Integral points on moduli schemes of elliptic curves

We combine the method of Faltings (Arakelov, Paršin, Szpiro) with the Shimura–Taniyama conjecture to prove effective finiteness results for integral points on moduli schemes of elliptic curves. For several fundamental Diophantine problems, such as for example S ‐unit and Mordell equations, this gives an effective method which does not rely on Diophantine approximation or transcendence techniques.


Introduction
Building on the work of Wiles [69] and Taylor-Wiles [68], the Shimura-Taniyama conjecture was finally established by Breuil-Conrad-Diamond-Taylor [14]. In this paper, we combine the method of Faltings (Arakelov, Paršin, Szpiro) with the Shimura-Taniyama conjecture to prove effective finiteness results for integral points on moduli schemes of elliptic curves. For several fundamental Diophantine problems, such as for example S-unit and Mordell equations, this gives an effective method which does not rely on Diophantine approximation or transcendence techniques. In what follows in the introduction, we describe in more detail the content of this paper.

Integral points on moduli schemes of elliptic curves
To provide some motivation for the study of integral points on moduli schemes of elliptic curves, we discuss in the following section fundamental Diophantine equations which are related to such moduli schemes. For any β ∈ Q, we denote by h(β) the usual (absolute) logarithmic Weil height of β defined for example in [12, p. 16].
1.1.1. S-unit and Mordell equations Let S be a finite set of rational prime numbers. We define N S = 1 if S is empty and N S = p with the product taken over all p ∈ S otherwise. Let O × denote the units of O = Z[1/N S ]. First, we consider the classical S-unit equation (1.1) The study of S-unit equations has a long tradition and it is known that many important Diophantine problems are encapsulated in the solutions of (1.1). For example, any upper bound for h(x) which is linear in terms of log N S is equivalent to a version of the (abc)-conjecture. Mahler [46], Faltings [26] and Kim [41] proved finiteness of (1.1) by completely different methods. Moreover, Baker's method [4] or a method of Bombieri [9] both allow in principle to find all solutions of any S-unit equation. We will briefly discuss the methods of Baker, Bombieri, Faltings, Kim and Mahler in Paragraph 7.2.1. In addition, we now point out that Frey remarked in [29, p. 544] that the Shimura-Taniyama conjecture implies finiteness of (1.1). It turns out that one can make Frey's remark in [29] effective and one obtains for example the following explicit result (see Corollary 7.2): Any solution (x, y) of the S-unit equation (1.1) satisfies h(x), h(y) 3 2 n S (log n S ) 2 + 65, n S = 2 7 N S . (After we submitted this paper, Hector Pasten informed us about his joint work with Murty [52] in which they independently obtain a (slightly) better version [52,Theorem 1.1] of the displayed height bound by using a similar method; see below Corollary 7.2 for more details. We would like to thank Hector Pasten for informing us.) Frey uses inter alia his construction of Frey curves. This construction is without doubt brilliant, but rather ad hoc and thus works only in quite specific situations. The starting point for our generalizations are the following two observations: The solutions of (1.1) correspond to integral points on the moduli scheme P 1 Z[1/2] − {0, 1, ∞}, and the construction of Frey curves may be viewed as an explicit Paršin construction induced by forgetting the level structure on the elliptic curves parameterized by the points of P 1 Z[1/2] − {0, 1, ∞}. We now discuss a second fundamental Diophantine equation which is related to integral points on moduli schemes. For any non-zero a ∈ O, one obtains a Mordell equation (1. 2) We shall see in Subsection 7.3 that this Diophantine equation is a priori more difficult than (1.1). In fact, the resolution of (1.2) in Z × Z is equivalent to the classical problem of finding all perfect squares and perfect cubes with given difference, which goes back at least to Bachet 1621. Mordell [49,50], Faltings [26] and Kim [42] showed finiteness of (1.2) by using completely different proofs, and the first effective result for Mordell's equation was provided by Baker [6]; see Paragraph 7.3.1 where we briefly discuss methods which show finiteness of (1.2). On working out explicitly the method of this paper for the moduli schemes corresponding to Mordell equations, we obtain a new effective finiteness proof for (1.2). More precisely, if a S = 2 8 3 5 N 2 S p min(2,ordp(a)) with the product taken over all rational primes p / ∈ S with ord p (a) 1, then Corollary 7.4 proves that any solution (x, y) of (1.2) satisfies h(x), h(y) h(a) + 4a S (log a S ) 2 .
This inequality allows in principle to find all solutions of any Mordell equation (1.2) and it provides in particular an entirely new proof of Baker's classical result [6]. Moreover, the displayed estimate improves the actual best upper bounds for (1.2) in the literature and it refines and generalizes Stark's theorem [65]; see Subsection 7.3 for more details.
We observe that S → Spec(Z) − S defines a canonical bijection between the set of finite sets of rational primes and the set of non-empty open subschemes of Spec(Z). In what follows in this paper (except Subsections 7.2-7.4), we will adapt our notation to the algebraic geometry setting and the symbol S will denote a base scheme.
1.1.2. Integral points on moduli schemes of elliptic curves More generally, we now consider integral points on arbitrary moduli schemes of elliptic curves. We denote by T and S non-empty open subschemes of Spec(Z), with T ⊆ S. Let Y = M P be a moduli scheme of elliptic curves, which is defined over S, and let |P| T be the maximal (possibly infinite) number of distinct level P-structures on an arbitrary elliptic curve over T ; see Section 3 for the definitions. We denote by Y (T ) the set of T -points of the S-scheme Y . Let h φ be the pullback of the relative Faltings height by the canonical forget P-map φ, defined in (3.3). Write ν T = 12 3 p 2 with the product taken over all rational primes p not in T . We obtain in Theorem 7.1 the following result.
This leads to Theorem A(i). To prove the explicit height bounds in Theorem A(ii), it suffices by (1.3) to control the relative Faltings height h(E) of E Q (see Section 2). We first work out explicitly an estimate of Frey [28] which relies on several non-trivial results, including [40]: If E Q is modular, then Frey estimates h(E) in terms of the modular degree m f of the newform f associated with E Q . The theory of modular forms allows one to bound m f in terms of the level N E of f , and (ST ) says that E Q is modular. Hence one obtains an estimate for h(E) in terms of N E , which then leads to Theorem A(ii).
To obtain upper bounds for heights on Y (T ) which are different to h φ , it remains to work out height comparisons. In the two examples discussed above, one can do this explicitly by using explicit formulas for certain (Arakelov) invariants of elliptic curves.
We emphasize that the crucial ingredients for Theorem A are (1.3) and the 'geometric' version (1.4) of (ST ) which relies inter alia on the Tate conjecture proved by Faltings [26]. The other tools, such as Frey's estimate, the theory of modular forms and the isogeny results of Mazur-Kenku [40], can be replaced by Arakelov theory and isogeny estimates; see [37]. In fact, the proof of Theorem A may be viewed as an application of the method of Faltings (Arakelov, Paršin, Szpiro) to moduli schemes of elliptic curves.

Plan of the paper
In Section 2, we recall the definition of the Faltings heights and the conductor of elliptic curves over number fields. In Section 3, we give Paršin constructions for moduli schemes of elliptic curves and in Section 4 we state a lemma which controls the variation of Faltings heights in an isogeny class of elliptic curves over Q. In Section 5, we use the theory of modular forms to bound the modular degree. Then we prove in Section 6 an explicit height conductor inequality for elliptic curves over Q and we derive some applications. In Section 7, we give our effective finiteness results for integral points on moduli schemes of elliptic curves. Here we begin with the general theorem and then we consider in detail the special cases of S-unit and Mordell equations. In addition, we compare our results with the literature and we discuss further Diophantine applications.
We mention that the setting of certain preliminary sections will be more general than is necessary for the proofs of the main results of this paper, since we wish also to look ahead to future work (see [37], our forthcoming paper 'Height and conductor of elliptic curves', and our forthcoming joint work 'Integral points on certain Shimura varieties' with Kret).

