THE MODULAR APPROACH TO DIOPHANTINE EQUATIONS (PRELIMINARY DRAFT)

SAMIR SIKSEK

Abstract. The aim of these lectures is to communicate Ribet’s Level Lowering Theorem and related ideas in an explicit and simplified (but hopefully still precise) way, and to explain how these ideas are used to derive information about solutions to Diophantine equations.

1. Facts about newforms

Think about newforms in terms of their q-expansions

(1)  f = q + ∑_{n≥2} c_{n}q^{n}.

Here are some facts about newforms:

(a) Associated to our newforms will be two integers: a weight k and a level N (positive integer). If we fix k and N then there are only finitely many newforms of weight k and level N. In these notes the weight k will always be 2.

(b) If f is a newform with coefficients c_{i} as in (1) and K = ℚ(c_{2}, c_{3}, …) then K is a totally real finite extension of ℚ.

(c) The coefficients c_{i} in fact belong to the ring of integers O_{K} of the number field K.

(d) If l is a prime then

|c_{l}^{σ}| ≤ 2\sqrt{l} for all σ ∈ Gal(\overline{ℚ}/ℚ).

This is a very deep result due to Deligne.

We shall only be concerned about newforms up to Galois conjugacy. The number of newforms (up to Galois conjugacy) at a particular level increases in a very erratic way with the level N.

Theorem 1. There are no newforms at levels

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 22, 25, 28, 60.

Example 1. The newforms at a fixed level N can be computed using the modular symbols algorithm [33], [12]. Thankfully, this has been implemented in MAGMA [5] by William Stein. To compute in MAGMA the newforms at level N, use the command Newforms(CuspForms(N)). For example, the newforms at level 110 are

Date: November 1, 2004.

Key words and phrases. Diophantine equations, Frey curves, level-lowering.

f_{1} = q − q^{2} + q^{3} + q^{4} − q^{5} − q^{6} + 5q^{7} + ⋯,
f_{2} = q + q^{2} + q^{3} + q^{4} − q^{5} + q^{6} − q^{7} + ⋯,
f_{3} = q + q^{2} − q^{3} + q^{4} + q^{5} − q^{6} + 3q^{7} + ⋯,
f_{4} = q − q^{2} + θq^{3} + q^{4} + q^{5} − θq^{6} − θq^{7} + ⋯,

where the first three have coefficients in ℤ and the last one has coefficients in ℤ[θ] where θ = (−1 + \sqrt{33})/2. Note that there is a fifth newform at level 110 which is the conjugate of f_{4}. As stated above, in these notes we will only need to worry about newforms up to Galois conjugacy.

2. Correspondence between rational newforms and elliptic curves

We call a newform rational if its coefficients are all in ℚ, otherwise we call it irrational.

Theorem 2. (The Modularity Theorem for Elliptic Curves) Associated to any rational newform f of level N is an elliptic curve E_{f}/ℚ of conductor N so that for all primes l ∤ N

c_{l} = a_{l}(E_{f})

where c_{l} is the l-th coefficient in the q-expansion of f and a_{l}(E_{f}) = l + 1 − ♯E_{f}(𝔽_{l}). For any given positive integer N, the association f ↦ E_{f} is a bijection between rational newforms of level N and isogeny classes of elliptic curves of conductor N.

The association f ↦ E_{f} is due to Shimura. The fact that this association is surjective was previously known as the Modularity Conjecture, and first proved for squarefree N (the semi-stable case) by Wiles [35], [34]. The proof was completed in a series of papers by Diamond [15], Conrad, Diamond and Taylor [11], and finally Breuil, Conrad, Diamond and Taylor [6].

Example 2. In example 1 we wrote down using MAGMA the four newforms at level 110. The first three are rational. For a rational newform f we can get the corresponding elliptic curve using the MAGMA command EllipticCurve(f). To get the reference in Cremona’s tables [12] for an elliptic curve E we can use the MAGMA command CremonaReference(E). We find that the three rational newforms at level 110 given above as f_{1}, f_{2} and f_{3} correspond respectively to the elliptic curves 110C1, 110B1 and 110A1 in Cremona’s tables.

3. Level-lowering

3.1. ‘arises from’.

Definition. Let E be an elliptic curve over the rationals of conductor N, and suppose that f is a newform (of weight 2 as always) and level N′ with q-expansion as in (1), and coefficients c_{i} generating the number field K/ℚ. We shall say ^{1} that the curve E arises modulo p from the newform f (and write E ∼_{p} f) if there is

ρ_{p}^{E} : Gal(\overline{ℚ}/ℚ) → Aut(E[p])

1Rather that saying that E arises modulo p from the newform f, it is usual here to say that the Galois representation arises from the newform f.

some prime ideal 𝔓 | p of K such that for almost all primes l, we have a_{l}(E) ≡ c_{l} (mod 𝔓).

In fact we can be a little more precise.

Proposition 3.1. Suppose E ∼_{p} f. Then there is some prime ideal 𝔓 | p of K such that for all primes l

(i) if l ∤ pNN′ then a_{l}(E) ≡ c_{l} (mod 𝔓), and (ii) if l ∤ pN′ and l || N then l + 1 ≡ ±c_{l} (mod 𝔓).

If f is rational newform, then we know that f corresponds to some elliptic curve F say (this is E_{f} in the notation of Theorem 2). If E arises modulo p from f then we shall also say that E arises modulo p from F (and write E ∼_{p} F).

Proposition 3.2. Suppose that E, F are elliptic curves over ℚ with conductors N and N′ respectively. Suppose that E arises modulo p from F. Then for all primes l

(i) if l ∤ NN′ then a_{l}(E) ≡ a_{l}(F) (mod p), and (ii) if l ∤ N′ and l || N then l + 1 ≡ ±a_{l}(F) (mod p).

It does seem that this Proposition is merely a restatement of Proposition 3.1 in this special case. However it does represent a slight but very important strengthening (due to Kraus and J. Oesterlé [19]); namely the assumption that l ≠ p is removed. This is important because later on p will be an unknown prime exponent in some equation that we would like to solve. It is thus awkward to have a conditions that depend on p.

We note in passing that the condition l ∤ NN′ is equivalent to saying that the two elliptic curves E and F have good reduction at l. The condition l ∤ N′ and l || N means that E has multiplicative reduction at l, whilst F has good reduction at l.

