). (Recall that a point x on a compact Riemann surface X is called a Weierstrass point of X if there exists a non-zero differential form whose order of vanishing at x exceeds the genus of X.) If N is a prime, Atkin [A] has shown that i∞ is not a Weierstrass point so that we get the sharper bound for D in this case. It is conjectured that for any squarefree N, i∞ is not a Weierstrass point.

§6. The Rankin-Selberg method.

We collect in this section the necessary facts and estimates derived from the Rankin - Selberg method. All of these facts are contained in the original papers of Rankin and Selberg though they perhaps did not pay much attention to the dependence on the level in the estimation of the error terms. Using standard estimates of the L-function of the symmetric square obtained by invoking the Phragmén - Lindelöf theorem as in [MM], we obtain:

Lemma 9. Let f be a newform of weight 2 and level N. Then,

\frac{1}{2} ∑_{n≤x} |a_{n}(f)|^{2} log^{2} \frac{x}{n} = c(f)x^{2} + O(x^{3/2}N^{1+ϵ})

where

c(f) = \frac{8π^{3}ϕ(N)(f, f)}{N^{2}}.

If g is another newform orthogonal to f, then

\frac{1}{2} ∑_{n≤x} a_{n}(f)\overline{a_{n}(g)} log^{2} \frac{x}{n} = O(x^{3/2}N^{1+ϵ}).

Remark. The constant c(f) is given explicitly in Stark [St, p. 90].

If we want estimates for

∑_{{n≤x \atop (n,M)=1}} |a_{n}(f)|^{2} log^{2} \frac{x}{n}

and

∑_{{n≤x \atop (n,M)=1}} a_{n}(f)\overline{a_{n}(g)} log^{2} \frac{x}{n},

then this involves removing the Euler factors for primes p|M in the Dirichlet series

∑_{n=1}^{∞} \frac{|a_{n}(f)|^{2}}{n^{s}}

and

∑_{n=1}^{∞} \frac{a_{n}(f)\overline{a_{n}(g)}}{n^{s}}

before applying the method to derive the above asymptotic formulas. This yields

Lemma 10. For f and g as in Lemma 9,

\frac{1}{2} ∑_{{n≤x \atop (n,M)=1}} |a_{n}(f)|^{2} log^{2} \frac{x}{n} = c(f, M)x^{2} + O(x^{3/2}N^{1+ϵ}M^{ϵ})

and

∑_{{n≤x \atop (n,M)=1}} a_{n}(f)\overline{a_{n}(g)} log^{2} \frac{x}{n} = O(x^{3/2}N^{1+ϵ}M^{ϵ})

where

c(f, M) = \frac{8π^{3}ϕ(N)(f, f)}{N^{2}} ∏_{p|M} ( ∑_{k=1}^{∞} \frac{|a_{p^{k}}(f)|^{2}}{p^{2k}} )^{−1}.

We can use this to deduce:

Lemma 11. If f and g are as in Lemma 9, then there is an n = O(N^{1+ϵ}M^{ϵ}) such that (n, M) = 1 and

a_{n}(f) ≠ a_{n}(g).

Proof. We use the estimate of Hoffstein and Lockhart for the Petersson inner product to estimate the residue from below. The result is now immediate.

We briefly indicate the proofs of these lemmas since they do not appear in the literature in the form we need it with the explicit dependence on the level N as stated above. If N is squarefree and M = 1, a variant of Lemma 11 is due to Goldfeld and Hoffstein [GH]. A simplified and generalized version of their work is found in [M2].

Let L(s) be a Dirichlet series satisfying the following properties:

(i) L(s) = ∏_{p} L_{p}(s) for σ > 1 where L_{p} is the inverse of a polynomial of degree m_{p} in p^{−s}. The product is over all finite primes.

(ii) There is a positive integer d so that m_{p} ≤ d for all p and m_{p} = d for all

but finitely many p. Let us write

L_{p}(s) = ∏_{i=1}^{m_{p}}(1 − π_{i,p}p^{−s})^{−1}

where π_{i,p} ∈ ℂ, |π_{i,p}| = 1. Define

L_{p}^{v}(s) = ∏_{i=1}^{m_{p}}(1 − \overline{π}_{i,p}p^{−s})^{−1}

and set

L^{v}(s) = ∏_{p} L_{p}^{v}(s).