Conventions and notations
We identify a non-zero prime ideal of the ring of integers O K of a number field K with the corresponding finite place v of K and vice versa. We write N v for the number of elements in the residue field of v, we denote by v(a) the order of v in a fractional ideal a of K and we write v | a (respectively, v a) if v(a) = 0 (respectively, v(a) = 0). If E is an elliptic curve over K with semi-stable reduction at all finite places of K, then we say that E is semi-stable.
By log, we mean the principal value of the natural logarithm and we define the maximum of the empty set and the product taken over the empty set as 1. For any set M , we denote by |M | the (possibly infinite) number of distinct elements of M . Let f 1 and f 2 be real-valued functions on M . We write f 1 f 2 if there exists a constant c such that f 1 cf 2 . For any map f : if for all > 0 there exists a constant c( ), depending only on , such that Let S be an arbitrary scheme. We often identity an affine scheme S = Spec(R) with the ring R. If T and Y are S-schemes, then we denote by Y (T ) = Hom S (T, Y ) the set of S-scheme morphisms from T to Y and we write Y T = Y × S T for the base change of Y from S to T . We say that a scheme Y is defined over S if Y is a S-scheme.

Height and conductor of elliptic curves
Let K be a number field and let E be an elliptic curve over K. In the first part of this section, we recall the definition of the relative and the stable Faltings height of E. In the second part, we define the conductor N E of E.

Faltings heights
We begin to define the relative and stable Faltings height of E following [26, p. 354]. Let B be the spectrum of the ring of integers of K. We denote by E the Néron model of E over B, with zero section e : B → E. Let Ω 1 be the sheaf of relative differential 1-forms of E/B. We now metrize the line bundle ω = e * Ω 1 on B. For any embedding σ : K → C, we denote by E σ the base change of E to C with respect to σ. We choose a non-zero global section α of ω. Let α σ σ be the positive real number that satisfies where α σ denotes the holomorphic differential form on E σ which is induced by α.
This definition does not depend on the choice of L , since the formation of the identity components of the corresponding semi-stable Néron models commutes with the induced base change. In particular, inequality (2.1) becomes an equality when h is replaced by h F . We shall state several of our results in terms of h F or h and therefore we now briefly discuss important differences between these heights. From (2.1), we deduce Further, as already observed, the height h F has the advantage over h that it is stable under base change. On the other hand, h F has in general weaker finiteness properties. For instance, there are only finitely many K-isomorphism classes of elliptic curves over K of bounded h, while h F is bounded on the infinite set given by the K-isomorphism classes of elliptic curves of any fixed j-invariant in K.

Conductor
We now define the conductor N E of an arbitrary elliptic curve E over any number field K.
Let v be a finite place of K. We denote by f v the usual conductor exponent of E at v, see, for example, [ [59,Subsection 4.9] and of Lockhart-Rosen-Silverman [45]. It follows from [15,Theorem 6.2] that Furthermore, the examples in [15] show that (2.2) is best possible in a strong sense.

Paršin constructions: forgetting the level structure
Paršin [53] discovered a link between the Mordell and the Shafarevich conjecture which is now commonly known as Paršin construction or Paršin trick. This link gives a finite map from the set of rational points of X into the integral points of a certain moduli space, where X is a curve of genus at least two which is defined over a number field.
In the first part of this section, we use the moduli problem formalism to obtain tautological Paršin constructions for moduli schemes of elliptic curves. In the second part, we explicitly work out this idea for P 1 − {0, 1, ∞} and once punctured Mordell elliptic curves. This results in completely explicit Paršin constructions for these hyperbolic curves.

Moduli schemes
We begin to introduce some notation and terminology. Let S be an arbitrary scheme. An elliptic curve over S is an abelian scheme over S of relative dimension 1. A morphism of elliptic curves over S is a morphism of abelian schemes over S. We denote by M (S) the set of isomorphism classes of elliptic curves over S. On following Katz-Mazur [39, p. 107], we write (Ell) for the category of elliptic curves over variable base-schemes: The objects are elliptic curves over schemes and the morphisms are given by cartesian squares of elliptic curves. Let (Sets) be the category of sets and let P be a contravariant functor from (Ell) to (Sets). An element α ∈ P(E) is called a level P-structure on an elliptic curve E over S. We say that P is a moduli problem on (Ell) and we define with the supremum taken over all elliptic curves E over S. In other words, |P| S is the maximal (possibly infinite) number of distinct level P-structures on an arbitrary elliptic curve over S. A scheme M P is called a moduli scheme (of elliptic curves) if there exists a moduli problem P on (Ell) which is representable by an elliptic curve over M P . The following lemma may be viewed as a tautological Paršin construction for moduli schemes. Proof. We note that the statement is intuitively clear, since Y (T ) is essentially the set of elliptic curves over T with 'level P-structure' and the map is essentially 'forgetting the level P-structure'. We now verify that this intuition is correct.
By assumption, there exists a contravariant functor P from (Ell) to (Sets) which is representable by an elliptic curve over Y . Suppose E and E are elliptic curves over a scheme Z, with α ∈ P(E) and α ∈ P(E ). Then the pairs (E, α) and (E , α ) are called isomorphic if there exists an isomorphism ϕ : E → E of objects in (Ell) with P(ϕ)(α ) = α. Let F (Z) be the set of isomorphism classes of such pairs (E, α). Then Z → F (Z) defines a contravariant functor from the category of schemes to (Sets), which is representable by Y since P is representable by an elliptic curve over Y . Thus we obtain an inclusion Y (T ) → F (T ), which composed with , 1 i n} is the fiber of this map over a point in M (T ). Then all E i are isomorphic objects of (Ell). Therefore, after applying suitable isomorphisms of objects in (Ell), we may and do assume that all E i coincide. This shows that n |P| T and then we conclude Lemma 3.1.
We call the map constructed in Lemma 3.1 the forget P-map. To discuss some fairly general examples of moduli schemes we consider an arbitrary scheme Y . If there exists an elliptic curve E over Y , then Y = M P is a moduli scheme with P = Hom (Ell) (−, E). This shows, in particular, that any Z[ 1 2 ]-scheme Y is a moduli scheme, since there exists an elliptic curve A over Z[ 1 2 ] and the base change A Y is an elliptic curve over Y . Next, we discuss a classical example of a moduli problem. Let N 1 be an integer and consider the 'naive' level N moduli problem P N from (Ell) to (Sets), defined by Here we view (Z/N Z) 2 as a constant S-group-scheme and E[N ] is the kernel of the Shomomorphism 'multiplication by N ' on the elliptic curve E over S. If P N (E/S) is non-empty and if S is connected, then we explicitly compute If N 3, then [39,Corollary 4.7.2] gives that P N is a representable moduli problem on (Ell), with moduli scheme Y (N ) = M PN a smooth affine curve over Spec(Z[1/N ]).
In the remaining of this section, we give two propositions. Their proofs consist essentially of working out explicitly Lemma 3.1 for particular moduli schemes, see the remarks given below the proofs of Propositions 3.2 and 3.4, respectively.