3.2. Ribet’s Level-Lowering Theorem. Let E be an elliptic curve over ℚ. Let Δ = Δ_{min} be the discriminant for a minimal model of E, and N be the conductor of E. Suppose p is a prime, and let ^{2}

(2)  N_{p} = \frac{N}{∏\begin{matrix}q||N, \\ p|ord_{q}(Δ) \end{matrix} q} .

We emphasize that the Δ appearing in the definition of N_{p} must be the minimal discriminant.

Theorem 3. (A simplified special case of Ribet’s Level-Lowering Theorem) Suppose E is an elliptic curve over ℚ and p ≥ 5 is prime. Suppose further that E does not have a p-isogenies. Let N_{p} be as defined above. Then there exists a newform f of level N_{p} such that E ∼_{p} f.

Ribet’s Theorem is much more general than this, but this is the only case that we need. Ribet’s Theorem has a modularity assumption, but since we are restricting ourselves to elliptic curves, this follows from Theorem 2 (the Modularity Theorem).

2A highbrow remark that should be omitted on first reading: This N_{p} is not always the same as the Serre conductor. If we denote the Serre conductor by N′_{p} then N′_{p} | N_{p} and the two can only differ by a power of p. In fact Ribet’s Theorem allows us to get a newform at level N′_{p} and weight k_{p} ≥ 2 (k_{p} is the Serre weight). However in these notes we have restricted ourselves to newforms of weight 2, and it turns out that we obtain a newform at level N_{p} and not at N′_{p}.

In my experience, for purposes of practical computation, this is a bonus and not a hindrance.

Example 3. Consider the elliptic curve

E : y^{2} = x^{3} − x^{2} − 77x + 330

with Cremona reference 132B1. The minimal discriminant and conductor are respectively

Δ_{min} = 2^{4} × 3^{10} × 11, N = 2^{2} × 3 × 11.

The only isogeny the curve E has is a 2-isogeny. Hence we may apply Ribet’s Theorem with p = 5. From the above recipe (2) for the level we find that N_{p} = 44. However, there is only one newform at level 44 which corresponds to the elliptic curve

F : y^{2} = x^{3} + x^{2} + 3x − 1

with Cremona reference 44A1. Thus E ∼_{5} F. We record here the traces for E and F for primes 2 ≤ l ≤ 19.

\begin{matrix}l & 2 & 3 & 5 & 7 & 11 & 13 & 17 & 19 \\ a_{l}(E) & 0 & −1 & 2 & 2 & −1 & 6 & −4 & −2 \\ a_{l}(F) & 0 & 1 & −3 & 2 & −1 & −4 & 6 & 8 \end{matrix}

The reader is invited to compare this table with what is expected from Proposition 3.2.

3.3. Absence of Isogenies. To be able to apply Ribet’s Theorem we must know that our elliptic curve does not have a p-isogeny. For this the following Theorem of Mazur is very helpful.

Theorem 4. (Mazur [24]) Suppose E/ℚ is an elliptic curve and that at least one of the following conditions holds.

• p ≥ 17 and j(E) ∉ ℤ[\frac{1}{2}],

• or p ≥ 11 and E is a semi-stable elliptic curve,

• or p ≥ 5, ♯E(ℚ)[2] = 4, and E is a semi-stable elliptic curve,

Then E does not have any p-isogenies.

Theorem 5. (Diamond and Kramer [16]) Suppose that E/ℚ is an elliptic curve with conductor N. If ord_{2}(N) = 3, 5, 7 then E does not have any isogenies of odd degree.

Example 4. Let E be a semi-stable elliptic curve and Δ be its minimal discriminant. Brumer and Kramer conjectured [7] that if |Δ| is a perfect p-th power for some prime p, then p ≤ 7 and E has a point of order p. Serre gave a proof of this in [29] that was dependent at the time on what is known as Serre’s conjectures. This dependency has now been removed thanks to the work of Ribet and Wiles. Let us follow some of the steps of Serre’s proof.

It is easy to see that N_{p} = 1. Suppose that p ≥ 11. By Theorem 4 the curve E, being semi-stable, does not have p-isogenies. Thus Ribet’s Theorem implies that E ∼_{p} f where f is a newform of level 1. But there are no newforms of level 1. This contradiction shows that p ≤ 7. With some extra work, Serre proves not only that p ≤ 7 but that the curve E has non-trivial p-torsion.

3.4. How to use Ribet’s Theorem. Well, given a Diophantine equation, we shall suppose that it has a solution and associate the solution somehow to an elliptic curve E called a Frey curve if possible. The key properties of a ‘Frey curve’ are that

• the coefficients of E depend on the solution to the Diophantine equation,

• The minimal discriminant of the elliptic curve can be written in the form Δ = C · D^{p} where D is an expression the depends on the solution of the Diophantine equation. The factor C does not depend on the solutions but only on the equation itself.

• E has multiplicative reduction at primes dividing D. The conductor N of E will be divisible by the primes dividing C and D, and those dividing D will be removed when we write down N_{p}. In other words we can make a finite list of possibilities for N_{p} that depend on the equation. Thus we are able to list a finite set of newforms f such that E ∼_{p} f. From knowing the newforms we deduce local information about E, and since the model for E has coefficients that depend on solutions of our original Diophantine equation, we get information about these solutions.

The rest of these notes is devoted to giving concrete examples of how Ribet’s Theorem is used in getting information about solutions to Diophantine equations and occasionally solving them.

4. Fermat’s Last Theorem

In this section we prove Fermat’s Last Theorem. In fact we solve a more general equation.

Theorem 6. Suppose p ≥ 5 is prime. The equation

(3)  x^{p} + 2^{r}y^{p} + z^{p} = 0

has no solutions with xyz ≠ 0, and x, y z pairwise coprime except r = 1 and (x, y, z) = ±(−1, 1, −1).

Proof. This Theorem is due to Wiles [35] for r = 0, Ribet [27] for r ≥ 2, and Darmon and Merel [13] for r = 1. Suppose that x, y, z is a solution to (3) that is non-trivial (meaning xyz ≠ 0) and primitive (meaning x, y, z are pairwise coprime). We may assume without loss of generality that

x^{p} ≡ −1 (mod 4),  2^{r}y^{p} ≡ 0 (mod 2),  0 ≤ r < p.