Let A be a positive real number and a, b non-negative integers with a + b ≤ d. Set

Λ(s) = A^{s/2}(π^{−s/2}Γ(s/2))^{a}(π^{(s+1)/2}Γ((s + 1)/2))^{b}L(s)
Λ^{v}(s) = A^{−s/2}(π^{−s/2}Γ(s/2))^{a}(π^{−(1+s)/2}Γ((1 + s)/2)^{b}L^{v}(s).

Suppose that

(iii) Λ(s) and Λ^{v}(s) have an analytic continuation to the entire complex plane

except possibly for a pole at s = 0 or s = 1, (iv) Λ(s) = wΛ^{v}(1 − s) with w ∈ ℂ, |w| = 1. Under assumptions (i) to (iv),

we have for 0 ≤ σ ≤ 1,

|L(σ + it)| ≤ (Aτ^{d})^{\frac{1−σ}{2}} (log Aτ^{d})^{d}

where τ = |t| + 2. This is an effective version of the Phragmén-Lindelöf theorem. The proof given in [MM2] applies to functions with no pole at s = 0 or 1. But it can easily be adapted to give the same result if L(s) has a pole at s = 0 or 1. (Let us also take this opportunity to point out a misprint in [MM2]. On page 251 of [MM2] in condition (i), L_{p} is the inverse of a polynomial in p^{−s}.)

We apply this result to the series

L_{f,g}(s) = ζ_{N}(2s) ∑_{n=1}^{∞} \frac{a_{n}\overline{b_{n}}}{n^{s+1}}

where

ζ_{N}(2s) = ∏_{p|/N} (1 − \frac{1}{p^{2s}} )^{−1}.

Rankin [R] established the analytic continuation of L_{f,g}(s). Ogg [Ogg], derived the functional equation in the case f and g have squarefree levels. Li[Li] has generalized this to arbitrary level and weight.

For the sake of simplicty, we will confine our attention to squarefree level. An easy adaptation of the method to the non-squarefree level is effected by using the (slightly more complicated) functional equation of Li [Li].

Let f, g be normalized cusp forms of squarefree level N_{1} and N_{2} respectively. Let N = lcm(N_{1}, N_{2}) and M = gcd(N_{1}, N_{2}). For p|M, let

c_{p} = a_{p}b_{p} = ±1

and set

Φ^{+}(s) = Φ(s) ∏_{p|M} (1 − c_{p}p^{−s})^{−1}

where

Φ(s) = (\frac{\sqrt{N}}{2π} )^{2s}Γ(s)Γ(s + 1)L_{f,g}(s).

Then

Φ^{+}(1 − s) = Φ^{+}(s).

We now consider the integral

\frac{1}{2πi} ∫_{c−i∞}^{c+i∞} ( ∑_{n=1}^{∞} \frac{a_{n}\overline{b_{n}}}{n^{s+1}} )\frac{x^{s}}{s^{3}} ds

which is equal to

\frac{1}{2} ∑_{n≤x} \frac{a_{n}\overline{b_{n}}}{n} log^{2} \frac{x}{n}.

We write the Dirichlet series in the integrand as

\frac{1}{ζ_{N}(2s)}(ζ_{N}(2s) ∑_{n=1}^{∞} \frac{a_{n}\overline{b_{n}}}{n^{s+1}} ),

and apply the Phragmén - Lindelöf theorem to the paranthetic term to obtain an estimate of

O(N^{1/2}T log^{4} NT)

on the line ℜ(s) = 1/2. The term 1/ζ_{N}(2s) can be estimated easily on ℜ(s) = 1/2. We derive the estimate

\frac{1}{2} ∑_{n≤x} \frac{a_{n}\overline{b_{n}}}{n} log^{2} \frac{x}{n} = O(x^{1/2}N^{1/2} log^{4} N).

when f = g, we find

∑_{n≤x} \frac{|a_{n}|^{2}}{n} log^{2} \frac{x}{n} = 2α_{f}x + O(x^{1/2}N^{1/2} log^{4} N)

where

α_{f} = \frac{4π^{3}ϕ(N)(f, f)}{N^{2}}.