Explicit constructions
We introduce and recall some notation. Let K be a number field and write B for the spectrum of the ring of integers O K of K. In the remaining of this section, we denote by either a non-empty open subscheme of B or the spectrum of the function field K of B and we write O = O S (S). Let E be an elliptic curve over S. We denote by h(E) and by h F (E) the relative and the stable Faltings height of the generic fiber E K of E, respectively, see Section 2 for the definitions. Let N E be the conductor of E K defined in Subsection 2.2 and let Δ E be the norm from K to Q of the usual minimal discriminant ideal of E K over K. We observe that h(E), h F (E), N E and Δ E define real-valued functions on M (S).
Let Y = M P be a moduli scheme defined over S, let T be a non-empty open subscheme of S and let φ : Y (T ) → M (T ) be the forget P-map from Lemma 3.1. On pulling back the relative Faltings height h by φ, we obtain a height h φ on Y (T ) defined by The height h φ has the following properties: If |P| T < ∞, then Lemma 3.1 together with Lemma 3.5 shows that there exist only finitely many P ∈ Y (T ) with h φ (P ) bounded. Furthermore, if P is given with |P| T < ∞, then the proof of Lemma 3.1 together with Lemma 3.5 implies that one can, in principle, determine, up to a canonical bijection, the set of T -points P of Y with h φ (P ) effectively bounded.
Let D K be the absolute value of the discriminant of K over Q, let d = [K : Q] be the degree of K over Q and let h K be the cardinality of the class group of B. We define with the product taken over all v ∈ B − T ; note that N T = ∞ if S = T = Spec(K). Further, we say that a non-zero β ∈ K is invertible on T if β and β −1 are both in O T (T ). For any vector β with coefficients in K, we denote by h(β) the usual absolute logarithmic Weil height of β which is defined in [12, 1.5.6].
3.2.1. S-unit equations We continue the notation introduced above and we now give an explicit Paršin construction for 'S-unit equations'. The solutions of such equations correspond to S-points of (iii) If B has trivial class group, then E extends to an elliptic curve over T and N E | 2 7d N T . If K = Q, then h(P ) 6h(E ) + 11.
In this article, we shall use Proposition 3.2 only for one-dimensional S and T . However, the height inequalities obtained in this proposition may be also of interest for S = T = Spec(K). We mention that the number 6 in these height inequalities is optimal.
To prove Proposition 3.2, we shall use inter alia the following lemma. Proof. For any embedding σ : K → C, we take τ σ ∈ C such that the base change of E K to C with respect to σ takes the form C/(Z + τ σ Z) and such that im(τ σ ) with the sum taken over all embeddings σ : K → C. Here |·| denotes the complex absolute value. Further, on using the elementary inequalities log |Δ(τ σ )/q| 24|q|/(1 − |q|) and |q| exp(−π √ 3), we deduce the estimate This together with the displayed formula for log Δ E implies the statement.
We remark that the proof shows in addition that Faltings' delta invariant δ(E C ) [27, p. 402] of a compact connected Riemann surface E C of genus one satisfies Indeed, this follows directly from (3.4) and Faltings' explicit formula [27,Lemma (c), p. 417] for δ(E C ). We mention that it is an important open problem to obtain explicit lower bounds, in terms of the genus, for the Faltings delta invariant of compact connected Riemann surfaces of arbitrary positive genus.
Proof of Proposition 3.2. We observe that if X(S) is empty, then all statements are trivial. Hence we may and do assume that X(S) is not empty. We denote by Y the spectrum of Z[λ, 1/(2λ(1 − λ))] for λ an 'indeterminate'. Then we observe that Then E is an elliptic curve over T and therefore we see that We now prove (i). In what follows, we write λ for z(P ) to simplify notation. The j-invariant j of the generic fiber E K of E satisfies This implies an upper bound for h(P ) as stated in (i). Next, we prove the claimed estimate for the conductor N E of E. This estimate holds trivially if T = Spec(K), and we now assume that T = Spec(K). In what follows, we denote by v a closed point of B. Let f v be the conductor exponent of E K at v. If v ∈ T , then E K has good reduction at v, since E → T is smooth and projective, and we obtain f v = 0. Thus the estimates in To show (ii), we observe that the statement is trivial if T = Spec(K). Hence we may and do assume that T = Spec(K). As in the proof of [36,Lemma 4.1], we see that Minkowski's theorem gives an open subscheme U of B with the following properties. There are at most Then we may and do take coprime elements l, m ∈ O U (U ) such that Let E be an elliptic curve over K defined by the Weierstrass equation We observe that E is geometrically isomorphic to E K . This implies that the j-invariant of E coincides with j and h F (E ) = h F (E). We now prove the claimed estimate for the conductor N E of E . Let Δ and c 4 be the usual quantities associated to the above Weierstrass equation of E , see [64, p. 42]. They take the form If v(Δ) 1, then it follows that v(c 4 ) = 0, since l, m ∈ O U (U ) are coprime and v 2. This implies that the above Weierstrass equation is minimal at v and then [64, p. 196] proves that E is semi-stable at v. We conclude f v 1. Next, we assume v ∈ U ∩ T . In the proof of (i), we showed f v = 0. This implies that f v = 0, since E K is geometrically isomorphic to E and E is semi-stable at v. On combining the above observations, we deduce Therefore, on using the properties of U , we see that the estimates in imply an upper bound for N E as claimed in (ii). It remains to prove (iii). We note that the first assertion of (iii) is trivial if T = Spec(K). If B has trivial class group, then we can take U = B in the proof of (ii): It follows that f v = 0 for any closed point v ∈ T and that This shows that E is the generic fiber of an elliptic curve over T and that N E 2 7d N T . If K = Q, then we obtain that h(P ) 1/2 log|Δ| − 2 log 2, and [64, p. 257] shows that |Δ| 2 12 Δ E . Therefore, Lemma 3.3 proves (iii). This completes the proof of Proposition 3.2.
We remark that the elliptic curve E over Y , which appears in the above proof, represents the moduli problem P = [Legendre] on (Ell) defined in [39, p. 111 in Lemma 3.1. However, to obtain our explicit inequalities in Proposition 3.2 it is necessary to take into account the particular shape of P = [Legendre].

3.2.2.
Mordell equations We continue the notation introduced above and we now give an explicit Paršin construction for Mordell equations. For any non-zero a ∈ O, we obtain that defines an affine Mordell curve over S. To state our next result, we have to introduce some additional notation. If P ∈ Z(S), then we write h(P ) = h(x(P )). Let R K be the regulator of K and let r K be the rank of the free part of the group of units and we observe that κ = 0 when K = Q. The origin of the constant κ shall be explained below Lemma 3.5. To measure the number a ∈ O, we use inter alia the quantity v with the product taken over all closed points v ∈ S with v(a) 1. We observe that log r 2 (a) dh(a) and if a ∈ O K , then r 2 (a) N K/Q (a) for N K/Q the norm from K to Q. (i) The map φ is finite. Furthermore, if ±1 are the only 12th roots of unity in K, then φ is injective.
To prove Proposition 3.4, we shall use a lemma which relates heights of elliptic curves. We recall that E K denotes the generic fiber of an elliptic curve E over S. Let W be a Weierstrass model of E K over B with discriminant Δ W , see, for example, [44,Paragraph 9.4.4] for a definition of W and Δ W . To measure W, we take the height for D the minimal discriminant ideal of E K and a ⊆ O K an ideal with N K/Q (a) 2 D K . For any non-zero β ∈ O K , an application of [32, Lemma 3] with n = 12 gives ∈ O × K such that dh( 12 β) log N K/Q (β) + 12dκ − 6 log(D K ). (This result relies on estimates for certain fundamental units of O K .) Hence, on using (3.7), we obtain a defining Weierstrass equation of W , with quantities c 4 , c 6 and discriminant Δ, such that , and Kodaira-Néron [64, p. 200] gives dh(j) = log Δ 2 + max(1, |j| σ ). Here the product and the sum are both taken over all embeddings σ : K → C and |β| σ denotes the complex absolute value of σ(β) for β ∈ K. Then, on splitting the product according to |Δ| −1 σ > |j| σ + 1728 and |Δ| −1 σ |j| σ + 1728, we deduce from (3.8) the estimate h(c 3 4 , c 2 6 ) log Δ 1 /d + h(j) + 12κ + log(2 · 1728). Hence, on combining 12h(W ) h(c 3 4 , c 3 6 ), (3.5) and log Δ 1 + 12dh F (E) = 12dh(E), we see that W has the desired property. This completes the proof of Lemma 3.5.
The proof shows in addition that one can take in Lemma 3.5 any Weierstrass model W of A defining Weierstrass equation of such a W is called a quasi-minimal Weierstrass equation of E K , see [64, p. 264].
Proof of Proposition 3.4. If Z(S) is empty, then all statements are trivial. Hence we may and do assume that Z(S) is not empty. We for c 4 and c 6 'indeterminates'. We observe that defines an (universal) elliptic curve E over Y . We take P ∈ Z(S). On using that To prove (i), we observe that E is a Weierstrass model of its generic fiber E K . Hence we see that if P ∈ Z(S) satisfies φ(P ) = φ(P ), then there is u ∈ K with u 4 x(P ) = x(P ) and u 6 y(P ) = y(P ), and thus u 12 a = a since P, P ∈ Z(S). Therefore, we deduce (i).
We now show (ii). Let W be the Weierstrass model of E K over B from Lemma 3.5. We denote by Δ, c 4 , c 6 the quantities of a defining Weierstrass equation of W , which we constructed in the proof of Lemma 3.5. We point out that one should not confuse these c 4 , c 6 ∈ O K with the 'indeterminates' which appear in the first part of the proof. On using that E is a Weierstrass Thus Lemma 3.5, (3.8) and Lemma 3.3 lead to an upper bound for h(P ) as stated in (ii).
To prove (iii), we may and do assume that T = Spec(O[1/(6a)]). Let U (respectively, U ) be the set of points v ∈ S − T with v 6 such that E K has (respectively, has not) semi-stable reduction at v. We define Ω = v∈U N v and Ω = v∈U N 2 v and then we deduce To control the unstable part Ω , we may and do assume that U is not empty. We take v ∈ U . The classification of Kodaira-Néron [64, p. 448] gives v(Δ) 2. Since P ∈ Z(S) we see that E extends to a Weierstrass model of E K over S, with discriminant 6 12 To estimate the stable part Ω, we use our assumption that T = Spec(O[1/(6a)]). This assumption implies that any v ∈ S − T with v 6 satisfies v(a) 1. Therefore, we obtain with the product taken over all v ∈ U such that v(a) 1. On combining the displayed inequalities, we deduce (iii). This completes the proof of Proposition 3.4.
We conclude this section with the following remarks. The elliptic curve E over Y , which appears in the proof of Proposition 3.4, represents a moduli problem [Δ = b] on (Ell). Here the moduli problem [Δ = b] is defined similarly as [Δ = 1] in [39, p. 70], but with 1 replaced by the number b which appears in the proof of Proposition 3.4.
The above propositions show that to solve S-unit and Mordell equations, it suffices to estimate effectively h(E) in terms of N E for any elliptic curve E over K. In this paper, we shall prove such estimates for K = Q and we refer to our forthcoming paper 'Height and conductor of elliptic curves' for arbitrary number fields K.
In the special case of S-unit and Mordell equations, it is possible to give ad hoc Paršin constructions which do not use the moduli problem formalism. For example, 'Frey-Hellegouarch curves' provide in principle such a construction for S-unit equations. However, using the moduli problem formalism gives more conceptual constructions, which generalize several known examples such as 'Frey-Hellegouarch curves'.