Associate to this solution the elliptic curve (called a Frey curve)

(4)  E : Y^{2} = X(X − x^{p})(X + 2^{r}y^{p}).

We write the associated invariants

c_{4} = 16(z^{2p} − 2^{r}x^{p}y^{p}),  Δ = 2^{2r+4}(xyz)^{2p},  j = \frac{(z^{2p} − 2^{r}x^{p}y^{p})^{3}}{2^{2r−8}(xyz)^{2p}} .

Applying Tate’s algorithm [32, pages 364–369] we compute the minimal discriminant and conductor: (5)

Δ_{min} = \begin{cases}2^{2r+4}(xyz)^{2p} & if 16 ∤ 2^{r}y^{p} \\ 2^{2r−8}(xyz)^{2p} & if 16 | 2^{r}y^{p}, \end{cases} N = \begin{cases}2 Rad_{2}(xyz) & r ≥ 5 or y is even \\ Rad_{2}(xyz) & r = 4 and y is odd \\ 8 Rad_{2}(xyz) & r = 2, 3 and y is odd \\ 32 Rad_{2}(xyz) & r = 1 and y is odd, \end{cases}

where for positive integer R and prime q we let

Rad_{q}(R) = ∏_{\begin{matrix}l | R prime, \\ l≠q \end{matrix}} l.

Applying the recipe in (2) we find that

N_{p} = \begin{cases}2 & r = 0 or r ≥ 5 \\ 1 & r = 4 \\ 2 & 1 ≤ r ≤ 3 and y is even \\ 8 & r = 2, 3 and y is odd \\ 32 & r = 1 and y is odd. \end{cases}

Before applying Ribet’s Theorem (Theorem 3) we must ensure that E does not have p-isogenies. Now we know that E(ℚ)[2] = 4. Thus by Mazur’s Theorem (Theorem 4), if the conductor N is squarefree then E does not have p-isogenies (recall that our p ≥ 5 and prime). Examining the formulae for N in (5) we see that if N is not squarefree then ord_{2}(N) = 3 or 5. In this case it follows from Theorem 5 that E has not p-isogenies.

Now Ribet’s Theorem tells us that there is newform f of level N_{p} such that E ∼_{p} f. Theorem 1 tells us that there are no newforms of levels 1, 2, 8. Thus we deduce that r = 1 and y is odd. We cannot yet eliminate this last possibility since there are newforms at level 32. In fact there is precisely one newform at level 32 corresponding to the elliptic curve

(6)  F : Y^{2} = X(X + 1)(X + 2)

with Cremona reference 32A2. Notice that we can get the elliptic curve F by letting x = −1, y = 1, r = 1, in the model for E given in (4). That is, we are substituting a solution to the equation (3) that satisfies the additional constraints placed above. At this point all that we can conclude is that E arises modulo p from F. The curve F is unusual in that it has complex multiplication. This fact enabled Darmon and Merel to solve the equation (3). The proof will be completed later after we take a closer look at the consequence for an elliptic curve to arise modulo p from another curve with complex multiplication. □

4.1. E arises from a curve having complex multiplication. To solve the equation x^{p} + 2y^{p} + z^{p} we shall use the following Theorem. For other Diophantine applications of this Theorem see [13], [22], [18].

Theorem 7. Suppose that E and F are elliptic curves over ℚ, and F has complex multiplication by an order in a number field L. Suppose that E ∼_{p} F for some prime p.

(i) (Halberstadt and Kraus [17]) If p = 11 or p ≥ 17, and p splits in L then the conductors of E and F are equal.

(ii) (Darmon and Merel [13]) If p ≥ 5 and p is inert in L, and E has a ℚrational subgroup of order 2 or 3, then j(E) ∈ ℤ[\frac{1}{p}].

Remark. In part (ii) of the Theorem, if we assume that p^{2} ∤ N where N is the conductor of E then we can deduce that j(E) ∈ ℤ. To see this suppose that j(E) ∈ ℤ[\frac{1}{p}]\ℤ. Then p || N, and so by Proposition 3.2

p + 1 ≡ ±a_{p}(F) (mod p).

But since p is inert in L, and F has complex multiplication by an order in L we see that a_{p}(F) = 0. This immediately gives a contradiction.

Completion of the Proof of Theorem 6. We now return to complete the proof of Theorem 6. We have shown that r = 1, that y is odd and E ∼_{p} F where E and F are as in (4) and (6). To simplify we assume that p ≠ 5, 13. However we note that Dénes [14] has solve the equation x^{p} + 2y^{p} + z^{p} = 0 for p ≤ 31 by classical means.

Now F has complex multiplication by ℤ[i]. By Theorem 7 if p ≡ 1 (mod 4) (p splits in ℚ(i)) we have that the conductor of E is 32. From the formula for the conductor in (5) we know that x, y and z are not divisible by any odd primes. But x, y, z are odd, and it follows (x, y, z) = ±(−1, 1, −1).

Suppose now that p ≡ 3 (mod 4) (p is inert in ℚ(i)). Note also that E has a point of order 2. Then j(E) ∈ ℤ. From the formula for the j-invariant above and the pairwise coprimality of x, y, z above see again that x, y, z is not divisible by odd primes and so (x, y, z) = ±(−1, 1, −1). □

Exercise 1. Consider the Diophantine equation

(7)  x^{2} = y^{p} + 2^{m}z^{p},  x, y, z are non-zero, odd and coprime,

where p ≥ 7 is prime, and m ≥ 2. Without loss of generality, assume that x ≡ 3 (mod 4) and associate a solution of this equation to the Frey curve

E : Y^{2} = X(X^{2} + 2xX + y^{p}).

Mimic the proof of Theorem 6 to show that the only solutions to (7) are m = 3, x = ±3, y = z = 1 and p arbitrary. Where does the proof breakdown for m = 1? [If you get stuck, then this equation is solved in [18] and [30]. You might find the paper of Papadopoulos [25] useful for calculating the conductor.]

5. An Occasional Bound for the Exponent

The proof of Theorem 6 is somewhat miraculous. In general, we cannot expect not to find newforms at the level predicted by Ribet’s Theorem. Moreover, complex multiplication is rather rare. In general what will probably find is a collection of newforms, some rational, and some irrational. It is however often possible to obtain a bound for the exponent p via the following Proposition.

