On ternary equations of Fermat type and relations with elliptic

curves

Gerhard Frey Institute for Experimental Mathematics University of Essen Ellernstraße 29 D-45326 Essen, Germany e-mail: frey@exp-math.uni-essen.de

1 Conjectures

The main purpose of the following paper is to show how arithmetical properties of elliptic curves E defined over global fields K and corresponding Galois representations are often related to interesting diophantine questions amongst the most prominent is without doubt Fermat’s Last Theorem which has become Wiles’ theorem now.

Of course, the most interesting case for us is that K is a number field or even equal to ℚ but many of the conjectured (or proved) assertions make sense for fields of finite type and become more convincing in the case that K is a function field or, in other words, for families of elliptic curves. The reason for this is that the conjectures predict the behaviour of “generic” elliptic curves and it is expected that each “special” property gives rise to interesting diophantine properties of K.

So assume from now on that K is a global field, i.e. that K is either a finite number field or a function field of one variable over a perfect field K_{0}. For simplicity we always assume that char(K_{0}) ≠ 2, 3. By Σ_{K} we denote the set of non archimedean places of K. For divisors D ∈ ℤ[Σ_{K}] we define deg(D) as the divisor degree in the usual sense if K is a function field, and as the logarithm of its norm in the number field case. For a subset S_{0} ⊂ Σ_{K} let O_{S_{0}} be the ring of S_{0}-integers, i.e. O_{S_{0}} = {x ∈ K; v_{𝔭}(x) ≥ 0 for 𝔭 ∉ S_{0}}, its units are called S_{0}-units.

K has two important numerical invariants: By g(K) we denote the genus of K which is the usual genus in the function field case, and equal to \frac{1}{2} log |Δ_{K/ℚ}| in the number field case (where Δ_{K/ℚ} is the discriminant of K/ℚ). By d(K) we denote the irrationality degree of K which is equal to [K : ℚ] if K is a number field. If K is a function field d(K) is equal to min_{f∈K\K_{0}}(deg(f)_{∞}) = min_{f∈K\K_{0}}([K :

K_{0}(f)] where (f)_{∞} is is the pole divisor of f; so d(K) is the minimal degree of a covering map from the curve corresponding to K to ℙ_{/K_{0}}^{1}.

a) Conjectures about diophantine equations

One of the most basic conjectures about solutions of ternary diophantine equations comes a little bit disguised: For x ∈ K^{∗} let h_{K}(x) denote its height, and supp(x) = ∏_{{v𝔭(x)≠0 \atop 𝔭∈Σ_{K}}} 𝔭 its support divisor.

Conjecture 1: There are numbers c and d such that the following holds: Assume that x ≠ 0, 1 and that K/K_{0}(x) is separable if K is a function field. Then h_{K}(x) ≤ c · deg supp(x(x − 1)) + d.

Of course here and in all of the conjectures which follow it is of great importance to specify how the constants depend on K. A refinement of Conjecture 1 is

Conjecture 1′: For all ε ∈ ℝ_{>0} c = 1 + ε and d = d(ε, g(K)) can be used in Conjecture 1, the dependence of d on g(K) is linear, and for K a function field ε = 0 is allowed.

In section 2 we shall see that Conjecture 1′ is true for function fields. Now we shall translate Conjecture 1 into the A-B-C-Conjecture due to Masser-Oesterlé for K = ℚ. First assume that K is a function field. We can asume that K has a prime 𝔭_{∞} of degree 1. Let (x)_{∞} be the pole divisor of x. The Riemann-Roch theorem implies that there is an element C ∈ K with (C)_{∞}|𝔭_{∞}^{Grad((x)_{∞})+2g(K)−1} and (x)_{∞}|(C)_{0} where (C)_{0} is the zero divisor of C. Take A = x · C. Then

(A)_{∞} = ((x)(C))_{∞}|(𝔭_{∞})^{Grad((x)_{∞})+2g(K)−1}

and (A)_{0}|(x)_{0} · (C)_{0} · (x)_{0}^{−1} = (x)_{0} · D′ with degD′ ≤ 2g(K) − 1.

Let S_{0}(x) be the set of primes 𝔭 ∈ Σ_{K} dividing 𝔭_{∞} · D′. Then A and C are relatively prime elements in O_{S_{0}(x)} and the same is true for B := A − C, which is equal to (x − 1)C, and deg(gcd(A)_{0}, (B)_{0}) ≤ 2g − 1. If K is a number field of genus g(K) we use the theorem of Minkowski to get a corresponding result: There are elements A, B, C ∈ O_{K} such that x = \frac{A}{C} , x − 1 = \frac{B}{C} with deg(gcd((A), (B)) and deg(gcd(A, C)) bounded linear by g(K). Hence Conjecture 1′ is translated to the

(A-B-C)-Conjecture : If K is a function field fix a place 𝔭_{∞}. Let O_{K} be the ring of integers of K (with respect to 𝔭_{∞} in the function field case). Let A and B be elements in O_{K}, let D(A, B) be their greatest common divisor in O_{K}, and assume that K/K(A/B) is separable. Then

h_{K}(A) ≤ (1 + ε)deg( ∏_{𝔭|AB(A−B)} 𝔭) + d(ε, deg(g(K), D(A, B)))

and the dependence on deg(D(A, B)) and g(K) should be linear.

For K = ℚ this means:

|A| ≤ d̃(ε)( ∏_{p|AB(A−B)} p)^{(1+ε)} whenever gcd(A, B) = 1

with d̃(ε) = e^{d(ε,1)} (conjecture of Masser-Oesterlé).

Remark 1.1 Of course one would be glad to have any bounds c and d at all. But this sharp version has the advantage that an effective version of Falting’s theorem about the finiteness of K-rational points on curves of genus ≥ 2 would follow (cf. [E]).

We continue to fix a prime 𝔭_{∞} of degree 1 if K is a function field : Now we fix a finite set S_{0} of non-archimedean places of K, s_{0} := deg( ∏_{𝔭∈S_{0}} 𝔭).

We assume that a, b, c ∈ K^{∗} with supp(a)supp(b)supp(c)| ∏_{𝔭∈S_{0}} 𝔭^{3}.

Let n be an integer prime to char(K) and define

L_{(a,b,c),n}(K) = { (x, y, z) ∈ K^{3}\{(0, 0, 0)}; ax^{n} − by^{n} = cz^{n} and K/K_{0}(\frac{a}{c}(\frac{x}{z})^{n}) is separable if K is a function field \}

We denote by h_{K}(x, y, z) the projective height in ℙ^{2}(K).

Conjecture 2: (“Asymptotic Fermat Conjecture”) There are numbers N(g(K), s_{0}), such that for all n > N(g(K), s_{0}) and all elements (x, y, z) ∈ L_{(a,b,c),n}(K) we get: xyz = 0 or \{\frac{x}{z}, \frac{y}{z} \} ⊂ O_{S_{0}}^{∗}.

Remark 1.2 If K is a number field or if K_{0} is finite it follows from Conjec-

ture 2 that  ⋃_{{n≥4 \atop supp(a)supp(b)supp(c)| ∏ ; 𝔭∈S_{0} 𝔭^{3}}} L_{(a,b,c),n}(K)

is a finite set, and for n large enough L_{(a,b,c),n}(K) consists of triples with coordinates in {0} ∪ roots of unities.

An easy observation is

Proposition 1.1 Conjecture 1 implies the Asymptotic Fermat conjecture.

