Modularity and integral points on moduli schemes

The purpose of this paper is to give some new Diophantine applications of modularity results. We use the Shimura-Taniyama conjecture to prove effective finiteness results for integral points on moduli schemes of elliptic curves. For several fundamental Diophantine problems (e.g. $S$-unit and Mordell equations), this gives an effective method which does not rely on Diophantine approximation or transcendence techniques. We also combine Faltings' method with Serre's modularity conjecture, isogeny estimates and results from Arakelov theory, to establish the effective Shafarevich conjecture for abelian varieties of (product) GL$_2$-type. In particular, we open the way for the effective study of integral points on certain higher dimensional moduli schemes.


Introduction
Starting with the key breakthroughs by Wiles [Wil95] and by Taylor-Wiles [TW95], many authors solved important Diophantine problems on using or proving modularity results.
The purpose of this paper is to give some new Diophantine applications of modularity results. We use 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.
We also combine Faltings' method with Serre's modularity conjecture, isogeny estimates and results from Arakelov theory, to establish the effective Shafarevich conjecture for abelian varieties of (product) GL 2 -type. In particular, we open the way for the effective study of Diophantine equations related to integral points on certain higher dimensional moduli schemes such as, for example, Hilbert modular varieties. 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 [BG06,p.16].

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 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 [Mah33], Faltings [Fal83b] and Kim [Kim05] proved finiteness of (1.1) by completely different methods. Moreover, Baker's method [Bak68c] or a method of Bombieri [Bom93] 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 Section 7.2.1. In addition, we now point out that Frey remarked in [Fre97,p.544] that the Shimura-Taniyama conjecture implies finiteness of (1.1). It turns out that one can make Frey's remark in [Fre97] 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 uploaded the present paper to the arXiv, Hector Pasten informed us about his joint work with Ram Murty [MP13] in which they independently obtain a (slightly) better version of the displayed height bound (see [MP13,Theorem 1.1]) by using a similar method; we refer to the comments below Corollary 7.2 for more details. We would like to thank Hector Pasten for informing us about [MP13].) 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 parametrized 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 nonzero a ∈ O, one obtains a Mordell equation (1.2) We shall see in Section 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 [Mor22,Mor23], Faltings [Fal83b] and Kim [Kim10] showed finiteness of (1.2) by using completely different proofs, and the first effective result for Mordell's equation was provided by Baker [Bak68b]; see Section 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 get 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 [Bak68b]. Moreover, the displayed estimate improves the actual best upper bounds for (1.2) in the literature and it refines and generalizes Stark's theorem [Sta73]; see Section 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 Sections 7.2-7.4), we will adapt our notation to the algebraic geometry setting and the symbol S will denote a base scheme.

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 Tpoints of the S-scheme Y . Let h M be the pullback of the relative Faltings height by the canonical forget P-map, 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.
Theorem A. 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 .
If the moduli problem P is given with |P| T < ∞, then the explicit upper bound for the height h M in (ii) has the following application: In principle one can determine the abstract set Y (T ) up to a canonical bijection; see the discussion surrounding (3.3). Part (i) gives a quantitative finiteness result for Y (T ) provided that |P| T < ∞. In fact most moduli schemes of interest in arithmetic, in particular all explicit moduli schemes considered in this paper, trivially satisfy |P| T < ∞. However, any scheme over an arbitrary Z[1/2]scheme is a moduli scheme of elliptic curves (see Section 3) and thus there exist many open subschemes S ⊂ Spec(Z) and moduli schemes Y over S such that Y (S) is infinite.
In addition, we show that the Shimura-Taniyama conjecture :=(ST ) allows to deal with other classical Diophantine problems. For example, we consider cubic Thue equations, we derive an exponential version of Szpiro's discriminant conjecture for any elliptic curve over Q, and we deduce an effective Shafarevich conjecture for elliptic curves over Q.
We remark that the theory of logarithmic forms gives more general versions of the results discussed so far, see [vK13, vK]. However, the approach via (ST ) has other advantages. For instance, in the two examples which we worked out explicitly, we obtained upper bounds with numerical constants that are smaller than those coming from the theory of logarithmic forms. Furthermore, in the forthcoming joint work with Benjamin Matschke [vKM13], we will estimate more precisely the quantities appearing in our proofs to further improve our final numerical constants. This will allow us to practically resolve S-unit and Mordell equations with "small" parameters. In fact the practical resolution of these Diophantine equations is still a challenging problem; see for example Gebel-Pethö-Zimmer [GPZ98] for partial results on Mordell's equation. We also point out that (ST ) has in addition the potential to find the solutions of Diophantine equations without using height bounds. For instance, we shall see in the proof of Theorem A that integral points on moduli schemes of elliptic curves correspond to elliptic curves over Q of bounded conductor, which in turn correspond by (ST ) to certain newforms of bounded level and such newforms can be computed by Cremona [Cre97]. We refer to [vKM13] for details.