Proposition 5.1. Let E/ℚ be an elliptic curve of conductor N, and suppose that t | ♯E(ℚ)_{tors}. Suppose that f is a newform of level N′. Let l be a prime such that l ∤ N′ and l^{2} ∤ N. Let

S_{l} = \{a ∈ ℤ : −2\sqrt{l} ≤ a ≤ 2\sqrt{l}, a ≡ l + 1 (mod t)\} .

Let c_{l} be the l-th coefficient of f and define

B′_{l}(f) = Norm_{K/ℚ}((l + 1)^{2} − c_{l}^{2}) ∏_{a∈S_{l}} Norm_{K/ℚ}(a − c_{l})

and

B_{l}(f) = \begin{cases}l · B′_{l}(f) & if f is not rational, \\ B′_{l}(f) & if f is rational. \end{cases}

If E ∼_{p} f then p | B_{l}(f).

Proof. This follows easily from Propositions 3.1, 3.2 and the fact the if l is a prime of good reduction for E then l + 1 − a_{l}(E) = ♯E(𝔽_{l}) ≡ 0 (mod t). □

Notice that this Proposition allows us to bound p provided we can find l such that B_{l}(f) ≠ 0. We are guaranteed to succeed in two cases:

(a) Suppose that f is irrational. Then for infinitely many primes l we have B_{l}(f) ≠ 0. This is true since c_{l} ∉ ℚ for infinitely many of the coefficients c_{l}.

(b) Suppose that f is rational and that t is prime or t = 4. Suppose that for every elliptic curve F in the isogeny class corresponding to f we have t ∤ ♯F (ℚ)_{tors}. Then there are infinitely many primes l such that B_{l}(f) ≠ 0.

Exercise 2. Let L = 2^{m} − 1 be a Mersenne prime with m ≥ 5. Show that there is some newform f having level 2L such that B_{l}(f) = 0 for all primes l ≠ 2, L. [Hint: Show that the elliptic curve

F : Y^{2} = X(X + 1)(X + 2^{m})

has conductor 2L. Now let f be the newform corresponding to F.]

6. An Example of Serre-Mazur-Kraus

Let L be an odd prime number. In this section we take a close look at the equation

(8)  x^{p} + L^{r}y^{p} + z^{p},  xyz ≠ 0, p ≥ 5 is prime,

studied by Serre in [29] and Kraus in [20] – the connection of this equation with Mazur will become apparent. We assume that

(9)  x, y, z are pairwise coprime,  0 < r < p.

Let A, B, C be some permutation of x^{p}, L^{r}y^{p} and z^{p} such that A ≡ −1 (mod 4) and 2 | B, and let E be the elliptic curve

(10)  E : Y^{2} = X(X − A)(X + B).

The minimal discriminant and conductor of E are

Δ_{min} = 2^{−8}L^{2r}(xyz)^{2p}, N = Rad(Lxyz).

The recipe for N_{p} in (2) shows that

N_{p} = 2L.

Notice that the elliptic curve E is semi-stable (squarefree conductor N) and that ♯E(ℚ)[2] = 4. By Theorem 4, E does not have p-isogenies. Applying Ribet’s Theorem we see that E arises modulo p from some newform f at level N_{p} = 2L.

The following result appears in Serre’s paper [29].

Theorem 8. (Mazur) Suppose that L is an odd prime that is neither a Mersenne prime nor a Fermat prime (hence L cannot be written in the form 2^{m} ± 1). Then there is a constant C_{L} such that if (x, y, p) is a solution to equation (8) satisfying condition (9) then p ≤ C_{l}.

Proof. The point of the proof is that for primes L that are neither Mersenne nor Fermat primes, there are no elliptic curves having full 2-torsion and conductor 2L. The Theorem then follows from the remarks made after Proposition 5.1.

We briefly sketch why there are no elliptic curves having full 2-torsion and conductor 2L unless L is a Mersenne or Fermat prime ^{3}. Suppose E is a curve with conductor 2L and full 2-torsion. It is possible to construct a model for E of the form

3There is no proof given of this in [29]. Here we follow [20, Lemme 7].

Y^{2} = X(X − a)(X + b)

which is minimal away from 2, where a, b are integers. Now the discriminant of this model must be of the form 2^{u}L^{v}. However the discriminant of this model is

16a^{2}b^{2}(a + b)^{2}.

As the model is minimal but has bad reduction at L, we find that precisely one of a, b, a + b is divisible by L. We quickly obtain a relation of the form

±2^{α} ± 2^{β} ± 2^{γ}L^{δ} = 0

where δ ≥ 1. From this it is easy to deduce that L is a Fermat or Mersenne prime. □

In fact Kraus [20] went even further proving the following:

Theorem 9. (Kraus) Suppose that L is an odd prime number that is neither a Mersenne prime nor a Fermat prime. Suppose that (x, y, p) is a solution to equation (8) satisfying the condition (9) where p is a prime. Then

p ≤ (\sqrt{\frac{L + 1}{2}} + 1)^{\frac{L+11}{6}}

The bound is rather large. However, in practice obtain a very tiny bound since we can, for any given newform f, compute B_{l}(f) for many primes l and take the greatest common divisor of them.

Theorem 10. Suppose 3 ≤ L < 100 is prime. Then the equation (8) has no solutions satisfying conditions (9) unless L = 31, in which case E ∼_{p} F where F is the curve 62A1.

Proof. The proof of this result depends on Proposition 5.1, the method of Kraus (Proposition 7.2 below), and the method of predicting exponents (Section 8). See Exercise 4 and Exercise 5.

For illustration we treat the case L = 19. From the above we know that E ∼_{p} f for some newform at level N_{p} = 38. There are two newforms at level 38:

f_{1} = q − q^{2} + q^{3} + q^{4} − q^{6} − q^{7} + ⋯
f_{2} = q + q^{2} − q^{3} + q^{4} − 4q^{5} − q^{6} + 3q^{7} + ⋯

We apply Proposition 5.1 with t = 4 and compute B_{3}(f_{1}) = −15 and B_{5}(f_{1}) = −144. By Proposition (5.1), if E ∼_{p} f_{1} then p must divide both. But this is impossible as we are assuming that p ≥ 5.