Proof We assume that xyz ≠ 0. We take t_{1} = \frac{a}{c}(\frac{x}{z})^{n} and t_{2} = \frac{b}{c}(\frac{y}{z})^{n} and assume that x ≠ 0, 1. So by Conjecture 1 we get:

n( ∑_{{v𝔭( \frac{x}{z})>0 \atop 𝔭∉S_{0}}} v_{𝔭}(\frac{x}{z}) + ∑_{{v𝔭( \frac{y}{z} )>0 \atop 𝔭∉S_{0}}} v_{𝔭}(\frac{y}{z})) ≤ h_{K}(t_{1}) + h_{K}(t_{2})

≤ 2(1 + ε)s_{0}deg supp(xyz) + d(g(K), ε). Hence for n large enough and v_{𝔭}(\frac{x}{z}) > 0 it follows that 𝔭 ∈ S_{0}, and the same is true for \frac{z}{x} and so \frac{x}{z} ∈ O_{S_{0}}^{∗}, and the analogous equations for \frac{b}{c}(\frac{y}{z})^{n} yield: (\frac{y}{z})^{n} ∈ O_{S_{0}}^{∗}.

Of course it is most interesting to find (a, b, c) such that one can determine exactly the elements in L_{(a,b,c),n}(K); in the last section we shall see how Wiles’ theorem about modularity of semi-stable elliptic curves can be used to find such examples if K = ℚ, and especially a = b = c = 1 will lead to Fermat’s Last Theorem.

b) Conjectures about elliptic curves

Now we shall state conjectures about elliptic curves and corresponding Galois representations which are, as we shall see below, related to the Conjectures 1 and 2. Since they seem to be interesting for their own sake we shall begin to describe a more general context.

So let A/K be an abelian variety with conductor N_{A}, Faltings height h_{K}(A) and geometric height h_{geom}(A). For us it is important that for elliptic curves E/K h_{K}(E) is is closely related to deg(Δ_{E}), where Δ_{E} is the minimal discriminant divisor of E, and h_{geom}(E) ∼ h_{K}(j_{E}) with j_{E} the absolute invariant of E (cf. [Si]). It should be noted that, if K is a number field or K_{0} is finite then there are only finitely many abelian varieties over K with fixed dimension and bounded height.

Let n be a natural number and let A_{n} be the kernel of the multiplication by n · id_{A} in A. We shall always assume that n is prime to char(K). The Galois group G_{K} = Aut(K_{sep}|K) acts on A_{n} and so we get a representation

ρ_{A,n} : G_{ℚ} → Gl(A_{n}) ≅ Gl(2 · dimA, ℤ/n)

induced by this action.

In general the image of ρ_{A,n} is ”as big as possible” the restrictions coming from the symplectic structure induced by the Weil pairing and the decomposition of A into simple factors.

Definition 1.1 Let H be a finite subgroup of A defined over K and of order prime char(K). H is exceptional, if

i) there is no subgroup H_{1} with 0 ≠ H_{1} < H such that A is isomorphic to A/H_{1}, and

ii) there is no proper Abelian subvariety B of A containing H.

Notation : An Abelian variety over K has multiplicity 1 if the (absolute) simple factors of A are pairwise non-isogenous.

Question 1: Is there a number N and a number M such that the existence of an exceptional subscheme H of order > N in an Abelian variety of multiplicity 1 implies: h_{geom}(A) < M ?

Of course this question makes only sense if one specifies how N and M should depend on A and K. A possible guess could be that both depend only on the maximum of the dimensions of the simple factors of A and on the genus of K or, even stronger, on the irrationality degree of K. It may be too soon to make more explicit guesses now, and we shall restrict ourselves to an observation depending on deep results of Faltings and (in an effective version) of Masser - Wüstholz (cf.[B]): Let K be a number field. Assume that we fix d ∈ ℕ and an finite set S_{0} ⊂ Σ_{K} and that we look at all Abelian varieties A/K of dimension d whose conductor N_{A} has support inside of S_{0}. Then A lies in a finite set of isomorphy classes A of Abelian varieties, and if A, A′ ∈ A are isogeneous then there is a K-rational isogeny η : A → A′ with deg(η) bounded by a number depending on K and S_{0} only. Therefore it follows immediately that numbers N and M as in the question exist if we restrict ourselves to Abelian varieties with simple factors whose dimension and conductors are bounded, and that these numbers depend on S_{0} and in fact, on g(K).

¿From now on we look at the special case that the factors of A are elliptic curves, and all essential features of the question can be found by looking at elliptic curves E or Abelian varieties of dimension 2 which are isogenous to E_{1} × E_{2} with E_{i}/K elliptic curves. We begin with A = E an elliptic curve defined over K.

Definition 1.2 E/K is admissible if char(K) = 0 or K/K_{0}(j_{E}) is separable.

Let H be an exceptional K-subgroup of E. It follows that H is cyclic and hence our question leads us to

Conjecture 3: There are numbers M(g(K)) and N(g(K)) such that for admissible elliptic curves E/K with separable cyclic K-rational isogeny of degree N > N(g(K)) we have: h_{geom}(E) < M(g(K)).

Remark 1.3 1. If K is a number field and if Conjecture 3 should hold then there exists a number N such that only elliptic curves with complex multiplication defined over K have cyclic K-isogenies of degree ≥ N. It seems to be not unreasonable to believe this statement.

2. In the next section we shall prove that Conjecture 3 is true if K is a function field.

3. If we sharpen the rationality condition to “H has a K-rational generator” then it is a theorem of Merel that in the number field case there is a number N(d(K)) with N ≤ M(d(K)).

4. A (much) weaker version of the stated conjectures is that M depends on N as well as on K, and then the corresponding statements are true over number fields, (cf. [Fr 2]. More precisely one gets:

Proposition 1.2 We assume that K is a number field, let p be a prime. There exists a number l_{p}(d(K)) such that for all n_{p} ≥ l_{p}(d(K)) and all E/K having a cyclic K-isogeny of degree l^{n_{p}} it follows that E has complex multiplication.

Proof: Corollary 2 in [Fr 2] states that for p^{l_{0}} > 120d(K) there are only finitely many j-invariants j_{E} in fields K with [K : ℚ] ≤ d(K) such that a corresponding elliptic curve E has a K-rational cyclic isogeny of degree p^{l_{0}}. Let E_{i} be an elliptic curve over K_{i} corresponding to j_{i}. If E_{i} has no complex multiplication then there is an l_{i} such that E_{i} (and no twist of E_{i}) has a K_{i}- rational cyclic isogeny of degree p^{l_{i}}. Then l_{p}(d(K)) = max{l_{i}; proposed that E_{i} has no complex multiplication } satisfies the assertion of the proposition.

Now we assume that dimA = 2. In this case the existence of exceptional subgroups is closely related to geometric fundamental groups of curves of genus 2 over K. For a discussion of some aspects of this relation we refer to [Fr 3].

In the following we restrict our attention to the case that A is not K-simple, i.e. A is K-isogeneous to E_{1} × E_{2} with E_{i}/K being elliptic curves. Let H < A be exceptional. Inside of A we have two non-isogeneous elliptic curves E1^{′}, E2^{′} intersecting in H′ := E1^{′}∩E2^{′} (which is not exceptional in our sense) and A/H′ ≅ E_{1} × E_{2}.

The case that H (or H′) contains rather large cyclic subgroups has been discussed above.

So we shall assume that H′ = Ei,n′^{′} for some n′ ∈ ℕ and so E_{i} ≅ Ei^{′}.

Let η : A → E_{1} × E_{2} be the quotient map and η̂ its dual map whose kernel Ĥ′ is isomorphic (as G_{K}-module) to H′, and both Ĥ′ and η(H) are exceptional, and since h_{K}(A) = h_{K}(E_{1}×E_{2}) we loose nothing by assuming that A = E_{1}×E_{2} and that H does not contain a proper cyclic subgroup.