Variation of Faltings heights under isogenies
In this section, we give a result which controls the variation of Faltings heights under isogenies of elliptic curves over Q. For any elliptic curve E over Q, we denote by h(E) the relative Faltings height of E defined in Section 2. We obtain the following lemma.
To prove this lemma, we combine a result of Faltings-Raynaud in [56] with the classification of cyclic Q-isogenies of elliptic curves over Q of Mazur-Kenku [40,48].
This isogeny ϕ is cyclic, since otherwise it factors through multiplication by an integer which contradicts the minimality of deg(ϕ). Therefore, [40,Theorem 1] gives that deg(ϕ) 163 and then we deduce Lemma 4.1.

Modular forms and modular curves
In the first part of this section, we collect results from the theory of modular forms. In the second part, we work out an explicit upper bound for the modular degree.

Cusp forms
We begin to collect standard results for cusp forms which are given, for example, in the books of Shimura [61] or Diamond and Shurman [21]. We take an integer N 1 and we consider the classical congruence subgroup Γ 0 (N ) ⊂ SL 2 (Z). Let S 2 (Γ 0 (N )) be the complex vector space of cusp forms of weight 2 with respect to Γ 0 (N ). We denote by X 0 (N ) and X(1) smooth, projective and geometrically connected models over Q of the modular curves associated to Γ 0 (N ) and SL 2 (Z), respectively. Throughout Section 5, we denote by d the degree of the natural projection X 0 (N ) → X(1). The dimension of S 2 (Γ 0 (N )) coincides with the genus g of X 0 (N ). Furthermore, it holds g d/12 and d = N (1 + 1/p) if N 2 (5.1) with the product taken over all rational primes p which divide N . Let f ∈ S 2 (Γ 0 (N )) be a non-zero cusp form. If div(f ) denotes the usual rational divisor on X 0 (N ) C of f , then deg(div(f )) = d/6.
For any rational integer n 1, we denote by a n (f ) the nth Fourier coefficient of the cusp form f . Further, we say that f is normalized if a 1 (f ) = 1. We next review properties of the basis of S 2 (Γ 0 (N )) constructed by Atkin and Lehner [3,Theorem 5]. Let S 2 (Γ 0 (N )) new be the new subspace of S 2 (Γ 0 (N )) and let S 2 (Γ 0 (N )) old be the old subspace of S 2 (Γ 0 (N )). There is a decomposition which is orthogonal with respect to the Petersson inner product (· , ·) on S 2 (Γ 0 (N )). We say that f is a newform of level N if f ∈ S 2 (Γ 0 (N )) new is normalized and if f is an eigenform for all Hecke operators on S 2 (Γ 0 (N )). The set B new of newforms of level N is an orthogonal basis of S 2 (Γ 0 (N )) new with respect to (· , ·). Moreover, there exists a basis B old of S 2 (Γ 0 (N )) old with the property that any f ∈ B old takes the form with M ∈ Z 1 a proper divisor of N , with m ∈ Z 1 a divisor of N/M and with f M a newform of level M . Conversely, any f ∈ S 2 (Γ 0 (N )) which is of the form (5.3) is in B old . We say that B = B new ∪ B old is the Atkin-Lehner basis for S 2 (Γ 0 (N )).

Modular degree
Let f ∈ S 2 (Γ 0 (N )) be a newform of level N 1, with all Fourier coefficients rational integers.
In this section, we estimate the modular degree of f in terms of N . We begin with the definition of the modular degree m f of f . Let J 0 (N ) = Pic 0 (X 0 (N )) be the Jacobian variety of X 0 (N ). We denote by T Z the subring of the endomorphism ring of J 0 (N ), which is generated over Z by the usual Hecke operators T n for all n ∈ Z 1 . Let I f be the kernel of the ring homomorphism T Z → Z[{a n (f )}] = Z which is induced by T n → a n (f ).
The modular degree m f of f is defined as the degree of the finite morphism ϕ f .
To estimate m f we shall use properties of the congruence number r f of f . We recall that r f is the largest integer such that there exists a cusp form f c ∈ S 2 (Γ 0 (N )), with rational integer Fourier coefficients, which satisfies (f, f c ) = 0 and a n (f ) ≡ a n (f c ) mod (r f ), n 1. (5.5) It is known that the modular degree m f and the congruence number r f are related. For example, the arguments in Zagier's article [70,Section 5] give For any real number r, we define r = max(m ∈ Z, m r), and for any integer n, we denote by τ (n) the number of positive integers which divide n. The author is grateful to Richard Taylor for proposing a strategy to prove an upper bound for m f . (ii) More precisely, let g be the genus of X 0 (N ) and let d be the degree of the natural projection X 0 (N ) → X(1). Then there exists a subset J ⊂ {1, . . . , d/6 } of cardinality g, which is independent of f, such that m f g g/2 j∈J τ (j)j 1/2 .
Proof. We first show (ii). It follows from (5.6) that m f r f . To estimate r f we reduce the problem to solve (by Cramer's rule) explicitly a system of linear Diophantine equations.
We claim that r f |det(F )|. To verify this claim, we take (k i ) ∈ C g such that f c ∈ S 2 (Γ 0 (N )) from (5.5) takes the form f c = i k i f i . Properties of B show that we may and do take f 1 = f and that (f, f i ) = 0 for any i 2. This implies that k 1 = 0, since (f, f c ) = 0 by (5.5). Therefore, on comparing Fourier coefficients, we see that where (h, x) = j∈J a j (h)x j for h ∈ S 2 (Γ 0 (N )) and (y, x) = j∈J y j x j . We write b = (−1, 0, . . . , 0) ∈ C g . It follows that if x = (x j ) ∈ C g satisfies F (x) = b, then (f, x) = −1, and (f c , x) = 0 by the first equality of (5.7). Hence, the second equality of (5.7) shows that any solution x = (x j ) ∈ C g of F (x) = b satisfies 1 = (y, x)r f . To prove that det(F ) 2 ∈ Z we use (5.3). It gives that a j (f i ) is a coefficient of a newform. Thus it is an eigenvalue of a certain Hecke operator. This implies that all a j (f i ) are algebraic integers. Hence det(F ) and all entries of ξ are algebraic integers. Further, Galois conjugates of newforms are newforms of the same level. Therefore, on using properties of the basis B discussed in Subsection 5.1, we see that any element σ of the absolute Galois group of Q 'permutes' the rows of the matrix F . Hence, we obtain that any such σ satisfies σ(det(F )) = ± det(F ) and we deduce that det(F ) 2 ∈ Z as desired. Then the formulas (5.8) and (5.9) imply r 2 f | det(F ) 2 which proves our claim r f |det(F )|.
To estimate |det(F )| we use the Ramanujan-Petersson bounds for Fourier coefficients, which hold in particular for any newform, and thus for all f i ∈ B by (5.3). These bounds give |a j (f i )| τ (j)j 1/2 for all i ∈ I and j ∈ Z 1 . Thus Hadamard's determinant inequality leads to |det(F )| g g/2 j∈J τ (j)j 1/2 and then the above inequalities imply (ii). It remains to prove (i). Any elliptic curve over Q has conductor at least 11. Therefore, we may and do assume that N 11. Next, we observe that any integer n 1 satisfies the elementary inequalities: (1/n) n k=1 τ (k) 1 + log n and (1 + 1/p) 1 + (log n)/(2 log 2) with the product taken over all rational primes p which divide n. Further, (5.1) shows that 2g d/6 = l and hence the proof of (ii) implies that m f |det(F )| (g!l!) 1/2 τ (j) with the product taken over the elements j of a set J ⊂ {1, . . . , l} of cardinality g. Then the above inequalities and (5.1) lead to (i). This completes the proof of Lemma 5.1.
Frey [29, p. 544] remarked without proof that it is easy to show the asymptotic bound log m f N log N . It seems that this estimate is still very far from being optimal. In fact, Frey [28] and Mai-Murty [47] showed that a certain polynomial upper bound for m f in terms of N is equivalent to a certain version of the abc-conjecture.
The above proof shows in addition that the inequalities of Lemma 5.1 hold with m f replaced by the congruence number r f of f . We note that Murty [51, Corollary 6] used a similar method to prove a slightly weaker upper bound for r f in terms of N . Further, we mention that Agashe-Ribet-Stein proved in [2, Theorem 2.1] that any rational prime number p, with ord p (N ) 1, satisfies ord p (m f ) = ord p (r f ). Moreover, they conjectured in [2, Conjecture 2.2] that ord p (r f /m f ) 1 2 ord p (N ) for all rational prime numbers p.