Also

B_{3}(f_{2}) = 15, B_{5}(f_{2}) = 240, B_{7}(f_{2}) = 1155, B_{11}(f_{2}) = 3360.

Thus if E ∼_{p} f_{2} then p = 5. It turns out that all of the B_{l}(f_{2}) are divisible by 5. To see why let F be the elliptic curve 38B1; this is the elliptic curve that corresponds to f_{2}. Now looking at Cremona’s tables [12] we see that this curve has a point of order 5. Hence 5 | ♯F (𝔽_{l}) for all primes l ∤ 38. In other words 5 | (l + 1 − a_{l}(F)) for all primes l ∤ 38. Now we see from the definition of B_{l}(f_{2}) that 5 | B_{l}(f_{2}) for all primes l ∤ 38. Thus we are unable to eliminate the possibility that p = 5 using Proposition 5.1. However we can turn the situation to our advantage as follows: suppose that E ∼_{5} f_{2} or equivalently E ∼_{5} F. Then a_{l}(E) ≡ a_{l}(F) (mod 5) for all but finitely many primes l. Hence 5 | (l +1−a_{l}(E)) for all but finitely many primes l. It follows from the Chebotarev Density Theorem that E has a 5-isogeny (do this as an exercise, or see [28, IV-6]). But E is semi-stable and has full 2-torsion; by Mazur’s Theorem (Theorem 4) we have reached a contradiction. Thus we know that equation (8) does not have any solutions with L = 19 and p ≥ 5 satisfying conditions (9). For the analogue of this trick when the newform is irrational see [20, pages 1155–1156].

□

Exercise 3. Let A, B, C be non-zero integers such that A + B + C = 0. Let E be the elliptic curve

E : Y^{2} = X(X − A)(X + B).

Show that any permutation of A, B, C will give a curve that is isomorphic to E or to its quadratic twist by −1.

7. The Method of Kraus

Proposition 5.1 is often capable of bounding p when our (hypothetical Frey) elliptic curve arises modulo p from a newform f. There is another rather interesting method, due to Kraus [21], that can often be used to derive a contradiction for a fixed value of p. Kraus used this method to prove that the equation

a^{3} + b^{3} = c^{p}, a, b, c non-zero and coprime,

has no solutions for 11 ≤ p ≤ 10000. A combination of Proposition 5.1, the method of Kraus and classical techniques for Diophantine equations recently lead to the complete solutions of equations x^{2} + D = y^{n} for n ≥ 3 and 1 ≤ D ≤ 100 (see [9], [31]). In this section we adapt the method of Kraus for equation (8).

We continue with the notation of the previous section. Recall that E is the curve (10) where A, B, C is some permutation of x^{p}, L^{r}y^{p}, z^{p} such that A ≡ −1 (mod 4) and 2 | B. It is somewhat awkward to work with the curve E since there are six possibilities for the triple A, B, C. However, letting

E′ : Y^{2} = X(X − x^{p})(X + z^{p}),

we see (from Exercise 3) that E and E′ are either isomorphic, or quadratic twists of each other by −1. Now E′ depends on two variables x, z. However if we write δ = (z/x)^{p} then we see that E′ is the quadratic twist of

E_{δ} : Y^{2} = X(X − 1)(X + δ),

by x^{p}. For prime l ∤ x it follows that a_{l}(E) = ±a_{l}(E_{δ}). From this and Proposition 3.2 we deduce the following.

Lemma 7.1. With notation as above, suppose that E ∼_{p} f for some newform f with level 2L. Suppose that l is a prime distinct from 2, L, p. Write c_{l} for the l-th coefficient of f as in (1).

• If l | xyz then p | Norm((l + 1)^{2} − c_{l}^{2}).

• If l ∤ xyz then p | Norm(a_{l}(E_{δ})^{2} − c_{l}^{2}).

Suppose l = np + 1 is prime. Let

(11)  μ_{n}(𝔽_{l}) = \{ζ ∈ 𝔽_{l} : ζ^{n} = \overline{1}\} .

Note that if l ∤ xyz then the reduction of δ = (x/z)^{p} modulo l belongs to μ_{n}(𝔽_{l}). The following proposition is now obvious.

Proposition 7.2. Suppose that p ≥ 5 be a fixed prime and E is as above. Suppose f is a newform at level 2L and c_{l} is its l-th coefficient. Suppose that there exists a positive integer n satisfying the following:

• l = np + 1 is prime.

• l ≠ L.
• p ∤ Norm((l + 1)^{2} − c_{l}^{2}).
• For all δ ∈ μ_{n}(𝔽_{l}) we have
p ∤ Norm(a_{l}(E_{δ})^{2} − c_{l}^{2}).

Then the equation (8) does not have any solutions satisfying conditions (9).

Theorem 11. Suppose L = 31. Then equation (8) does not have any solutions satisfying condition (9) for 11 ≤ p ≤ 10^{6}.

Proof. Suppose L = 31. By Theorem 10 we know that E ∼_{p} F where F is the elliptic curve 62A1 with equation

y^{2} + xy + y = x^{3} − x^{2} − x + 1.

We wrote a very short pari/gp [1] script which for a given prime p search for a prime l satisfying the conditions (i), (ii), (iii) of Proposition 7.2. This took about 18 minutes for p in the above range. Our program failed to find a suitable value of n ≤ 1000 for p = 5 and p = 7. □

Exercise 4. Using a combination of Proposition 5.1 and Proposition 7.2 to show that the equation (8) has no solutions for L = 23.

8. ‘Predicting Exponents of Constants’

The title is in quotes because it is rather vague. For various Diophantine equations the modular approach is very effective at predicting exponents of terms with constant base. This method is central to the recent determination of all perfect powers in the Fibonacci and Lucas sequences [8]. We would like to illustrate this method by studying the Diophantine equation

(12)  x^{2} − 2 = y^{p},  p ≥ 5 prime.

The exponent that we would like to predict will become known shortly. As a motivation for studying this equation let us note that the more general equation

x^{2} − 2^{m} = y^{p}

has been solved for all m ≥ 2 (if you did not do Exercise 1 then see [18], [30]). Besides, equation (12) is now considered to be one of the most difficult exponential Diophantine equations. This section presents a partial attempt at solving this equation by Bugeaud, Mignotte and myself.