It follows from these assumptions that H is isomorphic to p_{i}(H) = E_{i,N} (where p_{i} is the projection to E_{i}) for i = 1, 2 and some N ∈ ℕ. So H is given as graph of an element α ∈ Iso_{G_{K}} (E_{1,N}, E_{2,N}): H = Δ_{α} = {(P, αP); P ∈ E_{1,N}} and the representation ρ_{E_{1,N}} of G_{K} induced by the action on E_{1,N} is equivalent to ρ_{E_{2,N}}. Hence the question about the existence of exceptional subgroups in the special case under consideration is a question about equivalence of representations on torsion elements of elliptic curves raised by Mazur in [Maz.-Isog] for K = ℚ.

We can give an geometric interpretation (cf. [K-S]): Let X(N) be the modular curve parametrizing level-N structures of elliptic curves. Its group of automorphismus is equal to Sl(2, ℤ/N)/ ± id. The triple (E_{1}, E_{2}, α) gives rise to a K-rational point on α_{ε}(Sl(2, ℤ))\X(N) × X(N) =: Z_{N,ε} where α_{ε}(Sl(2, ℤ/N))/ ± id is a diagonal embedding of (Sl(2, ℤ/N))/ ± id) into (Sl(2, ℤ/N))/ ± id)^{2} induced by the conjugation with a matrix g ∈ Gl(2, ℤ/N) with det g = ε ∈ (ℤ/N)^{∗} where ε is determined by e_{2,N}(αP, αQ) = e_{1,N}(P, Q)^{ε} for P, Q ∈ E_{1,N} and e_{i,N} the Weil pairing on points of order N of E_{i}.

It is to hope that geometrical properties of the surface Z_{N,ε} will yield results for the raised question which can now be expressed as follows: Let ZN,ε^{′}(K) ⊂ Z_{N,ε}(K) be the set of K-rational points P corresponding to (E_{1}, E_{2}, α) with E_{i} admissible and E_{1} not isogenous to E_{2}.

Conjecture 4: There are numbers N(g(K)), M(g(K)) such that for all N > N(g(K)) and P ∈ ZN,ε^{′}(K) with corresponding curves E_{1}, E_{2} one has:

max_{i=1,2}(h_{geom}(E_{i})) < M(g(K)).

It is obvious but worth to remark that for K a number field it follows from Conjecture 4 that for N large enough ZN,ε^{′}(K) = ∅. In the conjecture both N and M depend on K. Kani-Schanz proved in [K-S] that for N ≥ 13 the surface Z_{N,ε} is of general type. Thinking at Lang’s conjecture about rational points on such surfaces one is lead to:

Conjecture 4′: There is a N_{0}(= 13?) independent of K such that for N ≥ N_{0} there is a number M(g(K)) such that for all P ∈ ZN,ε^{′}(K) with corresponding curves E_{1}, E_{2} one has:

max_{i=1,2}(h_{geom}(E_{i})) ≤ M(g(k)).

Remark 1.4 Conjectures 4 and 4’ are essentially due to Darmon ([Da]).

As stated already Conjecture 4′ would follow if Lang’s conjecture would be true for Z_{N,ε} and if all curves of genus ≤ 1 on Z_{N,ε} would be birationally equivalent to modular curves X_{0}(n). We shall see in the next section that in the “generic” case (e.g. K a function field over ℚ) it would be enough to have an analogous result for curves of higher genus resp. irrationality degree on Z_{N,ε} to prove Conjecture 4′.

The situation becomes much simpler if we \underline{fix} one of the curve E_{i} which we denote by E_{0} from now on.

Conjecture 5: LetE_{0}/K be admissible. Then there exist numbers N_{0}(g(K), E_{0}), M(g(K), E_{0}) such that for N ≥ N_{0}(g(K), E_{0}) and (E_{0}, E) corresponding to a point P ∈ ZN,ε^{′}(K) we get: h_{K}(E) < M(g(K), E_{0}).

Again Conjecture 5 is true over function fields. A crucial point is the fact that ρ_{E_{1,N}} ≅ ρ_{E_{2,N}} imposes strong arithmetical conditions on E:

Proposition 1.3 Assume that H < E_{1} × E_{2} is exceptional over K of order N^{2} with N ≥ 5. Let N_{E_{1}} be the conductor of E_{1}.

1. If 𝔭 ∈ Σ_{K} does not divide N · N_{E_{1}} then E_{2} is semi-stable at 𝔭.

2. If 𝔭 does not divide N_{E_{1}} then max(−v_{𝔭}(j_{E_{2}}), 0) ≡ 0 mod N.

Proof : If 𝔭 |̸N · N_{E_{1}} then K(E_{1,N}) = K(E_{2,N}) is unramified at 𝔭 and so E_{2} has to be semi-stable at 𝔭 (cf. [Si]). If 𝔭 |̸N_{E_{1}} then E_{1,N} is a finite group scheme at 𝔭, and using the theory of Tate curves (cf. [Si] again) it follows that if v_{𝔭}(j_{E_{2}}) < 0 then v_{𝔭}(j_{E_{2}}) ≡ 0 mod N.

As a consequence we see that prime divisors of N_{E_{2}} which don’t divide N_{E_{1}} give a large contribution to the height of E_{2}, and this should contradict the conjectured relation between the height of elliptic curves and the degree of their conductors which is stated as

Conjecture 6 (Height Conjecture for Elliptic Curves): There are constants c and d(g(K)) such that for all admissible elliptic curves over K we get: h_{K}(E) ≤ c degN_{E} + d(g(K)). Moreover for any ε > 0 the number c = \frac{1}{2} + ε and d = d(g(K), ε) should work, with ε = 0 allowed if K is a function field.

Remark 1.5 If we replace h_{K}(E) by \frac{1}{12}degΔ_{E} (i.e. if we take care only of the contribution of the non-archimedean places to the height of E we get

Szpiro’s Conjecture: There exist constants c′, d′ such that degΔ ≤ c′degN_{E} + d′.

This conjecture would suffice for many applications.

Here are some consequences of Conjecture 6:

Proposition 1.4 Assume that Conjecture 6 is true over K with constants c and d. Let δ_{K} be equal to 1 if K is a function field, and equal to [K : ℚ] + 6 if K is a number field.

i) Let S_{0} be a finite set of places of K, s_{0} = deg ∏_{𝔭∈S_{0}} 𝔭. There is a number

N_{0}(s_{0}, c, d, d(K)) such that all admissible elliptic curves E/K which are semi-stable outside of S_{0} and have a K-rational cyclic isogeny of degree N > N_{0} have potentially good reduction at all 𝔭 ∈ Σ_{K}. ii) Let P be an K-rational point of an admissible E with order N prime to

char(K). Then either N < max(5δ_{K}, 2c + 2d) or E has good reduction at all non archimedean places of K and so j_{E} ∈ K_{0} if K is function field, and h_{k}(E) is bounded (depending on d and d(K)) if K is a number field.

iii) Fix an admissible E_{0}/K. There is a number N_{0}(c, d, N_{E_{0}}, d(K)) =: N_{0}, such that for all N > N_{0} and all admissible elliptic curves E/K with (E_{0}, E) corresponding to a point P ∈ Z(N, ε)(K) we get:

h_{geom}(E)_{fin} ≤ M(c, d, NN_{E_{0}}, d(K))

where h_{geom}(E)_{fin} is the non-archimedean part of the geometric height of E. If E is semi-stable at divisors of N we can replace h_{geom}(E)_{fin} by h_{K}(E). If c = \frac{1}{2} +ε and d depends on g(K) and ε only we get Conjecture 5 for pairs (E_{0}, E) corresponding to P ∈ Z_{N,ε}(K) with E admissible and semi-stable at divisors of N.