Principal ideas of Theorem A
We continue the notation of the previous section. Let T ⊆ S ⊆ Spec(Z) be as above and suppose that Y = M (P) is a moduli scheme over S with |P| T < ∞. 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 [Fal83b]. The other tools, such as Frey's estimate, the theory of modular forms and the isogeny results of Mazur-Kenku [Ken82], can be replaced by Arakelov theory and isogeny estimates; see Section 1.2.1 below. In fact the proof of Theorem A may be viewed as an application of a refined Arakelov-Faltings-Paršin method to moduli schemes of elliptic curves.

Effective Shafarevich conjecture
In 1983, Faltings [Fal83b] proved the Shafarevich conjecture [Sha62] for abelian varieties over number fields. It is known that an effective version of the Shafarevich conjecture would have striking Diophantine applications. For example, we show in Section 9 that the following effective Shafarevich conjecture (ES) implies the effective Mordell conjecture for any curve of genus at least 2, defined over an arbitrary number field.
Let S be a non-empty open subscheme of Spec(Z), and let g ≥ 1 be an integer. We denote by h F the stable Faltings height, defined in Section 2.
Conjecture (ES). There exists an effective constant c, depending only on S and g, such that any abelian scheme A over S of relative dimension g satisfies h F (A) ≤ c.
We mention that Conjecture (ES) is widely open if g ≥ 2 and we point out that (ES) implies in particular the "classical" effective Shafarevich conjecture for curves over arbitrary number fields; we refer to Section 9 for a discussion of Conjecture (ES).
Let A be an abelian scheme over S of relative dimension g. We say that A is of GL 2type if there exists a number field F of degree [F : Q] = g together with an embedding F ֒→ End(A) ⊗ Z Q. Here End(A) denotes the ring of S-group scheme morphisms from A to A. More generally, we say that A is of product GL 2 -type if A is isogenous to a product of abelian schemes over S of GL 2 -type; see Section 8 for a discussion of abelian schemes of product GL 2 -type. Write N S = p with the product taken over all rational primes p not in S. We prove in Theorem 9.2 the following result.
Theorem B. If A is of product GL 2 -type, then h F (A) ≤ (3g) 144g N 24 S .
This explicit Diophantine inequality establishes in particular the effective Shafarevich conjecture (ES) for all abelian schemes of product GL 2 -type. In addition, we deduce in Corollary 9.4 new cases of the "classical" effective Shafarevich conjecture for curves, and we derive in Corollary 9.5 new isogeny estimates for any A of product GL 2 -type. Next, we consider the set M GL 2 ,g (S) formed by the isomorphism classes of abelian schemes over S of relative dimension g which are of product GL 2 -type. We obtain in Theorem 9.6 the following quantitative finiteness result for M GL 2 ,g (S).
We deduce in Corollary 9.7 explicit finiteness results for Q-isogeny classes of abelian varieties over Q of product GL 2 -type. Further, we mention that Brumer-Silverman [BS96], Poulakis [Pou00] and Helfgott-Venkatesh [HV06] established the important special case g = 1 of Theorem C. They used completely different arguments which in fact give a better exponent for N S if g = 1. However, their methods crucially depend on the explicit nature of elliptic curves and they do not allow to deal with higher dimensional abelian varieties; see the discussion surrounding Proposition 6.4 for more details.
We remark that the above results open the way for the effective study of classes of Diophantine equations which appear to be beyond the reach of the known effective methods. For instance, Theorems B and C are the main tools of the joint paper with Arno Kret [vKK]. Therein we combine these results with canonical forgetful maps in the sense of (1.3), and we prove quantitative and effective finiteness results for integral points on higher dimensional moduli schemes which parametrize abelian schemes of GL 2 -type. In particular, we work out the case of Hilbert modular varieties.