Equation (12) is a special case of the more general equation Ax^{n} + By^{n} = Cz^{2} and so applying the recipes in Section 9 we may associate any solution (x, y) of (12) to the Frey curve

E : Y^{2} = X^{3} + 2xX^{2} + 2X.

We find that

Δ_{min} = 2^{8}y^{p},  N = 2^{7} Rad(y),  N_{p} = 128.

From Theorem 5 we know that the curve E does not have p-isogenies. We deduce that E arises from a newform of level 128. There are four newforms at level 128—all rational—corresponding to the four elliptic curves

F_{1} = 128A1, F_{2} = 128B1, F_{3} = 128C1, F_{4} = 128D1.

Hence E ∼_{p} F_{i} for some i. Notice that the equation (12) has the solutions (x, y) = (±1, −1) for all exponents p. Hence any attempt to prove that p is bounded by some result similar to Proposition 5.1 will fail. So will mimicking Kraus’s method. However we can still use the modular approach to derive non-trivial information about (12).

The classical line of attack for an equation such as (12) is to factorize the lefthand side and deduce that

(13)  x + \sqrt{2} = (1 + \sqrt{2})^{r}(U + V \sqrt{2})^{p}

for some U, V ∈ ℤ and

(14)  \frac{−(p − 1)}{2} < r ≤ \frac{p − 1}{2}.

We deduce that

(15)  \frac{1}{2\sqrt{2}} ((1 + \sqrt{2})^{r}(U + V \sqrt{2})^{p} − (1 − \sqrt{2})^{r}(U − V \sqrt{2})^{p}) = 1.

Notice that the polynomial on the left-hand side is homogeneous of degree p in U, V with coefficients in ℤ. Thus to solve equation (12) for any particular exponent p we are forced to solve p Thue equations (15), one for each value of r in the range (14). As p gets larger, the coefficients of these equations become very unpleasant, making it difficult to solve them. However, we believe, based on a short search, that the only solutions are x = ±1, y = −1. Thus from (13) we suspect the the only values of r that should correspond to solutions are r = ±1. We prove this using the modular approach together with a result proved by classical means.

Proposition 8.1. With notation as above, r = ±1.

Proof. Fix F to be one of the four elliptic curves F_{1}, …, F_{4} above, and suppose that E ∼_{p} F. Now let l be a prime satisfying the following conditions:

(a) l = np + 1 for some integer n,

(b) (\frac{2}{l}) = 1,

(c) l + 1 ≢ ±a_{l}(F) (mod p),

Let θ ∈ 𝔽_{l} be some fixed choice of \sqrt{2} modulo l. We impose on l yet one more conditions, namely:

(d) (1 + θ)^{n} ≢ 1 (mod l). We note that if l | y then l will be a prime of multiplicative reduction for E and so condition (a) contradicts Proposition 3.2. We deduce that l ∤ y. Hence \overline{y}^{p} ∈ μ_{n}(𝔽_{l}) where μ_{n}(𝔽_{l}) is given in (11). Let

𝔛′_{l} = \{δ ∈ 𝔽_{l} : δ^{2} − \overline{2} ∈ μ_{n}(𝔽_{l})\} .

We see that \overline{x} ∈ 𝔛′_{l}. Notice that 𝔛′_{l} that has cardinality at most 2n. We would like to refine 𝔛′_{l} to obtain better information on the value of x modulo l. For δ ∈ 𝔛′_{l}, let E_{δ} be the elliptic curve over 𝔽_{l}

E_{δ} : Y^{2} = X^{3} + 2δX^{2} + 2X.

We let

𝔛_{l} = {δ ∈ 𝔛′_{l} : a_{l}(E_{δ}) ≡ a_{l}(F) (mod p)} .

It is clear from Proposition 3.2 that \overline{x} ∈ 𝔛_{l} where 𝔛_{l} is now a set that we hope is much smaller than 𝔛′_{l}. Now we want to obtain information about r from knowing that \overline{x} ∈ 𝔛_{l}. It follows from (13) that, for some δ ∈ 𝔛_{l},

(16)  δ + θ ≡ (1 + θ)^{r}(U + V θ)^{p} (mod l);

We note that U + V θ ≢ 0 (mod l) since U^{2} − 2V^{2} = ±y and we know that l ∤ y. To get information about r we need to use the discrete logarithm modulo l. Fix once and for all some primitive root g of 𝔽_{l}. The discrete logarithm with respect to g is the isomorphism 𝔽_{l}^{∗} → ℤ/(l − 1) given by g^{k} ↦ k (mod (l − 1)). Write L for the composite of the discrete logarithm with the natural projection ℤ/(l − 1) → ℤ/p. We apply L to both sides of (16) and deduce that,

L(δ + θ) ≡ rL(1 + θ) (mod p).

It follows from (d) that L(1 + θ) ≢ 0 (mod l). Hence

r (mod p) ∈ R_{l}(F) := \{\frac{L(δ + θ)}{L(1 + θ)} : δ ∈ 𝔛_{l}\} .

Now we would like to show that r = ±1. Since r lies in the interval (14), it is enough to show that r ≡ ±1 (mod p). Thus we look for primes l_{1}, …, l_{k} satisfying conditions (a)–(d) so that

∩_{j=1}^{k}R_{l_{j}}(F) ⊆ {±1 (mod p)} .

If we can do this for each of the F = F_{1}, …, F_{4} we will have proved that r = ±1. We wrote a short pari/gp script to carry out the above proof for all 5 ≤ p < 10^{6}. The proof for this range took about 3 hours.

Going up to p < 10^{6} is indeed overkill, since a careful application of linear forms in logarithms [23] to this problem shows that p < 8200 if y ≠ −1. Thus we know for any p that is not in our range (and indeed for p > 8200) that y = −1 and we easily see that r = ±1 in all cases. □

It is possible to improve the estimate p < 8200 mentioned in the proof above now that we know that r = ±1 and using another interesting piece of information given below.

Lemma 8.2. Suppose y ≠ −1. Then y ≥ (\sqrt{p} − 1)^{2}.

Proof. It is clear that if y ≠ −1 then y > 1. Clearly y is odd. Hence there is some odd prime l | y. By Proposition 3.2 we see that

l + 1 ≡ ±a_{l}(F) (mod p),

where F is one of F_{1}, …, F_{4}. However |a_{l}(F)| < 2\sqrt{l}. Thus we see that

p ≤ l + 1 + 2\sqrt{l} ≤ y + 1 + 2\sqrt{y} = (1 + \sqrt{y})^{2}.