Proof :

i) Let N_{E} = NE^{′} ∏_{𝔭∈S_{0}} 𝔭^{n𝔭} be the conductor of E and hence of η(E) where

NE^{′} is prime to S_{0} and n_{𝔭} are bounded by d(K). Using the theory of Tate curves one gets (cf. [Fr 1]):

If 𝔭|NE^{′} then −v_{𝔭}(j_{E}) − v_{𝔭}(j_{η(E)}) ≥ N + 1, and so

2c degN_{e} + 2d ≥ 2h_{K}(E) ≥ \frac{1}{6}(N + 1)degNE^{′} + d̃(d(K)) with d̃(d(K)) ∈ ℝ.

It follows that for N > 12(deg(∏_{𝔭∈S_{0}} 𝔭^{n𝔭} + d + c) − d̃(d(K)) ) we have: deg(N_{E′}) = 0.

ii) If E has a K-rational point of order N ≥ 5 then E has semi-stable reduction at all places 𝔭 |̸N. So if N = p^{α}N′ and N′ ≥ 5 then E is semi-stable at all divisors of p. If this condition is not satisfied but p^{α} > v_{𝔭}(p)+6 then again E is semi-stable at q|p. Hence E/K is semi-stable at all 𝔭 ∈ Σ_{K} if

N ≥ 5δ_{K} = \{ \begin{matrix}5 & if K is a function field \\ 5([K : ℚ] + 6) & if K is a number field. \end{matrix}

Hence by using i) (with η corresponding to < P >) we get: If N ≥ max(5δ_{K}, 24c + 24d) then E has good reduction at all places of K, and the assertion follows.

iii) Assume that (E_{0}, E) corresponds to P ∈ Z_{N,ε}(K). If N ≥ 5 it follows

from Proposition 1.3 that N_{E} = NE^{′} ∏_{{𝔭|N \atop v𝔭(j_{E})≥0}} 𝔭^{δ𝔭} ∏_{𝔭|N_{E_{0}}} 𝔭^{δ𝔭} with NE^{′} prime

to N_{E_{0}} and 0 ≤ δ_{𝔭} bounded by numbers depending on d(K). Moreover

v_{𝔭}(j_{E}) ≡ 0 mod N if 𝔭|NE^{′}.

Hence \frac{N}{12}degNE^{′} ≤ c(degNE^{′} + deg ∏_{𝔭|N} 𝔭^{δ𝔭}) + d′(N_{E_{0}}, c, d).

Since deg ∏_{𝔭|N} 𝔭 is bounded by a number depending on d(K) it follows

that for large N (depending on N_{E_{0}}, d(K), c, d) NE^{′} is trivial and so the non-archimedean part of h_{geom}(E) is bounded. If E is semi-stable at all divisors of N then N_{E} = NE^{′} ∏_{𝔭|N_{E_{0}}} 𝔭^{δ𝔭} and hence, for large N: N_{E} =

∏_{𝔭|N_{E_{0}}} 𝔭^{δ𝔭}. So in this case h_{K}(E) is bounded.

Corollary 1.1 Assume that Conjecture 6 is true over K and that K is a number field or a function field over a finite field K_{0}. Then for N large enough we get: If E/K is semi-stable at all divisors of N and ρ_{E,N} ≅ ρ_{E_{0},N} then E is isogenous to E_{0}. Especially : Conjecture 6 for function fields implies Conjecture 5 for function fields.

c) Relations between elliptic curves and Fermat equations The considerations in the last section have shown that the arithmetical

properties of the representations ρ_{E,n} are in close connection with arithmetical properties of Δ_{E}. If E has K-rational points of order 2 these properties can be translated into properties of the X-coordinates of such points which was done in various papers by Hellegouarch [H] and the author. The procedure is described in detail in [Fr 1] but since it is a crucial point we shall repeat it for the convenience of the reader.

Take x ∈ K^{∗}\{0, 1} and a prime 𝔭_{∞} ∈ Σ_{K} if K is a function field, and let O_{K} be the ring of integers of K (with respect to 𝔭_{∞} in the function field case). Now we define E_{(x)} : Y^{2} = X(X − 1)(X − x).

Since j_{E_{(x)}} =: j_{x} = 2^{8} \frac{(x^{2}−x+1)^{3}}{x^{2}(x−1)^{2}} it follows that E_{(x)} is admissible if K/K_{0}(x) is

separable, and that h_{K}(j_{E_{(x)}} ≥ c′ + 6h_{K}(x). E_{(x)} has good reduction at all primes 𝔭 not dividing 2·supp(x(x−1)), and since for 𝔭|supp(x(x − 1)) but 𝔭 |̸2

v_{𝔭}(j_{(x)}) = \{ \begin{matrix}2v_{𝔭}(x(x − 1)) & if v_{𝔭}(x) ≥ 0 \\ 2v_{𝔭}(x) & if v_{𝔭}(x) < 0 \end{matrix}

we have bad reduction at all divisors of supp(x(x − 1)) which are prime to 2. Now choose A, C such that x = \frac{A}{C} with A, C ∈ O_{K} (where O_{K} = O_{𝔭_{∞}} with 𝔭_{∞} ∈ Σ_{K} if K is a function field, and 𝔭_{∞} = 1 else), deg(gcd(A, C)) ≤ s_{0}(g(K)) (see the discussion of the A-B-C-Conjecture), and take B = A − C. Then E_{(A,B)} : Y^{2} = X(X − A)(X − B) is a twist of E_{(x)}. Define S_{0} = {𝔭 ∈ Σ_{K}, 𝔭|2𝔭_{∞} · gcd(A, C)}. Let Δ_{(A,B)} be the discriminant of E_{(A,B)}. Then for 𝔭 ∉ S_{0} we have:

v_{𝔭}(Δ_{A,B}) = −v_{𝔭}(j_{x}) = −v_{𝔭}(2^{8} \frac{(A^{2}+B^{2}−AB)^{3}}{A^{2}B^{2}C^{2}} ) = 2v_{𝔭}(ABC).

Moreover we get for 𝔭 ∉ S_{0} that X(X − A)(X − B) mod 𝔭 has at least two different zeros and hence E_{(A,B)} has semi-stable reduction modulo 𝔭 for all 𝔭 ∉ S_{0}, and has good reduction if 𝔭 ∉ S_{0} ∪ supp(ABC). Hence

N_{(A,B)} := N_{E_{(A,B)}} = ∏_{𝔭∈S_{0}} 𝔭^{δ𝔭} ∏_{𝔭|̸2 ; {𝔭∉S_{0} \atop 𝔭|supp(x(x−1))}} 𝔭 = ∏_{𝔭∈S_{0}} 𝔭^{δ𝔭} ∏_{{𝔭|(ABC) \atop 𝔭∉S_{0}}} 𝔭

Now we are ready to state

Proposition 1.5 Assume that Conjecture 6 holds with constants c and d. Then h(x) ≤ 2(c + ε)deg supp(x(x − 1)) + d̃(d, ε, d(K)) for each ε ∈ ℝ_{>0}, and ε = 0 allowed if K is a function field.

So the height conjecture for elliptic curves implies the A−B −C-Conjecture and the Asymptotic Fermat Conjecture. We continue to look at the curve E_{(A,B)} and get with C = A − B:

Proposition 1.6 For N ∈ ℕ define S_{N} := S_{0} ∪ {𝔭 ∈ supp(ABC); 2v_{𝔭}(ABC) ≢ 0 mod N}. Then E_{(A,B),N} is a finite groupscheme at all 𝔭 ∉ S_{N}, and the Artin conductor

N_{ρ_{(A,B),N}} of ρ_{E_{(A,B),N}} is equal to N_{ρ(A,B),N} = ∏_{𝔭∈S_{N} ∪supp(N)} 𝔭^{f𝔭}.