Principal ideas of Theorems B and C
We continue the notation of the previous section. Let S ⊆ Spec(Z) and g ≥ 1 be as above, and let h F be the stable Faltings height. Suppose that A is an abelian scheme over S of relative dimension g which is of product GL 2 -type. Write A Q for the generic fiber of A.
To prove Theorem B we combine ideas of Faltings [Fal83b] with the following tools:  [Car86] allows to control the number N in (i). This together with (i)-(iii) leads to an effective bound for h F (A) in terms of N A and g, and then in terms of N S and g since A is an abelian scheme over S.
To reduce the general case of Theorem B to the case when A Q is Q-simple, we use inter alia Poincaré's reducibility theorem and the isogeny estimates in (ii).
We now describe the principal ideas of Theorem C. Following Faltings [Fal83b] we divide our quantitative finiteness proof for M GL 2 ,g (S) into two parts: (a) Finiteness of M GL 2 ,g (S) up to isogenies and (b) finiteness of each isogeny class of M GL 2 ,g (S). To prove (a) we use (i) and we show that any Q-simple "factor" A i of A Q is a quotient where ν is an integer depending only on S and g. To show (b) we combine Theorem B with an estimate of Masser-Wüstholz [MW93,MW95] for the minimal degree of isogenies of abelian varieties which is based on transcendence theory. In fact we use here the most recent version of the Masser-Wüstholz estimate, due to Gaudron-Rémond [GR12].
We remark that results from transcendence theory are not crucial to prove Conjecture (ES) for abelian schemes A over S of product GL 2 -type (resp. to effectively estimate |M GL 2 ,g (S)|). However, they lead to an upper bound for h F (A) (resp. for |M GL 2 ,g (S)|) which is exponentially (resp. double exponentially) better in terms of N S and g, than the estimate which would follow by using the results in (ii) based on Faltings' method.

Plan of the paper
In Section 2 we discuss properties of Faltings heights and of the conductor of abelian varieties over number fields. In Section 3 we give Paršin constructions for moduli schemes of elliptic curves and in Section 4 we collect results which control the variation of Faltings heights in an isogeny class. In Section 5 we use the theory of modular forms to bound the modular degree of elliptic curves over Q. We also estimate the stable Faltings heights of certain classical modular Jacobians. 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. In Section 8 we prove a height conductor inequality for abelian varieties over Q of product GL 2 -type. Finally, we establish in Section 9 the effective Shafarevich conjecture (ES) for abelian schemes of product GL 2 -type and we deduce some 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 [vK, vKK]. I would like to thank Richard Taylor for answering several questions, in particular for proposing a first strategy to prove Lemma 5.1. Many thanks go to Bao le Hung, Arno Kret, Benjamin Matschke, Richard Taylor, Jack Thorne and Chenyan Wu for motivating discussions. Parts of the results were obtained when I was a member (2011/12) at the IAS Princeton, supported by the NSF under agreement No. DMS-0635607. I am grateful to the IAS and the IHÉS for providing excellent working conditions. Also, I would like to apologize for the long delay between the first presentation (2011) of the initial results on S-unit and Mordell equations and the completion (2013) of the manuscript. The delay resulted from the attempt to understand the initial examples in a way which is more conceptual and which is suitable for generalizations.