The Lemma follows. □

Using this information, another careful application of linear forms in logarithms [23] shows that p < 2810.

We can also try to solve the Thue equations with r = ±1. In fact if we let F_{r}(U, V) be the polynomial on the left-hand side of equation (15), we see that F_{−1}(U, V) = F_{1}(U, −V). Hence it is sufficient to solve the Thue equation F_{1}(U, V) = 1. Solving this with pari/gp for 5 ≤ p ≤ 37 we get that (U, V) = (1, 0) is the only solution. Thus we have proved the following modest result.

Lemma 8.3. If 5 ≤ p ≤ 37 then (x, y) = (±1, −1) are the only solutions to (12).

Exercise 5. Adapt the method of predicting exponents to equation (8). Use this to prove that equation (8) has no solutions with p = 7 and L = 31.

9. Recipes for Ternary Diophantine Equations

By ternary Diophantine equations we mean equations of the form Ax^{l} + By^{m} = Cz^{n}; the triple of exponents (l, m, n) is called the signature of the equation. How to associate such an equation to a Frey curve is detailed for three important signatures (p, p, p), (p, p, 2) and (p, p, 3) respectively by Kraus [20], by Bennett and Skinner [2], and by Bennett, Vatsal and Yazdani [3]. For convenience of the reader we reproduce the recipes appearing in these papers for the Frey curves and levels. We must however point out that there is much more in these papers than just the recipes and the reader is particularly urged to pursue them. This section is influenced heavily by Bennett’s paper [4].

9.1. Recipes for signature (p, p, p). Suppose that A, B, C are non-zero pairwise coprime integers, and p ≥ 5 is prime. Let

R = ABC,

and suppose that

ord_{q}(R) < p

for every prime number q. Consider the equation

(17)  Ax^{p} + By^{p} + Cz^{2} = 0,

where we assume that

Ax, By, Cz are non-zero and pairwise coprime.

Without loss of generality we also suppose that

Ax^{p} ≡ −1 (mod 4), By^{p} ≡ 0 (mod 2).

The Frey curve is

E : Y^{2} = X(X − Ax^{p})(X + By^{p}).

The minimal discriminant is

Δ_{min} = \begin{cases}2^{4}R^{2}(xyz)^{2p} & if 16 ∤ By^{p}, \\ 2^{−8}R^{2}(xyz)^{2p} & if 16 | By^{p}, \end{cases}

and the conductor N is given by

N = \begin{cases}2 Rad_{2}(Rxyz) & if ord_{2}(R) = 0 or ord_{2}(R) ≥ 5, \\ 2 Rad_{2}(Rxyz) & if 1 ≤ ord_{2}(R) ≤ 4 and y is even, \\ Rad_{2}(Rxyz) & if ord_{2}(R) = 4 and y is odd, \\ 2^{3} Rad_{2}(Rxyz) & if ord_{2}(R) = 2 or 3 and y is odd, \\ 2^{5} Rad_{2}(Rxyz) & if ord_{2}(R) = 1 and y is even. \end{cases}

Theorem 12. (Kraus [20]) Under the above assumptions, E ∼_{p} f for some newform f of level N_{p} where

N_{p} = \begin{cases}2 Rad_{2}(R) & if ord_{2}(R) = 0 or ord_{2}(R) ≥ 5, \\ Rad_{2}(R) & if ord_{2}(R) = 4, \\ 2 Rad_{2}(R) & if 1 ≤ ord_{2}(R) ≤ 3 and y is even, \\ 2^{3} Rad_{2}(R) & if ord_{2}(R) = 2 or 3 and y is odd, \\ 2^{5} Rad_{2}(R) & if ord_{2}(R) = 1 and y is odd. \end{cases}

The proof is left as an exercise to the reader. Note that you must show that E does not have any p-isogenies.

Example 5. If E has no p-isogenies then we know from Ribet’s Theorem that E ∼_{p} f for some newform f at level N_{p}. At level N_{p} there may be rational and nonrational newforms, and some of the non-rational ones can be defined over number fields of rather large degree. There is no reason to suppose that f is rational, or even that the degree of number field over which f is defined is small as we will show in this example.

Let p be a prime and let L = 2^{p+4} + 1; here we do not assume that L is prime. Write

E : Y^{2} = X(X + 1)(X − 2^{p+4})

The minimal discriminant and conductor are respectively given by

Δ_{min} = 2^{2p}L^{2}, N = 2 Rad(L).

From Mazur’s Theorem E has no p-isogenies, and so applying Ribet’s Theorem we see that E ∼_{p} f for some newform at level N_{p}, defined over some number field K. We cannot calculate N_{p} exactly since we do not know if L has any p-th powers in its prime–power decomposition, but we observe that 2 ∤ N_{p}. Moreover 2 || N. Applying Proposition 3.1 we see that

p | Norm_{K/ℚ}(3 ± c_{2}),

where c_{i} are the coefficients of the q expansion of f. From the facts we have stated about newforms we know that all the conjugates of c_{2} are bounded by 2\sqrt{2} < 3. Hence

p < 6^{[K:ℚ]}

or in other words

[K : ℚ] > \frac{log p}{log 6},

showing indeed that an elliptic curve can arise from a newform whose degree (of field of definition) is arbitrarily large.

9.2. Recipes for signature (p, p, 2). Consider the equation

Ax^{p} + By^{p} = Cz^{2}, p ≥ 7 is prime,

where we assume that

Ax, By, Cz are non-zero and pairwise coprime.

We moreover suppose that

ord_{q}(A) < p,  ord_{q}(B) < p,  for all primes q

and

C is squarefree.

Without loss of generality we may suppose that we are in one of the following situations:

(i) ABCxy ≡ 1 (mod 2) and y ≡ −BC (mod 4).

(ii) xy ≡ 1 (mod 2) and either ord_{2}(B) = 1 or ord_{2}(C) = 1.

(iii) xy ≡ 1 (mod 2), ord_{2}(B) = 2 and z ≡ −By/4 (mod 4).

(iv) xy ≡ 1 (mod 2), ord_{2}(B) ∈ {3, 4, 5} and z ≡ C (mod 4).