Corollary 1.2 Assume that (x, y, z) ∈ L_{(a,b,c),N}(K) and that S_{0} ⊂ Σ_{K} such that x, y, z ∈ O_{S_{0}}, {a, b, c} ⊂ O_{S_{0}}^{∗} and supp(gcd(x, y, z)) has only prime divisors

in S_{0}. Then N_{ρ_{E_{(ax^{N} ,by^{N}),N}}} = ∏_{𝔭∈S_{0}∪supp(N)} 𝔭^{f𝔭},

and at 𝔭|N, 𝔭 ∉ S_{0} the subgroup scheme E_{(ax^{N} ,by^{N}),N} is finite.

The disturbing fact is that in our general frame the set S_{0} can depend on (x, y, z). But since the height conjecture and hence the asymptotic Fermat conjecture are true in the function field case the interesting application of the Corollary 1.2 is to the case that K is a number field. In this case we find a fixed set S_{K} ⊂ Σ_{K} such that O_{S_{K}} is a principal domain and such that supp(S) ⊂ Σ_{K} (where the degree of ∏_{𝔭∈S_{K}} 𝔭 can be bounded by a number depending on g(K)),

and hence we can represent elements in L_{a,b,c,N}(K) up to projective equivalence by (x, y, z) ⊂ O_{S_{K}} which are relatively prime. Hence we get

Corollary 1.3 Assume that K is a number field and S_{K} as above. To (x′, y′, z′) ∈ L_{(a,b,c),N}(K) with x′y′z′ ≠ 0 we find a projectively equivalent

triple (x, y, z) such that N_{ρ_{E_{(ax^{N} ,by^{N}),N}}} = ∏_{𝔭∈S_{K}∪supp(abcN)} 𝔭^{f𝔭}

and E_{(ax^{N} ,by^{N}),N} is a finite groupscheme at 𝔭|N but 𝔭 |̸abc.

Conversely assume that we have an elliptic curve E|K, a finite set S_{0} of primes containing all divisors of 2 and N ≥ 5 such that O_{S_{0}} is a principal domain and such that E_{2N} is finite at 𝔭 ∉ S_{0}. It follows that g(K(E_{2})) is controlled by deg ∏_{𝔭∈S_{0}} 𝔭 and hence in our context we can assume without great loss of

generality that E_{2} ⊂ E(K). Then we find an equation Y^{2} = X(X − A)(X − B) for E with A, B ∈ O_{S_{0}} relatively prime. (Here we use that E has to have semi-stable reduction outside of S_{0}.) So for 𝔭 ∉ S_{0} we get: v_{𝔭}(A) ≡ v_{𝔭}(B) ≡ v_{𝔭}(A−B) ≡ 0 mod N and hence A = ax^{N}, B = by^{N}, C = cz^{N} with a, b, c ∈ O_{S_{0}}^{∗} and (x, y, z) relatively prime in O_{S_{0}}. So (x, y, z) ∈ L_{(a,b,c)}(K), and we can summarize:

Proposition 1.7 Conjecture 2, the Asymptotic Fermat Conjecture, implies Conjecture 5 for even N, i.e. for given E_{0} there is a number N_{0} = N_{0}(g(K), E_{0}) such that for N > N_{0} and ρ_{E_{0},N} ≅ ρ_{E,N} it follows that E_{0} is isogenous to E. Conversely the Asymptotic Fermat Conjecture for a given set S_{0} holds if there are only finitely many odd irreducible two-dimensional representations of G_{K} into Gl(2, ℤ/N) (with some N ∈ ℕ, N large enough) which are unramified outside of S_{0} and finite in divisors of N. Especially there is no solution (x, y, z) in L_{(a,b,c);N}(K) with xyz ≠ 0 and (x, y, z) ∈ O_{K} relatively prime if there is no odd representation

ρ : G_{K} → Gl(2, ℤ/Nℤ)

satisfying the conditions above with S_{0} = {𝔭; 𝔭|2abc}.

It is obvious that the verification of such non-existence theorems as discussed above for G_{K}-representations ρ needs strong number theoretical tools. Since in general the image of ρ is not solvable “classical” methods of class field theory cannot be applied. Looking for other methods it is quite natural that one gets the idea to use “Langland’s philosophy” or over ℚ, its beautiful concretisation given by Serre’s conjecture, and hence one is naturally lead to the theory of modular forms.

d) Dependence on d(K)

Above we have stated conjectures involving constants which should depend on g(K). This seems to be quite natural for the A-B-C-Conjecture and the height conjecture of elliptic curves but for other conjecture, for instance about the existence of cyclic isogenies, one could hope that the constants depend only on d(K). (Cf. Merel’s theorem about the order of torsion points of elliptic curves.) Especially this seems to be convincing if the objects one looks for are parametrized by a family of curves C_{n} and if one can prove that the irrationality degree of C_{n} grows with n. For simplicity we restrict the discussion to the case that K is a number field.

Example 1.1 Let p be an odd prime such that the residue field K(𝔮) has n_{𝔮} elements. Let d_{K}(X_{0}(p)) be the irrationality degree of X_{0}(p) × K. Then d_{K}(X_{0}(p)) ≥ max(\frac{p}{12(n𝔮^{2}+1)}, \frac{p}{120d(K)}) (cf. [Fr 2]). To get the second inequality we use that X_{0}(p) is defined over ℚ and has irrationality degree ≥ \frac{p}{60} over ℚ. By a theorem of Faltings (cf. [Fr 2]) it has only finitely many points of degree ≤ \frac{p}{120} over ℚ and hence of degree ≤ \frac{p}{120d(K)} over K, and so its irrationality degree over K is ≥ \frac{p}{120d(K)}.

Take E_{0}/K and p an odd prime. Then the set of elliptic curves E over extension fields L of K such that (E_{0}, E, α) with α : E_{0,p} ⟶ E_{p} corresponds to a point P ∈ Z_{p,ε}(L) are parametrized by a twisted modular curve X_{ρ_{E_{0},p}} defined over K. Let 𝔮 be a place of K with v_{𝔮}(j_{E_{0}}) < 0, let C_{p} < E_{0,p} be a cyclic group of order p and let K(C_{p}) be the smallest extension of K such that C_{p} becomes Galois invariant. Then the theory of Tate-curves implies that 𝔮 has an extension 𝔮_{p} to K(C_{p}) with K(𝔮) = K((C_{p})(𝔮)). Hence the irrationality degree of X_{0}(p) × K(C_{p}) is bounded from below by \frac{p}{12(|K(𝔮)|^{2}+1)} (see example 1.1). Now assume that L/K has degree d and that (E_{0}, E, α) corresponds to a point in X_{ρ_{E_{0,p}}} (L). Then E has a L(C_{p})-rational isogeny η of degree p and hence (E, η) ∈ X_{0}(p)(L(C_{p})) is a point of K(C_{p}) of degree ≤ d. It follows (see [Fr 2]) that the irrationality degree of X_{0}(p) × K(C_{p}) ≤ 2d and so \frac{p}{12(K(𝔮)^{2}+1)} ≤ 2d.

Hence we get:

Proposition 1.8 The irrationality degree of X_{ρ_{E_{0,p}}} ×K is larger than \frac{p−1}{24(K(𝔮)^{2}+1)} for any prime 𝔮 ∈ Σ_{K} with v_{𝔮}(j_{E_{0}}) < 0 and v_{𝔮}(p) = 0.