Height and conductor of elliptic curves over Q
In the first part of this section, we give explicit exponential versions of Frey's height conjecture and of Szpiro's discriminant conjecture for elliptic curves over Q. We also derive an effective version of Shafarevich's conjecture for elliptic curves over Q. In the second part, we prove Propositions 6.1 and 6.4 on combining the Shimura-Taniyama conjecture with lemmas obtained in previous sections.

Height, discriminant and conductor inequalities
Let E be an elliptic curve over Q. We denote by N E the conductor of E, and we denote by h(E) the relative Faltings height of E. See Section 2 for the definitions of N E and h(E). We now can state the following proposition which gives an exponential version of Frey's height conjecture [28, p. 39] for all elliptic curves over Q.
Proposition 6.1. If E is an elliptic curve over Q, then Let K be a number field. On using a completely different method, which is based on the theory of logarithmic forms, we established (see our forthcoming paper 'Height and conductor of elliptic curves') a version of Proposition 6.1 for arbitrary elliptic curves over K. However, in the case of elliptic curves E over Q, this version provides only the weaker inequality h(E) (25N E ) 162 .
As in Subsection 3.2, we denote by Δ E the norm of the usual minimal discriminant ideal of E. Our next result provides an explicit exponential version of Szpiro's discriminant conjecture [67, p. 10] for elliptic curves over Q.

Corollary 6.2. Any elliptic curve E over Q satisfies
Proof. This follows from Proposition 6.1, since log Δ E 12h(E) + 16 by Lemma 3.3.
On combining Arakelov theory for arithmetic surfaces with the theory of logarithmic forms, we obtained in [38] versions of Corollary 6.2 for all hyperelliptic (and certain more general) curves over K. In the case of elliptic curves E over Q, we see that Corollary 6.2 improves the inequality log Δ E (25N E ) 162 provided by [38,Theorem 3.3].
To state our next corollary, we denote by h(W ) the height of a Weierstrass model W of E over Spec(Z), defined in (3.6). Let S be a non-empty open subscheme of Spec(Z) and with the product taken over all rational primes p not in S. We say that an arbitrary elliptic curve E over Q has good reduction over S if E has good reduction at all rational primes in S (This definition is equivalent to the classical notion of good reduction outside a finite set S of rational prime numbers. Indeed S = Spec(Z)S has the structure of a non-empty open subscheme of Spec(Z), and E has good reduction outside S if and only if E has good reduction over S.). It turns out that the number ν S has the property that any elliptic curve E over Q, with good reduction over S, has conductor N E dividing ν S . The Diophantine inequality in Proposition 6.1 leads to the following fully effective version of the Shafarevich conjecture [60] for elliptic curves over Q.