Thus,

\frac{1}{2} ∑_{n≤x}(a_{n} − b_{n})^{2} log^{2} \frac{x}{n} = (α_{f} + α_{g})x + O(x^{1/2}N^{1/2} log^{4} N).

This leads to the result that a_{n} ≠ b_{n} for some n = O(N log^{8} N). (In [M2], the slightly better estimate O(N log log N) is obtained. ) This completes the sketch of the proof of Lemmas 9, 10 and 11.

§7. Proof of Theorem 7.

If we have a congruence between newforms f and g

a_{n}(f) ≡ a_{n}(g)( mod ℓ)

for all (n, N) = 1, then by Lemma 11, we deduce that for some n = O(N^{1+ϵ}ℓ^{ϵ}), we have

a_{n}(f) ≠ a_{n}(g)

and (n, N) = 1. Therefore,

ℓ ≪ N^{\frac{1}{2} +ϵ}.

This completes the proof of Theorem 7.

§8. Proof of Theorem 8.

By Serre [Se], we know that if ℓ ≥ N^{c} for some constant c, then

Gal(ℚ(E_{1}[ℓ])) ≃ GL_{2}(𝔽_{ℓ})

and

Gal(ℚ(E_{2}[ℓ])) ≃ GL_{2}(𝔽_{ℓ})

We now apply results of Ribet [Ri] to deduce that the only obstruction to the assertion of the theorem is the set of congruence primes linking f_{1} and f_{2} where f_{1} and f_{2} are the newforms associated to E_{1} and E_{2}. We will need the following lemma which is a special case of Theorem 3.8 of [Ri2].

Lemma 12. For a prime ℓ ≥ 5, let e be a divisor of ℓ − 1, let V, V′ be vector spaces of dimension 2 over 𝔽_{ℓ}, let B, B′ be the subgroups of GL(V) and GL(V′) respectively consisting of all elements whose determinants are e-th powers. Let E be the subgroup of B × B′ consisting of all (b, b′) with det b = det b′ and suppose H is a subgroup of E in B × B′ whose projections to each factor are surjective. Then, if H ≠ E, there is an isomorphism ϕ from V to V′ and a character χ of H with χ^{2} = 1 such that

b′ = χ(h)ϕbϕ^{−1}

for all (b, b′) ∈ H.

We apply this lemma in the case e = 1. We suppose ℓ is a prime such that ℓ ≥ N^{c} with c sufficiently large and G_{i} = Gal(ℚ(E_{i}[ℓ])/ℚ) ≃ GL_{2}(𝔽_{ℓ}) for i = 1, 2. Then, let H = Gal(ℚ(E_{1}[ℓ], E_{2}[ℓ])/ℚ) ⊆ G_{1} × G_{2} and E the subgroup of elements (b, b′) of G_{1} × G_{2} with det b = det b′. Clearly H satisfies the conditions of the lemma. If H ≠ E, there is an isomorphism ϕ and a quadratic character χ of H such that

b′ = χ(h)ϕbϕ^{−1} ∀(b, b′) ∈ H.

Thus, evaluating this on the powers of the Frobenius automorphism gives

a_{p^{m}}(E_{1}) ≡ χ(p^{m})a_{p^{m}}(E_{2})( mod ℓ)

for all

p|/ℓN_{1}N_{2}.

Since E_{1} and E_{2} are modular, there are cusp forms f_{1} and f_{2} such that

a_{p^{m}}(E_{i}) = a_{p^{m}}(f_{i}), i = 1, 2.