Corollary 1.4 Let K be a number field and E_{0}/K an elliptic curve with nonintegral j-invariant. For d ∈ ℕ there exists a number N_{0} = N_{0}(d, d(K), E_{0}) such that for N > N_{0} the set

⋃_{[L:K]≤d}{P ∈ Z_{N,ε}(L); P corresponds to a tripel (E_{0}, E, α) with E ≠ E_{0}}

is finite, and for all primes p there is a number l_{p} = l(p, d, d(K), E_{0}) such that for N = p^{l_{p}} the corresponding set is empty.

It would be interesting to know whether analoguous results can be proved for the sets L_{(a,b,c),n} and so one would like to find bounds for the irrationality degree of Fermat curves:

Question 2: Let C_{a,b,c,n} be the curve given by aX^{n} − bY^{n} = cZ^{n}. Is d(C_{a,b,c,n} × K) bounded (at least asymptotically) by n^{1+ε} ?

For K = ℚ this would follow from a conjecture of Chowla that the minimal prime 𝔮 ≡ −1 mod n is bounded by O(n^{1+ε}) (cf. [C]).

2 The generic case

In this section we shall assume that K in a function field in one variable on a perfect field K_{0}.

Proposition 2.1 Let E/K be an admissible elliptic curve. Then h_{K}(E) ≤ \frac{1}{2}degN_{E} + (g(K) − 1), hence Conjecture 6 is true.

Proof There is a nice proof of Szpiro using the geometry of the elliptic surface E defined by E over K_{0} using the Bogomolov-Miyaoka-Yau inequality between Chern classes of E, and it is an idea and hope of Parshin [P] that this kind of proof can possibly be “translated” into the frame of arithmetical surfaces over number fields. Here we shall give a very elementary proof which uses only Hurwitz’ genus formula for function fields, it can be found in [Fr 1] already, and we repeat it for the convenience of the reader.

First we can assume that j_{E} ∉ K_{0} and that K_{0} is algebraically closed. The next observation is that it is enough to prove the inequality for a quadratic twist of E and so we can assume that E is semi-stable at all 𝔭 ∈ Σ_{K} with v_{𝔭}(j_{E}) < 0. We have to prove:

degΔ_{E} ≤ 6degN_{E} + 12(g − 1).

Our assumptions imply: If 𝔭^{2}|N_{E} then v_{𝔭}(j_{E}) ≥ 0 and 0 < v_{𝔭}(Δ_{E}) ≤ 10. If j_{E} ≡ 12^{3} mod 𝔭 then v_{𝔭}(Δ_{E}) ≤ 9, and if j_{E} ≡ 0 mod 𝔭 then v_{𝔭}(Δ_{E}) ≤ 8.

If v_{𝔭}(j_{E}) < 0 then v_{𝔭}(Δ_{E}) = −v_{𝔭}(j_{E}) and [K : K_{0}(j_{E})] = deg ∏_{v_{𝔭}(j_{E})<0} 𝔭^{−v_{𝔭}(j_{E})} =:

d_{∞}. Let NE^{′} be the square free part of N_{E}. Now use Hurwitz genus formula to get:

2g(K) − 2 ≥ −2d_{∞} + d_{∞} − degNE^{′} + deg( ∏_{{𝔭|̸N_{E} \atop v𝔭(j_{E} −12^{3})>0}} 𝔭) + 2deg( ∏_{{𝔭|̸N_{E} \atop v𝔭(j_{E})>0}} 𝔭) +

deg( ∏_{{𝔭^{2}|N_{E} \atop v𝔭(Δ_{E})=10}} 𝔭).

Define s^{(0)} = deg( ∏_{{𝔭|N_{E} \atop v𝔭(j_{E} −12^{3})>0}} 𝔭), s^{(1)} = deg( ∏_{{𝔭|N_{E} \atop v𝔭(j_{E})>0}} 𝔭), s^{(2)} = deg( ∏_{{𝔭|N_{E} \atop v𝔭(Δ_{E})=10}} 𝔭).

Then:

2g − 2 ≥ −d_{∞} − degNE^{′} + \frac{d_{∞}−s^{(0)}}{2} + \frac{d_{∞}−s^{(1)}}{3} + \frac{1}{3}s^{(2)}

or:

\frac{1}{6}d_{∞} ≤ degNE^{′} + \frac{s^{(0)}}{2} + \frac{2}{3}s^{(1)} − \frac{1}{3}s^{(2)}

and so degΔ_{E} − 12(g − 1) ≤

≤ 6degNE^{′}+10deg(  ∏_{{𝔭|N_{E} \atop v𝔭(j_{E} (j_{E} −12^{3}))=0}} 𝔭)+12deg(  ∏_{{𝔭|N_{E} \atop v𝔭(j_{E} −12^{3})>0}} 𝔭)+12deg( ∏_{{𝔭|N_{E} \atop v𝔭(j_{E})>0)}} 𝔭) ≤

≤ 6degN_{E}.

Corollary 2.1 There are constants d_{0} and d_{1} such that for all function fields K over perfect fields K_{0} and for all x ∈ K\{0, 1} with K/K_{0}(x) separable we get: h_{K}(x) ≤ \frac{1}{2}degsupp(x(x − 1)) + d_{1}g(K) + d_{0}, hence Conjecture 1’ is true, as well as Conjecture 2 and Conjecture 5.

We can use the height formula to bound the order K-rational torsion subgroups for elliptic curves defined over K but we can do much better: Assume that j_{E} ∉ K_{0} and that E is admissible. Let η : E → E′ be a separable cyclic isogeny of order N defined over K. Then (j_{E}, jE^{′}) generate a separable subfield in K which is isomorphic to the function field of X_{0}(N) × K_{0}, hence g(K) ≥ g(X_{0}(N)) ∼ \frac{N}{6} and so N ≤ 6g(K), and Conjecture 3 is true.

Next assume for simplicity that N is an odd prime. If K_{0} is finite then the irrationality degree of X_{0}(N) × K_{0} is bounded from below by \frac{N}{120(|K_{0}|^{2}+1)} and so d(K) ≥ \frac{N}{120(|K_{0}|^{2}+1)}, and hence the Conjecture 3 remains true if we replace g(K) by (d(K) · (120)|K_{0}|^{2} + 1). It is easy to see that an analogous result can be obtained if K_{0} is a number field and |K_{0}| is replaced by 2^{d(K_{0})}.

What about Darmon’s conjectures ? To give a flavour of the kind of problems which arise we simplify by assuming that char(K_{0}) = 0 and interprete Z_{N,ε} as threefold over K_{0}. For P ∈ Z_{N,ε}(K) let C_{P} be a component of its Zariski closure. If P is non-constant then g(C_{P}) ≤ g(K) and d(C_{P}) ≤ d(K). Hence Conjecture 4 and 4′ leads to

Conjecture 7: Let g (resp. d) be non-negative integers. There exist numbers N_{0}(g) (resp. N0^{′}(d)) such that for N ≥ N_{0}(g) (N > N0^{′}(d)) and irreducible curves C ⊂ Z_{N,ε} it follows that C is birationally equivalent to X_{0}(n) for some n.

Let us repeat: The conjecture of Darmon in Kani’s version predicts that N_{0}(1) = 13.

3 K = ℚ

What can we say about our conjectures if K = ℚ ? There is one essential tool to use: The arithmetic of modular curves and its applications to elliptic curves over ℚ. There are two major reasons for the relations between modular curves and elliptic curves. Firstly level-n-structures are parametrized by points on these curves and secondly elliptic curves appear as factors of their Jacobians.

The great power of the “modular method” is due to major contributions of many mathematicians. In our context we have to mention especially B. Mazur, J.P. Serre, K.Ribet and above all A. Wiles.