Corollary 6.3. If [E] is a Q-isomorphism class of elliptic curves over Q with good reduction over S, then there exists a Weierstrass model W of E over Spec(Z) that satisfies
In particular, there exist only finitely many Q-isomorphism classes of elliptic curves over Q with good reduction over S and these classes can be determined effectively.
Proof. We take a Q-isomorphism class [E] of elliptic curves over Q, with good reduction over S. Lemma  The first effective version of the Shafarevich conjecture for elliptic curves over Q is due to Coates [17, p. 426]. He applied the theory of logarithmic forms. This theory is also used in [36,Theorem] which provides a version of Corollary 6.3 for arbitrary hyperelliptic curves over K. In the case of elliptic curves over Q, Corollary 6.3 improves the actual best bound h(W ) (2N S ) 1296 which was obtained in [36,Theorem].
We mention that in our forthcoming paper 'Height and conductor of elliptic curves', Section 2) and we give in addition effective asymptotic versions of the above results: S . Further, we there discuss that the exponent 21 + is optimal for the known methods which are based on the theory of logarithmic forms. Thus these methods cannot produce inequalities as strong as those in Proposition 6.1, Corollaries 6.2 and 6.3.
We denote by N (S) the number of Q-isomorphism classes of elliptic curves over Q, with good reduction over S. The explicit height estimate in Corollary 6.3 implies an explicit upper bound for N (S). However, this bound would be exponential in terms of ν S . The following Proposition 6.4 gives an explicit upper bound for N (S) which is polynomial in terms of ν S . The proof uses inter alia the Shimura-Taniyama conjecture and a result of Mazur-Kenku [40] on Q-isogeny classes of elliptic curves.  [55,Theorem 2] an explicit upper bound for N (S). One observes that Proposition 6.4 is better than Poulakis' result when N S 2 65 , and is worse when N S is sufficiently large. However, for sufficiently large N S the actual best estimate is due to Ellenberg, Helfgott and Venkatesh [23,34]. Namely on refining the proof of [34,Theorem 4.5] with the upper bound in [23,Proposition 3.4], one obtains Furthermore, Brumer-Silverman [16] observed that one can considerably improve (6.1) on assuming ( * ): If E is an elliptic curve over Q, with vanishing j-invariant, then the L-function L(E, s) of E satisfies the 'Generalized Riemann Hypothesis' and the rank of E(Q) is at most the order of vanishing of L(E, s) at s = 1. More precisely, [16,Theorem 4] gives that ( * ) implies N (S) N S ; note that the 'Generalized Riemann Hypothesis' together with the 'Birch and Swinnerton-Dyer conjecture' implies ( * ).
We point out that the methods of Brumer-Silverman, Helfgott-Venkatesh and Poulakis are entirely different from the method which is used in the proof of Proposition 6.4. For example, to obtain Diophantine finiteness, they use the following tools: Brumer-Silverman [16] apply an estimate of Evertse-Silverman [25] based on Diophantine approximation, Helfgott-Venkatesh [34] use a bound of Hajdu-Herendi [33] relying on the theory of logarithmic forms, and Poulakis [55] applies an estimate of Evertse [24] based again on Diophantine approximation. 6.2. Proof of Propositions 6.1 and 6.4 Our main tool in the proof of Proposition 6.1 is the Shimura-Taniyama conjecture. Building on the work of Wiles [69] and Taylor-Wiles [68], Breuil-Conrad-Diamond-Taylor [14] proved this conjecture for all elliptic curves over Q. The modularity result in [14] implies the following version of the Shimura-Taniyama conjecture. For any integer N 1, let X 0 (N ) be the modular curve defined in Subsection 5.1. Suppose E is an elliptic curve over Q with conductor N = N E . Then there exists a finite morphism of curves over Q. We mention that the implication [14] ⇒ (6.2) uses inter alia the Tate conjecture which was established by Faltings [26]. To prove Proposition 6.1, we use a strategy of Frey [28]. In the first part, we apply Lemma 4.1 to pass to an elliptic curve over Q, which is Q-isogenous to E and which is an 'optimal quotient'. In the second part, we consider a formula which involves inter alia h(E), the modular degree of the newform attached to E by (6.2), and the 'Manin constant' of E. In the third part, we estimate the quantities which appear in this formula. Here we use inter alia the bound for the modular degree in Lemma 5.1 and a result of Edixhoven in [22] which says that the 'Manin constant' of E is an integer.
Proof of Proposition 6.1. Let E be an elliptic curve over Q with conductor N = N E .
(1) The version of the Shimura-Taniyama conjecture in (6.2) gives a finite morphism ϕ : X 0 (N ) → E of smooth projective curves over Q. We recall that J 0 (N ) = Pic 0 (X 0 (N )) denotes the Jacobian of X 0 (N ). By Picard functoriality, the morphism ϕ induces a surjective Q-morphism of abelian varieties Let A be the identity component of the kernel of ψ. It is an abelian Q-subvariety of J 0 (N ) of codimension 1. We denote by (·) ∨ = Pic 0 (·) the dual. The kernel of the inclusion morphism i : A → J 0 (N ) is geometrically connected. Thus ker(i ∨ ) is connected and has dimension 1 by the dimension formula. It follows that E = (ker(i ∨ )) ∨ is an elliptic curve over Q. The functor Pic 0 (·) is exact for abelian varieties over fields. Hence, on dualizing twice, we obtain a surjective Q-morphism ψ : We deduce that E is Q-isogenous to E (For example, Poincaré's reducibility theorem gives an abelian Q-subvariety E of J 0 (N ) such that addition induces a Q-isogeny A × Q E → J 0 (N ). Then on using that A is a Q-subgroup scheme of the kernels of the surjective Q-morphisms ψ : J 0 (N ) → E and ψ : J 0 (N ) → E , one obtains Q-isogenies E → E and E → E which implies that E and E are Q-isogenous.).
Therefore, E has conductor N and Lemma 4.1 gives The kernel of ψ : J 0 (N ) → E is A, which is connected. An elliptic curve over Q with this property is called an optimal quotient of J 0 (N ) or a (strong) Weil curve. As in Subsection 5.2, we denote by ι : X 0 (N ) → J 0 (N ) the usual embedding which maps ∞ to the zero element of J 0 (N ). To simplify the exposition we write E and ϕ for E and ψ • ι respectively.
(2) It is known by Frey [28, pp. 45-47] that the degree deg(ϕ) of ϕ is related to h(E). A precise relation can be established as follows. We denote by E the Néron model of E over B = Spec(Z). Since Z is a principal ideal domain, the line bundle ω = ω E/B on B from Section 2 takes the form ω ∼ = αZ with a global differential one form α of E. Then, on recalling the definition of the relative Faltings height h(E) in Section 2, we compute As in Subsection 5.1, we denote by S 2 (Γ 0 (N )) the cusp forms of weight 2 for Γ 0 (N ) and by (· , ·) the Petersson inner product on S 2 (Γ 0 (N )). The pullback ϕ * α of α under ϕ defines a differential on X 0 (N ). It takes the form ϕ * α = c · 2πif dz with c ∈ Q × and f ∈ S 2 (Γ 0 (N )) a newform of level N with Fourier coefficients a n (f ) ∈ Z for all n ∈ Z 1 . After adjusting the sign of α, we may and do assume that c is positive. The number c is the Manin constant of the optimal quotient E. By definition, it holds The elliptic curve E f over Q, which is associated to f in (5.4), is Q-isogenous to E. Indeed, this follows for example from [26,Korollar 2] since by construction the L-functions of E and f , of f and E f , and thus of E and E f , have the same Euler product factors for all but finitely many primes. Furthermore, E is an optimal quotient of J 0 (N ) by 1. and E f is an optimal quotient of J 0 (N ), since the kernel I f J 0 (N ) (see Subsection 5.2) of the natural projection J 0 (N ) → E f is connected. Therefore, it follows that the modular degree m f of f , defined in Subsection 5.2, satisfies m f = deg(ϕ). Then, on using that ϕ * α = c · 2πif dz and on integrating over X 0 (N )(C), we see that the change of variable formula and the above displayed formulas for h(E) and (f, f ) lead to We now estimate the quantities which appear on the right-hand side of this formula.
(3) It follows from [1, Lemme 3.7], or from [63, p. 262], that (f, f ) e −4π /(4π). Further, Edixhoven showed in [22,Proposition 2] that the Manin constant c of the optimal quotient E of J 0 (N ) satisfies c ∈ Z and thus we obtain that log(2πc) log(2π). Then the above lower bound for (f, f ), the formula (6.5) and the estimate for m f in Lemma 5.1(i) prove Proposition 6.1 for the optimal quotient E of J 0 (N ). Finally, on using the reduction in (1) and (6.4), we deduce Proposition 6.1 for all elliptic curves over Q.
The main ingredients for the following proof of Proposition 6.4 are the Shimura-Taniyama conjecture and a result of Mazur-Kenku [40] on Q-isogeny classes of elliptic curves over Q.
Proof of Proposition 6.4. Let E be an elliptic curve over Q, with good reduction over S. We write N E for the conductor of E, and we denote by J 0 (N ) = Pic 0 (X 0 (N )) the Jacobian of the modular curve X 0 (N ) for N 1 (see Section 5). There exists a finite morphism X 0 (ν S ) → X 0 (N E ) of curves over Q, since N E divides ν S by (2.2). Picard functoriality gives a surjective morphism J 0 (ν S ) → J 0 (N E ) of abelian varieties over Q, and as in (6.3) we see that the Shimura-Taniyama conjecture provides that E is a Q-quotient of J 0 (N E ). Thus there exists a surjective morphism of abelian varieties over Q. Then Poincaré's reducibility theorem shows that E is Q-isogenous to a Q-simple 'factor' of J 0 (ν S ). Furthermore, the dimension of J 0 (ν S ) coincides with the genus g of the modular curve X 0 (ν S ), and the abelian variety J 0 (ν S ) has at most g Q-simple 'factors' up to Q-isogenies. Therefore, we see that there exists a set of elliptic curves over Q with the following properties: This set has cardinality at most g and for any elliptic curve E over Q, with good reduction over S, there exists an elliptic curve in this set which is Q-isogenous to E. Further, Mazur-Kenku [40,Theorem 2] give that each Q-isogeny class of elliptic curves over Q contains at most eight distinct Q-isomorphism classes of elliptic curves over Q. On combining the results collected above, we deduce that N (S) 8g and then the upper bound for g in (5.1) implies Proposition 6.4.
In the following section, we shall combine Propositions 6.1 or 6.4 with the Paršin constructions from Section 3 to obtain explicit Diophantine finiteness results.

Integral points on moduli schemes
In the first part of this section, we give in Theorem 7.1 an effective finiteness result for integral points on moduli schemes of elliptic curves. In the second and third part, we refine the method of Theorem 7.1 for the moduli schemes corresponding to P 1 − {0, 1, ∞} and to once punctured Mordell elliptic curves. This leads to effective versions of Siegel's theorem for P 1 − {0, 1, ∞} and once punctured Mordell elliptic curves, which provide explicit height upper bounds for the solutions of S-unit and Mordell equations. We also give additional Diophantine applications. In particular, we consider cubic Thue equations.