Therefore,

a_{n}(f_{1}) = χ(n)a_{n}(f_{2})( mod ℓ)

for all n coprime to ℓN_{1}N_{2}. Moreover, the quadratic character χ can only be ramified at most at the prime dividing ℓ, since the kernel of χ cuts out a quadratic extension contained in ℚ(E_{i}[ℓ]) and the quadratic subfield of ℚ(ζ_{ℓ}) is the only such. By Lemma 11, we deduce that for some n = O(N^{1+ϵ}ℓ^{ϵ}), and (n, ℓN_{1}N_{2}) = 1, we deduce

a_{n}(f_{1}) ≠ χ(n)a_{n}(f_{2}).

Therefore, by the Ramanujan estimate, we deduce ℓ ≪ N^{1/2+ϵ}. The first assertion now follows from Theorem 7.

To deduce the statement on the GRH, we know from Serre that the Galois groups generated by the ℓ-division points of E_{1} and E_{2} are as large as possible for ℓ ≥ (log N)^{2}. By the Rankin-Selberg method, we know that for i = 1, 2,

∑_{p≤x} |a_{p}(f_{i})|^{2} log p = x^{2} + O(x(log Nx)^{2})

and

∑_{p≤x} a_{p}(f_{1})\overline{a_{p}(f_{2})} = O(x(log Nx)^{2})

from which we deduce that

∑_{p≤x} (a_{p}(f_{1}) − a_{p}(f_{2}))^{2} = 2x^{2} + O(x(log Nx)^{2})

so that for some p_{0} ≪ (log N)^{2}, we have

a_{p_{0}}(f_{1}) ≠ a_{p_{0}}(f_{2}).

Hence,ℓ divides

|a_{p_{0}}(f_{1}) − a_{p_{0}}(f_{2})|

so that

ℓ ≪ (log N)

which completes the proof of Theorem 8.

9. An alternate method.

We make some remarks using an alternate method. At the outset, let us note that optimal results are obtained by the Rankin-Selberg method as in Theorem 7. Also, assuming the generalized Riemann hypothesis, one can bound the congruence prime ℓ by O(log N) using the method outlined in the proof of Theorem 8. If instead of the generalised Riemann hypothesis, we assume only the generalized Lindelöf hypothesis for the Rankin-Selberg convolutions that intervene in the proof of Theorem 7, then the term N^{1/2+ϵ} appearing in the error terms in Lemmas 9, 10, and 11 can be replaced by N^{ϵ}. This would then give an estimate of ℓ ≪ N^{ϵ} in Theorem 7.

Without the Rankin-Selberg method, one can prove:

Theorem 7^{∗}. Let f and g be as in Theorem 7 and suppose that

f ≡ g( mod ℓ).

Then, ℓ ≪ N^{3/2+ϵ}.

Proof. We may suppose ℓ > N. We have

a_{n}(f) ≡ a_{n}(g)( mod ℓ)

for all (n, N) = 1. We twist by a quadratic character χ_{N} whose conductor is divisible by all the primes dividing N. Then,

a_{n}(f)χ_{N}(n) ≡ a_{n}(g)χ_{N}(n)( mod ℓ)

for all n. The level of the twist of f by χ_{N} divides N^{3}. Therefore, if

a_{n}(f)χ_{N}(n) = a_{n}(g)χ_{N}(n)

for all

n ≤ ∑_{d|N^{3}} dim S_{2}(d) = O(N^{3+ϵ})

then

a_{n}(f) = a_{n}(g)

for all (n, N) = 1 contradicting the assumption (f, g) = 0. Hence, for some n = O(N^{3+ϵ}), we have

a_{n}(f)χ_{N}(n) ≠ a_{n}(g)χ_{N}(n).

Reasoning now as before, we deduce that

ℓ ≪ N^{3/2+ϵ}.

This argument may be of some independent interest. It is similar to arguments used in [M2].

10. Concluding remarks.

Masser and Wusthölz [MW] have obtained results similar to Theorem 8. Their bounds on how large ℓ should be to ensure that the Galois groups be large as possible are given in terms of the Faltings height of the elliptic curves whereas the results above are in terms of the conductor of N.

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Department of Mathematics, Queen’s University, Kingston, Ontario, K7L 3N6, Canada e-mail: murty@mast.queensu.ca