That elliptic curves with cyclic isogenies of degree N correspond to points on X_{0}(N) we used already in the previous sections. Over ℚ Mazur in [Ma 2] succeeded to prove:

Theorem 3.1 Let E/ℚ be an elliptic curve with ℚ-rational cyclic isogeny of degree N. Then N ≤ 163. Hence Conjecture 3 is true over ℚ.

Moreover Mazur gives the entire list of such curves E.

In [Ma 1] Mazur already emphasizes the importance of congruence primes, these primes play a major role for the study of elliptic curve E which are factors of J_{0}(N_{E}), i.e. which are \underline{modular elliptic curves}; they link arithmetical properties with attached Galois representations.

Here is the definition of these primes corresponding to a modular curve E: Let φ_{E} : X_{0}(N_{E}) → E be a ℚ-rational non-trivial morphism of minimal degree. Let ω_{E} be the Néron differential of E.

Then φ_{E}^{∗}ω_{E} = c_{φ}f_{E}(z) dz with c_{φ} ∈ ℤ\{0} and f_{E}(z) = q + ∑_{i=2}^{∞} a_{i}q^{i} with f_{E} ∈

S_{2}(N_{E})(ℤ) a new form. For primes l |̸pN_{E} one has: T r(ρ_{E,p}(σ_{l}) ≡ a_{l} mod p where σ_{l} is a Frobenius element at l. We call f_{E} the cusp down attached to E.

Now assume that K is a number field and g(z) = q + ∑_{i=2}^{∞} b_{i}q^{i} ∈ S_{2}(N, O_{K})

is an eigenform such that there is a prime divisor 𝔭 of p with b_{l} ≡ a_{l} mod 𝔭 for all l |̸pNN_{E}. Then we say by abuse of language that g is congruent to f_{E} modulo p and if g ≠ f_{E}, that p is a congruence prime for f_{E} resp. for E.

The following theorem uses Corollary 1.3 and is a special case of a beautiful result of K. Ribet [Ribet] in which he proved a large part of Serre’s conjecture for odd two-dimensional representations of G_{ℚ} (cf. [Se]) in the modular case:

Theorem 3.2 (Ribet): Let p be an odd prime. Assume that E is a modular elliptic curve which is semi-stable outside of S_{0}, let f_{E} be the attached new form in S_{2}(N_{E})(ℤ). Assume that ρ_{E,p} is irreducible.

Let  N_{ρ′_{E,p}} := ∏_{l∈S_{0}} l^{v_{l}(N_{E})} ∏_{{l∉S_{0} \atop v_{l}(Δ_{E})≢0modp}} l

Then there is a number N_{0} | Nρ_{E,p}^{′} and a new form g ∈ S_{2}(N_{0}) congruent to f_{E} modulo p.

Finally we come to Andrew Wiles [W] celebrated result:

Theorem 3.3 (A. Wiles): Assume that E has a twist E′ which is semi-stable at 3 and 5. Then E is a modular elliptic curve. Especially it follows that E is modular if E_{2} ⊂ E(ℚ).

As a first application we shall give a “modular version” of the height conjecture for elliptic curve and hence of the A-B-C-Conjecture: Assume that E is modular with attached eigenform f_{E}. Then (cf. [Fr 1]) we have

h_{ℚ}(E) = \frac{1}{2} log degφ_{E} − log |c_{φ}| − \frac{1}{2}log (\frac{1}{2π} ∫_{X_{0}(N_{E})⊗ℂ} |f_{E}|^{2} dσ)

with dv = 4π dx dy.

Lemma 3.1: i) There is a number k > 0 (independent of E) such that

\frac{1}{2π} ∫_{X_{0}(N_{E})⊗ℂ} |f_{E}|^{2} dv ≥ k.

ii) log ( ∫_{X_{0}(N_{E})⊗ℂ} |f_{E}|^{2} dv) ≤ O (log N_{E}).

The (easy) proof of i) can be found in [Fr 1], the much more involved proof of ii) is a result due to Mai-Murty (cf. [MM]).

Proposition 3.1 i) Assume that E is modular. Then

2h_{ℚ}(E) ≤ log(deg φ_{E}) + O(1).

ii) Assume that E is semi-stable. Then

log degφ ≤ 2h_{ℚ}(E) + O(log N_{E}).

Proof: Obvious using Lemma 3.1 and for the second part, Theorem 3.3 and the fact that |c_{φ}| is bounded if E is semi-stable (conjecturally |c_{φ}| = 1).

Hence the height conjecture for modular elliptic curves E is true with constant c (and computable d) if for such curves log(deg φ_{E}) ≤ 2c log N_{E} + d′ and conversely for semi-stable elliptic curves the existence of such constants follows from the height conjecture for elliptic curves.

Corollary 3.1 The A-B-C-Conjecture for relatively prime numbers A and B holds with constants c, d, if for modular elliptic curves E we have: degφ_{E} ≤ c log N_{E} + d + log |c_{φ}| − \frac{1}{2} log k, and its truth (with some constants c̃, d̃) is equivalent with an estimate of degφ_{E} of this type for all E with E_{2} contained in E(ℚ).

Remark 3.1 It is not difficult to show that log deg φ_{E} ≤ O(N_{E} log N_{E}) and hence we get an exponential version of the A-B-C-Conjecture but this is not surprising since by transcendental methods one gets this type of results, too.

There is obvious relation between φ_{E} and congruence primes, and more general, Question 1: φ_{E} induces a morphism ψ_{E} : J_{0}(N_{E}) ⟶ E. Let ψ̂_{E} its dual map. Using theorem 1 we can assume that ψ_{E} does not factor through another elliptic curve and so ψ̂_{E} is injective and ker(ψ_{E}) = A_{E} ⊂ J_{0}(N_{E}) is an abelian variety. Since ψ_{E} ∘ ψ̂_{E} = degψ_{E} ∘ id it follows that ψ̂_{E}(E) ∩ A_{E} = ψ̂_{E}(E)_{degφ_{E}}, and so any prime divisior of degφ_{E} is a congruence prime. Moreover we find a subvariety B in A_{E}/ψ̂_{E}(E)_{degφ_{E}} and an monomorphism α : E_{degφ_{E}} ⟶ B such that Δ_{α} = {(P, αP); (P ∈ E_{degφ})} is an exceptional subgroup of E × B.

So a positive answer to Question 1 would have strong consequences for the height conjecture for elliptic curves, the A-B-C-Conjecture and the Asymptotic Fermat Conjecture (remember that dimB ≤ O(N_{E})).

Example 3.1 Assume that E/ℚ has conductor N_{E} = l ∈ ℙ. Using Theorem 3 and Theorem 2 and checking small primes one gets: Δ_{E}|l^{6}, and so Szpiro’s conjecture is true. To get the height conjecture one would need more, namely the \underline{Hall Conjecture} (cf. [Si]) which predicts that log(g_{2}(E)) ≤ d′(ε)+(2+ε) log |Δ_{E}| for all ε ∈ ℝ_{>0}.