(v) ord_{2}(By^{p}) ≥ 6 and z ≡ C (mod 4).

In cases (i) and (ii) we consider the curve

E_{1} : Y^{2} = X^{3} + 2CzX^{2} + BCy^{p}X.

In cases (iii) and (iv) we consider

E_{2} : Y^{2} = X^{3} + CzX^{2} + \frac{BCy^{p}}{4} X,

and in case (v) we consider

E_{3} : Y^{2} + XY = X^{3} + \frac{Cz − 1}{4}X^{2} + \frac{BCy^{p}}{64} X.

Theorem 13. (Bennett and Skinner [2]) With assumptions and notation as above, we have:

(a) The minimal discriminant of E_{i} is given by

Δ_{i} = 2^{δ_{i}}C^{3}B^{2}A(xy^{2})^{p},

where

δ_{1} = 6, δ_{2} = 0, δ_{3} = −12.

b The conductor of the curve E_{i} is given by

N = 2^{α}C^{2} Rad(ABxy),

where

α = \begin{cases}5 & if i = 1, case (i) \\ 6 & if i = 1, case (ii) \\ 1 & if i = 2, case (iii), ord_{2}(B) = 2 and y ≡ −BC/4 (mod 4) \\ 2 & if i = 2, case (iii), ord_{2}(B) = 2 and y ≡ BC/4 (mod 4) \\ 4 & if i = 2, case (iv) and ord_{2}(B) = 3 \\ 2 & if i = 2, case (iv) and ord_{2}(B) = 4 or 5 \\ −1 & if i = 3, case (v) and ord_{2}(By^{7}) = 6 \\ 0 & if i = 3, case (v) and ord_{2}(By^{7}) ≥ 7. \end{cases}

(c) suppose that E_{i} does not have complex multiplication (This would follow if we assume that xy ≠ ±1). Then E_{i} ∼_{p} f for some newform f of level

N_{p} = 2^{β}C^{2} Rad(AB)

where

β = \begin{cases}α & cases (i)–(iv), \\ 0 & case (v) and ord_{2}(B) ≠ 0, 6, \\ 1 & case (v) and ord_{2}(B) = 0, \\ −1 & case (v) and ord_{2}(B) = 6. \end{cases}

(d) The curves E_{i} have non-trivial 2-torsion.

(e) Suppose E = E_{i} is a curve associated to some solution (x, y, z) satisfying the above conditions. Suppose that F is another curve defined over ℚ such that E ∼_{p} F. Then the denominator of the j-invariant j(F) is not divisible by any odd prime q ≠ p dividing C.

Part (d) is included to help with the application of Proposition 5.1. Part (e) is often very useful in eliminating rational newforms (which correspond to elliptic curves). See for example Exercise 7.

Exercise 6. Determine all the solutions of the equation

x^{p} + 2^{α}y^{p} = 3z^{2}, α ≥ 2, p ≥ 7 prime

in coprime integers x, y, z.

Exercise 7. The Fibonacci and Lucas sequences F_{n}, L_{n} are defined by

F_{0} = 0, F_{1} = 1, F_{n+2} = F_{n} + F_{n+1} for all n ≥ 0,
L_{0} = 2, L_{1} = 1, L_{n+2} = L_{n} + L_{n+1} for all n ≥ 0.

(a) Show that 5F_{n}^{2} + 4(−1)^{n} = L_{n}^{2}.

(b) Prove that the equation L_{n} = y^{p} has no solution with n even. [ Hint for (b): You should follow the recipes above and use part (e) of Theorem 13 to deduce a contradiction.]

9.3. Recipes for signature (p, p, 3). Consider the equation

Ax^{p} + By^{p} = Cz^{3}, p ≥ 5 is prime,

where we suppose that

Ax, By, Cz are non-zero and pairwise coprime.

We suppose without loss of generality that

ord_{q}(A) < p, ord_{q}(B) < p, ord_{q}(C) < 3,

for all primes q, and that

Ax ≢ 0 (mod 3), By^{p} ≢ 2 (mod 3).

Let

E : Y^{2} + 3CzXY + C^{2}By^{p}Y = X^{3}.

Theorem 14. (Bennett, Vatsal and Yazdani [3]) With notation and assumptions as above:

(a) The conductor N of the curve E is given by

N = Rad_{3}(ABxy) Rad_{3}(C)^{2}ϵ_{3}

where

ϵ_{3} = \begin{cases}3^{2} & if 9 | (2 + C^{2}By^{p} − 3Cz), \\ 3^{3} & if 3 || (2 + C^{2}By^{p} − 3Cz), \\ 3^{4} & if ord_{3}(By^{p}) = 1, \\ 3^{3} & if ord_{3}(By^{p}) = 2, \\ 1 & if ord_{3}(By^{p}) = 3, \\ 3 & if ord_{3}(By^{p}) > 3, \\ 3^{5} & if 3 | C. \end{cases}

(b) Suppose that xy ≠ 1 and the curve E does not correspond to one of the equations

1 · 2^{5} + 27 · (−1)^{5} = 5 · 1^{3}, 1 · 2^{7} + 3 · (−1)^{7} = 1 · 5^{3}.

Then E ∼_{p} f for some newform f of level

N_{p} = Rad_{3}(AB) Rad_{3}(C)^{2}ϵ′_{3},

where

ϵ′_{3} = \begin{cases}3^{2} & if 9 | (2 + C^{2}By^{p} − 3Cz), \\ 3^{3} & if 3 || (2 + C^{2}By^{p} − 3Cz), \\ 3^{4} & if ord_{3}(By^{p}) = 1, \\ 3^{3} & if ord_{3}(By^{p}) = 2, \\ 1 & if ord_{3}(B) = 3, \\ 3 & if ord_{3}(By^{p}) > 3 and ord_{3}(B) ≠ 3, \\ 3^{5} & if 3 | C. \end{cases}

(c) The curve E has a point of order 3, namely the point (0, 0).

(d) Suppose F is an elliptic curve defined over ℚ such that E ∼_{p} F. Then the denominator of the j-invariant j(F) is not divisible by any odd prime q ≠ p dividing C.

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Samir Siksek, Department of Mathematics and Statistics, College of Science, Sultan Qaboos University, P.O. Box 36, Al-Khod 123, Oman

   E-mail address: siksek@squ.edu.om