Moduli schemes
To state our result for integral points on moduli schemes of elliptic curves, we use the notation and terminology which was introduced in Section 3.
Let T and S be non-empty open subschemes of Spec(Z), with T ⊆ S. We write ν T = 12 3 p 2 with the product taken over all rational primes p not in T . For any moduli problem P on (Ell), we denote by |P| T the maximal (possibly infinite) number of distinct level P-structures on an arbitrary elliptic curve over T ; see (3.1). We suppose that Y = M P is a moduli scheme of elliptic curves, which is defined over S. Let h φ be the pullback of the relative Faltings height by the canonical forget P-map φ, defined in (3.3).
Theorem 7.1. The following statements hold.
(i) The cardinality of Y (T ) is at most 2 3 |P| T ν T (1 + 1/p) with the product taken over all rational primes p which divide ν T .
We refer to Paragraph 1.1.2 for a discussion of this theorem. In addition, we now mention that for many classical moduli problems P on (Ell) it is possible to express |P| T in terms of more conventional data, where T is an arbitrary scheme which is connected. For example, if P N is the 'naive' level N moduli problem on (Ell) considered in Section 3, then (3.2) shows that |P N | T is an explicit function in terms of the level N 1.
It is quite difficult, when not impossible, to compare Theorem 7.1 with quantitative or effective finiteness results in the literature, since these results hold in different settings. One can mention for example the quantitative result of Corvaja-Zannier [19] for hyperbolic curves which relies on Schmidt's subspace theorem, or the effective result of Bilu [8] for certain modular curves which is based on the theory of logarithmic forms. It holds that f p 2 for p 5 and (2.2) gives that f 2 8 and f 3 5. Furthermore, if p ∈ T, then we obtain that f p = 0 since E Q extends to an abelian scheme over T . On combining the above results, we deduce that N E | ν T . An application of Proposition 6.1 with E Q gives that h φ (P ) 1 4 N E (log N E ) 2 + 9 which together with N E ν T implies assertion (ii). This completes the proof of Theorem 7.1.
On replacing in the proof of Theorem 7.1(i) the explicit estimate from Proposition 6.4 by the asymptotic bound (6.1) of Ellenberg-Helfgott-Venkatesh, we obtain the following version of We note that the complement of S in Spec(Z) is a finite set of rational prime numbers. For the remaining of Section 7, we will adapt our notation to the classical number theoretic setting, and in (Subsections 7.2-7.4) the symbol S denotes a finite set of rational prime numbers.
Before we apply the method of this paper to S-unit equations (1.1), we briefly review in the following subsection alternative methods which give finiteness of (1.1). 7.2.1. Alternative methods The first finiteness proof for S-unit equations (1.1) goes back to Mahler [46]. He used the method of Diophantine approximations (Thue-Siegel). Another proof of Mahler's theorem was obtained by Faltings, whose general finiteness theorems in [26] cover in particular (1.1). Faltings studied semi-simple -adic Galois representations associated to abelian varieties. Recently, Kim [41] gave a new finiteness proof of (1.1). He used Galois representations associated to the unipotentétale and de Rham fundamental group of P 1 − {0, 1, ∞}. The methods of Faltings, Kim and Thue-Siegel (Mahler) are a priori not effective. The first effective finiteness proof of (1.1) was given by Baker's method, using the theory of logarithmic forms; see, for example, Győry [31] or Baker-Wüstholz [7]. (For instance, Coates explicit result [17], which was published in 1970, implies an effective height upper bound for the solutions of (1.1).) Another effective finiteness proof of (1.1) is due to Bombieri-Cohen [10]. They generalized Bombieri's method in [9], which uses effective Diophantine approximations on the multiplicative group G m (Thue-Siegel principle). The methods of Baker and Bombieri both give explicit upper bounds for the heights of the solutions of (1.1) in terms of S, and they both allow to deal with S-unit equations in any number field. So far, the theory of logarithmic forms, which was extensively polished and sharpened over the last 47 years, produces slightly better bounds than Bombieri's method. On the other hand, Bombieri's method is relatively new and is essentially self-contained; see Bombieri-Cohen [11].

Effective resolution
To state and discuss our effective result for S-unit equations, we put n S = 2 7 N S . Let h(β) be the usual absolute logarithmic Weil height of any β ∈ Q. This height is defined for example in [12, p. 16]. We obtain the following corollary. h(x), h(y) 3 2 n S (log n S ) 2 + 65.
Proof. We use the notation and terminology of Section 3. The discussion in (7.1) shows that we may and do assume that 2 is invertible on T = Spec(O). Write for z an 'indeterminate'. We suppose that (x, y) satisfies (1.1). Then we see that there exists P ∈ X(T ) with z(P ) = x. Thus an application of Proposition 3.2 with P and T gives an elliptic curve E over T that satisfies (write E = E ) h(x) 6h(E) + 11 and N E n S .
Here N E is the conductor of E and h(E) is the relative Faltings height of E, see Subsection 3.2 for the definitions. Proposition 6.1 provides that h(E) 1 4 N E (log N E ) 2 + 9. Then the displayed inequalities imply the claimed upper bound for h(x), and then for h(y) by symmetry. This completes the proof of Corollary 7.2.
As already mentioned in the introduction, this corollary is an effective version of Frey's remark in [29, p. 544 [52]. The main results of [52] independently establish versions of Corollary 7.2 (and of Lemma 5.1, Proposition 6.1 and Corollary 6.2 which are used in the proof of Corollary 7.2) with effective bounds of the form N log N , while our corresponding bounds are of the (slightly) weaker form N (log N ) 2 . The method used in [52] is similar to our proof of Corollary 7.2. To conclude the discussion, we point out that the results were obtained completely independently: We obtained the results of this paper without knowing anything of the related work of Hector Pasten and Ram Murty, and they obtained the results of [52] without knowing anything of our related work).
Corollary 7.2 allows in principle to find all solutions of any S-unit equation (1.1). To discuss a practical aspect of Corollary 7.2, we observe that any u ∈ O × satisfies u = p up with the product taken over all p ∈ S and u p = ord p (u). Therefore, any S-unit equation may be viewed as an exponential Diophantine equation of the form p∈S p xp + p∈S p yp = 1, ((x p ), (y p )) ∈ Z s × Z s for s = |S|. If ((x p ), (y p )) satisfies this exponential Diophantine equation, then Corollary 7.2 implies that max p∈S |x p | and max p∈S |y p | are at most (3/2 log 2)n S (log n S ) 2 + 94. On using additional tricks, we improve in von Känel and Matschke (the forthcoming paper 'Solving Sunit and Mordell equations via Shimura-Taniyama conjecture') the absolute constants 3/2 log 2 and 94 and we will transform the proof of Theorem 7.2 into a practical algorithm to solve S-unit equations.

Comparison to known results
Next, we compare Corollary 7.2 with the actual best effective results in the literature for S-unit equations (1.1). We note that (1.1) has no solutions when |S| = 0, and ( 1 2 , 1 2 ), (2, −1) and (−1, 2) are the only solutions of (1.1) when |S| = 1. Further, we see that if (1.1) has a solution, then 2 ∈ S. Thus, for the purpose of the comparison, we may and do assume s = |S| 2 and 2 ∈ S. (7.1) Let (x, y) be a solution of the S-unit equation (1.1). The actual best explicit height upper bound for (x, y) in the literature is due to Győry-Yu [32]. They used the state of the art in the theory of logarithmic forms. In the case of ( Proof. We use the terminology and notation introduced in Section 3. The discussion in (7.1) shows that we may and do assume that 2 is invertible on T = Spec with all fibers having cardinality at most |P| T . Here |P| T is defined in (3.1) and M (T ) is the set of isomorphism classes of elliptic curves over T . The arguments of Proposition 3.2(iii) and of Theorem 7.1 imply that the cardinality of φ(Y (T )) is at most the number of Q-isomorphism classes of elliptic curves over Q, with conductor dividing n S = 2 7 N S . Therefore, on replacing ν S by n S in the proof of Proposition 6.4, we deduce (1 + 1/p).
Here we used that 2 ∈ S. It follows that Y (T ) has at most 4n S p∈S (1 + 1/p) elements, since the fibers of φ have cardinality at most |P| T 6. Then we conclude Corollary 7.3.
We now compare Corollary 7.3 with results in the literature. Evertse [24, Theorem 1] used the method of Diophantine approximations to prove that any S-unit equation (1.1) has at most 3 · 7 3+2|S| solutions. We mention that Evertse's result holds for more general unit equations in any number field, and it provides, as far as we know, the actual best upper bound in the literature for the number of solutions of (

Once punctured Mordell elliptic curves: Mordell equations
In the first part of this section, we briefly review alternative methods which give finiteness for integral points on once punctured Mordell elliptic curves, or equivalently for the number of S-integer solutions of Mordell equations. In the second and third part, we state and prove Corollary 7.4 on Mordell equations and we compare it with the actual best effective results in the literature. In the fourth and fifth part, we refine a result of Stark and we discuss explicit upper bounds for the number of solutions of Mordell equations.
We continue to denote by S an arbitrary finite set of rational prime numbers and we write O = Z[1/N S ] for N S the product of all p ∈ S. For any non-zero a ∈ O, we recall that Mordell's equation (1.2) is of the form This Diophantine equation is a priori more difficult than S-unit equations (1.1). Indeed, elementary transformations reduce (1.1) to (1.2), while the known (unconditional) reductions of (1.2) to controlled S-unit equations require to solve (1.1) over field extensions. 7.3.1. Alternative methods As already mentioned in the introduction, the resolution of Mordell's equation in Z × Z is equivalent to the classical problem of finding all perfect squares and perfect cubes with given difference. We refer to Baker's introduction of [6] for a discussion (of partial resolutions) of this classical problem, which goes back at least to Bachet 1621. Mordell [49,50] showed that (1.2) has only finitely many solutions in Z × Z. He reduced the problem to Thue equations and then he applied Thue's finiteness theorem which is based on Diophantine approximations. More generally, the completely different methods of Siegel, Faltings [26] and Kim [41,42] give finiteness of (1.2). Siegel's method uses Diophantine approximations, and the methods of Faltings and Kim are briefly described in Paragraph 7.2.1. We mention that these methods, which in fact allow to deal with considerably more general Diophantine problems, are all a priori not effective. See also Bombieri [9] and Kim's discussions in [43]. The first effective finiteness result for solutions in Z × Z of Mordell's equation (1.2) was provided by Baker [6]. Baker's result is based on the theory of logarithmic forms.