Let n be a non zero integer and define

δ_{n} = \{ \begin{matrix}0 & if & v_{2}(n) ≥ 4 \\ 2 & if & 2 ≤ v_{2}(n) ≤ 3 \\ 4 & if & v_{2}(n) = 1 \\ 0 & if & v_{2}(n) = 0 \end{matrix}

Now take A, B ∈ ℤ relatively prime and assume that A is even and that B ≡ 3 mod 4. We use again theorem 3.1,3.2,3.3 and get:

Proposition 3.2 The curve E_{(A,B)} : Y^{2} = X(X − A)(X − B) is modular, and for primes p ≥ 5 with ρ_{E_{(A,B)},p} irreducible ( e.g. p > 163) we get: f_{E_{(A,B)}} is congruent modulo p to g ∈ S_{2}(N_{(A,B),p}) with

N_{(A,B),p} = 2^{δ_{A}} ∏_{v_{l}(2^{−4}AB(A−B)≢0modp} l

This result implies strong conditions for solutions of equations of Fermattype. Take a, b, c ∈ ℤ\{0} relatively prime. We want to study L_{(a,b,c),p}(ℚ) for large primes p . We can assume without loss of generality that v_{2}(a) ≥ v_{2}(b) + v_{2}(c) and that p > v_{2}(a). Define L̃_{(a,b,c),p}(ℚ) to be the set of triples (x, y, z) ∈ L_{(a,b,c),p}(ℚ) with xyz ≠ 0 and gcd(x, y, z) = 1. Now apply Proposition 3.2 with A = ax^{p}, B = by^{p} to get:

Corollary 3.2 Let p be larger than 163. L̃_{(a,b,c),p}(ℚ) is empty if the genus of

X_{0} (2^{δ_{a}} ∏_{v_{l}(2^{−4}abc)≢0modp} l)

is equal to 0, and there is no triple (x, y, z) ∈ L̃_{(a,b,c),p}(ℚ) with v_{2}(x) > 0 if the genus of

X_{0} ( ∏_{v_{l}(2^{−4}abc)≢0modp} l)

is equal to 0.

Since g(X_{0}(N)) = 0 if N ≤ 10 orN = 12, 13, 16, 18, 25 one can use this corollary together with a careful discussion of small exponents to determine L_{(a,b,c),p}(ℚ) for suitable a,b,c (cf.[R 2]).

We look only at one special case: a = b = c = 1 and so N = 2 and get

Fermat’s Last Theorem: Let n be a natural number ≥ 3 and x, y, z ∈ ℤ with x^{n} − y^{n} = z^{n}. Then xyz = 0.

Another interesting case is that f_{E_{(A,B)}} (with A = ax^{p}, B = by^{p}, A−B = cz^{p} as above) is congruent modulo p to a new form g_{0} belonging to an elliptic curve

E_{0} with conductor N_{0}|N_{(A,B),p}|2^{δ_{a}+1}abc, with g_{0}(z) = q + ∑_{i=2}^{∞} b_{i}q^{i}.

Let l be a prime not dividing pN_{0} but dividing ABC.Then there is a quadratic extension K_{1} of ℚ_{l} such that E_{(A,B)}(K_{1}) contains a point of order p. Hence E_{0}(K_{1}) contains a point of order p and since it has good reduction at l we get:l ≥ p^{1/2} + 1. So for l ≤ p^{1/2} the curve E_{(A,B)} has good reduction and a_{l} ≡ b_{l} mod p. Since |a_{l}| ≤ 2\sqrt{l} + 1, |b_{l}| ≤ 2\sqrt{l} + 1 we have a_{l} = b_{l} for p > 4\sqrt{l} + 2 . Especially E_{0} mod l has 4 rational points of two-power order . Hence we get for l prime to N_{0} and l < Min((p − 2)^{2}/16, p^{1/2}) that E_{0} mod l has rational points of order 2. For p large enough (depending on N_{0} only) it follows that E_{0} is isogenous to an elliptic curve E0^{′} with E0,2^{′} ⊂ E0^{′}(ℚ). Replace E_{0} by E0^{′}. Then E_{0} is given by an equation

Y^{2} = X(X − a′)(X − b′) with supp(a′b′(a′ − b′))|2^{δ_{a}+1} ( ∏_{l|abc} l)

There is one case in which we can be sure that g_{0} belongs to an elliptic curve:

Proposition 3.3 Assume that for N_{0} = 2^{δ_{a}+1} ( ∏_{l|abc} l) we have g(X_{0}(N_{0})) = 1

and that X_{0}(N_{0}) has not 4 ℚ-rational points of 2-power order. Then there is an effectively computable number p_{0}(N_{0}) such that for p ≤ p_{0}(N_{0}) we have: L̃_{(a,b,c),p}(ℚ) = ∅.

Now there are only very few modular curves of genus 1 and hence the assumption of the proposition is not often satisfied. But it is an observation of Mazur that one can enforce the existence of E_{0} by choosing p large: The situation is as above. We assume that p > 163 and apply theorem 3.3 to get a new form g_{0} ∈ S_{2}(N_{0})(K) with Fourier coefficient b_{i} ∈ O_{K} such that for a divisor 𝔭 of p and prime numbers l |̸pN_{0} we get: a_{l} ≡ b_{l} mod 𝔭. Now the degree of K/ℚ can be bounded by g(X_{0}(N_{0})) ∼ N_{0}/6, and since the absolute values of the conjugates of b_{l} are bounded by 2l^{1/2} it follows that for log(p) > O(N_{0}log(N_{0})) we have: a_{l} = b_{l} for all l ≤ 2g(X_{0}(N_{0})) and hence g_{0} ∈ S_{2}(N_{0})(ℚ).

We summarize:

Proposition 3.4 Assume that a,b,c are relatively prime integers different from 0 and define N_{0} as above. Then there is an effectively computable number p_{0}(N_{0})(< O(N_{0}log(N_{0})) such that for p ≥ p_{0}(N_{0}) and L̃_{(a,b,c),p}(ℚ) ≠ 0 we get: X_{0}(N_{0}) has an elliptic factor E_{0} with E_{0,2} ⊂ E_{0}/(ℚ) and ρ_{E_{0,p}} ≅ ρ_{E_{(ax^{p},by^{p}),p}} for (x, y, z) ∈ L̃_{(a,b,c),p}(ℚ).

Corollary 3.3 (Mazur)Let q be a prime. If the equation 2^{4}X^{p} − Y^{p} = qZ^{p} has solutions for infinitely many p then q = 17.

Proof X_{0}(q) has to have an elliptic factor given by Y^{2} = X(X − a′)(X − b′) and (since it has good reduction outside of q) a′ = 2^{4} and |b′(a′ − b′)| = q .

Finally we observe a consequence of proposition 3.4 which brings our conjectures about solutions of equations of Fermat-type and about exceptional subgroups of E_{1} × E_{2} nicely together:

Proposition 3.5 Conjecture 2 is equivalent with Conjecture 5, i.e. for any finite set S_{0} ∈ ℙ the set

⋃_{{supp(abc)| ∏ ; l∈S_{0} l \atop p≥5}} L_{(a,b,c),p}(ℚ) is finite if and only if the set

{E/ℚ; E_{2} ⊂ E(ℚ) and there is a prime p ≥ 5 and a curve E_{0} with N_{E_{0}}| ∏_{l∈S_{0}} l

and ρ_{E_{0,p}} ≅ ρ_{E,p}} is finite.

Proof: One direction is true in general and was proved in proposition 1.7. Conversely assume that Conjecture 5 is true and that there are infinitely many solutions (x, y, z) ∈ L̃_{(a,b,c),p}(ℚ) for p ≥ 5, supp(abc)| ∏_{l∈S_{0}} l. Hence there are

infinitely many non-isogenous elliptic curves E_{(ax^{p},by^{p})} which are modular with N_{(ax^{p},by^{p}),p}|2^{n_{2}} ∏_{l∈S_{0}} l with n_{2} ≤ 5. By proposition 3.4 there is one elliptic curve

E/ℚ of level N_{0} dividing 2^{n_{2}} ∏_{l∈S_{0}} l with ρ_{E_{0,P}} ≅ ρ_{E_{(ax^{p},by^{p}),p}} for infinitely many

primes p which contradicts Conjecture 5.

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