Effective resolution
We now state and prove our effective result for Mordell equations. We continue to denote by h(β) the absolute logarithmic Weil height of any β ∈ Q. To measure the set S and the non-zero number a ∈ O, we use inter alia the quantity with the product taken over all rational primes p / ∈ S with ord p (a) 1. The following corollary allows in principle to find all solutions of any Mordell equation (1.2). Proof. The proof is completely analogous to the proof of Corollary 7.2. We use the notation and terminology of Section 3. Write T = Spec(O[1/(6a)]) and define for x 0 and y 0 'indeterminates'. We suppose that (x, y) is a solution of (1.2). Then there exists a T -integral point P ∈ Z(T ) with x 0 (P ) = x and thus an application of Proposition 3.4 with K = Q, P and T gives an elliptic curve E over T that satisfies  Over the last 45 years, many authors improved the explicit bound provided by Baker [6], using refinements of the theory of logarithmic forms; see Baker-Wüstholz [7] for an overview. The actual best explicit upper bound is due to Hajdu-Herendi [33], and due to Juricevic [35] in the important special case O = Z. We first discuss the classical case O = Z. If S = {(10 181 , 4), (10 23 , 5), (10 19 , 6)} and if a ∈ Z − {0}, then Juricevic [35] gives that any solution (x, y) ∈ Z × Z of (1. On using (7.2), we see that Corollary 7.4 improves this inequality and therefore our corollary establishes the actual best result for (1.2) in the classical case O = Z. It remains to discuss the case of arbitrary O. To state the rather complicated bound in [33], we have to introduce some notation. As in [33], we define c 1 = 32 3 Δ 1/2 (8 + 1 2 log Δ) 4 , c 2 = 10 4 · 256 · Δ 2/3 , Δ = 27|a| 2 . Write c S = 7 · 10 38s+86 (s + 1) 20s+35 q 24 max(1, log q) 4s+2 for s = |S| and q = max S. If s 1 and if a ∈ Z − {0}, then the result of Hajdu-Herendi [33, Theorem 2], which in fact holds more generally for any elliptic equation, gives that any solution (x, y) of (1.2) satisfies h(x), h(y) c S c 1 (log c 1 ) 2 (c 1 + 20sc 1 + log(ec 2 )).
It follows from (7.2) that the dependence on a ∈ Z of Corollary 7.4 is of the form |a|(log|a|) 2 , while [33,Theorem 2] is of the weaker form |a| 2 (log|a|) 10 . Further, on using again (7.2), we see that Corollary 7.4 improves [33, Theorem 2] for 'small' sets S, in particular, for all sets S with N S 2 1200 or with s 12. This improvement might be useful for the practical resolution of Mordell equations (1.2), see, for example, von Känel and Matschke (the forthcoming paper 'Solving S-unit and Mordell equations via Shimura-Taniyama conjecture'). However, if N S |a|, then one cannot say which bound is better. The point is that there are sets S with N S |a| for which our result is better than [33,Theorem 2], and vice versa. Finally, we mention that (so far) all effective results for (1.2) in the literature are based on the theory of logarithmic forms, and this theory allows us to deal with more general Diophantine equations over arbitrary number fields; see [7]. This concludes our comparison. for M (T ) the set of isomorphism classes of elliptic curves over T . We notice that the map φ coincides with the map φ constructed in Proposition 3.4. Further, we see that the arguments of Proposition 3.4(iii) and of Theorem 7.1 imply that |φ(Y (T ))| is at most the number of Q-isomorphism classes of elliptic curves over Q, with conductor dividing a S . Therefore, on replacing in the proof of Proposition 6.4 the number ν S by a S , we deduce |φ(Y (T ))| 2 3 a S (1 + 1/p) with the product taken over all rational primes p dividing a S . Proposition 3.4(i) shows that φ is injective and then the displayed inequality implies Corollary 7.6.
We now compare Corollary 7.6 with results in the literature. In the classical case O = Z, the actual best explicit upper bound for the number of solutions of (1.2) is due to Poulakis [55].
We see that Corollary 7.6 is better than Poulakis' result when a S 2 180 , and is worse when a S is sufficiently large. However, for large a S the actual best bound follows from Ellenberg-Helfgott-Venkatesh [23,34]. On combining their results (see, for example, our forthcoming paper 'Height and conductor of elliptic curves'), one obtains that the number of solutions of (1.2) is c s 0 (1 + log q) 2 rad(a) 0.1689 for c 0 an absolute constant, s = |S| and q = max S. This asymptotic bound is better than the asymptotic estimate implied by Corollary 7.6. We point out that the methods of Poulakis [55] and Helfgott-Venkatesh [34] are fundamentally different from the method of Corollary 7.6; see the end of Subsection 6.1 for a brief discussion of the Diophantine results used in the proofs of [34,55]. To conclude our comparison, we mention that Evertse-Silverman [25] applied Diophantine approximations to obtain an explicit upper bound for the number of solutions of (1.2). Their bound involves inter alia a quantity which depends on a certain class number.

Additional Diophantine applications
In this section, we discuss additional Diophantine applications of the Shimura-Taniyama conjecture. In particular, we consider cubic Thue equations.
There are many Diophantine equations which can be reduced to S-unit or Mordell equations, such as for example (super-) elliptic Diophantine equations. Usually these reductions consist of elementary, but ingenious, manipulations of explicit equations and they often require to solve S-unit and Mordell equations over controlled field extensions K of Q. Unfortunately, we cannot use most of the standard reductions, since our results in the previous sections only hold for K = Q. However, we now discuss constructions which allow to reduce certain classical Diophantine problems without requiring field extensions. The famous result of Thue, based on Diophantine approximations, gives that (7.3) has only finitely many solutions. Moreover, Baker [5] used his theory of logarithmic forms to prove an effective finiteness result for Thue equations; see [7] for generalizations. We now suppose that n = 3. To prove (effective) finiteness results for (7.3), we may and do assume by standard reductions that (7.3) has at least one solution and that mΔ = 0. Thus we obtain a smooth, projective and geometrically connected genus one curve X = Proj(Q[x, y, z]/(f − mz 3 )).
See, for example, Silverman [62, p. 401] for details. Moreover, if (u, v) satisfies (7.3) and if P denotes the corresponding Q-point of X, then the definition of ϕ shows that x(ϕ(P )) and y(ϕ(P )) are both in Z and z(ϕ(P )) = 1. In other words, the finite Q-morphism ϕ : X −→ Pic 0 (X) of degree 3 reduces any cubic Thue equation (7.3) to a Mordell equation (1.2) of the form (v ) 2 = (u ) 3 + a, (u , v ) ∈ Z × Z. Therefore, we see that Corollary 7.6 gives a quantitative finiteness result for any cubic Thue equation (7.3). In fact, the above arguments prove more generally that any cubic Thue equation ( At the time of writing, it is not clear to the author how to generalize the method in order to deal with the cases n 4. Such generalizations would be interesting for various reasons. For example, any elliptic Diophantine equation over Z can be reduced to certain controlled Thue equations (7.3) of degree n = 4. This well-known reduction is ingenious, but completely elementary. It only requires the classical reduction theory of quartic binary forms over Z which goes back (at least) to Hermite 1848.