Conventions and notations
We identify a nonzero 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 (resp. v ∤ a) if v(a) = 0 (resp. v(a) = 0). If A is an abelian variety over K with semi-stable reduction at all finite places of K, then we say that A is semi-stable.
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 . Further, if A and B are abelian schemes over S, then we denote by Hom(A, B) the abelian group of S-group scheme morphisms from A to B and we write End(A) = Hom(A, A) for the endomorphism ring of A. Following [BLR90], we say that S is a Dedekind scheme if S is a normal noetherian scheme of dimension 0 or 1.
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 , 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 .
Finally, for any map f :

Height and conductor of abelian varieties
Let K be a number field and let A be an abelian variety over K. In the first part of this section, we recall the definition of the relative and the stable Faltings height of A, and we review fundamental properties of these heights. In the second part, we define the conductor N A of A and we recall useful properties of N A .

Faltings heights
We begin to define the relative and stable Faltings height of A following [Fal83b,p.354].
If A = 0 then we set h(A) = 0. We now assume that A has positive dimension g ≥ 1. Let B be the spectrum of the ring of integers of K. We denote by A the Néron model of A over B, with zero section e : B → A. Let Ω g be the sheaf of relative differential g-forms of A/B. We now metrize the line bundle ω = e * Ω g on B. For any embedding σ : K ֒→ C, we denote by A σ the base change of A to C with respect to σ. We choose a nonzero 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 A σ which is induced by α. To obtain a stable height we may (see [GR72]) and do take a finite extension 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 .
Further, we define h F (0) = 0. We shall need an effective lower bound for h F (A) in terms of the dimension g of A. An explicit result of Bost [Bos96a] gives See for example [GR13,Corollaire 8.4] and notice that h F (A) = h B (A) − g 2 log π where h B denotes the height which appears in the statement of [GR13,Corollaire 8.4].
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 that h F (A) ≤ h(A). 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.
More generally, let S be a connected Dedekind scheme with field of fractions K. If A is an abelian scheme over S, then we define the stable and relative Faltings height of A by h F (A) = h F (A K ) and h(A) = h(A K ) respectively. Here A K is the generic fiber of A.

Conductor
We first define the conductor N A of an arbitrary abelian variety A over any number field K. Let v be a finite place of K. We denote by f v the usual conductor exponent of A at v, see for example [Ser70, Section 2.1] for a definition. The conductor N A of A is defined by with the product taken over all finite places v of K. In particular, f v (0) = 0 and N 0 = 1.
We now recall some useful properties of f v and N A . It holds that f v = 0 if and only if A has good reduction at v. Furthermore, if A ′ is an abelian variety over K which is K-isogenous We shall need an explicit upper bound for f v in terms of g = dim(A) and K. Brumer-Kramer [BK94] obtained such a bound by refining earlier work of Serre [Ser87,Section 4.9] and of Lockhart-Rosen-Silverman [LRS93]. To state the main result of [BK94] we have to introduce some notation. Let p be the residue characteristic of v, let e v = v(p) be the ramification index of v, and let n be the largest integer that satisfies n ≤ 2g/(p − 1).
We define λ p (n) = ir i p i for r i p i the p-adic expansion of n = r i p i with integers 0 ≤ r i ≤ p − 1. Then [BK94,Theorem 6.2] gives Furthermore, the examples in [BK94] show that (2.4) is best possible in a strong sense.
More generally, if S is a connected Dedekind scheme with field of fractions K and if A is an abelian scheme over S, then we define the conductor 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 one. A morphism of elliptic curves over S is a morphism of abelian schemes over S. We denote by  Proof. We notice 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 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 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 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 The height h M has the following properties: If |P| T < ∞, then Lemma 3.1 together with Lemma 3.5 below shows that there exist only finitely many P ∈ Y (T ) with h M (P ) bounded. Furthermore, if P is given with |P| T < ∞, then the proof of Lemma 3.1 together with Lemma 3.5 below implies that one can in principle determine, up to a canonical bijection, the set of points P ∈ Y (T ) with h M (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 Further, we say that any nonzero 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 [BG06, 1.5.6].

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 To simplify notation we write X = P 1 S − {0, 1, ∞}. For any P ∈ X(S), we define h(P ) = h(z(P )). 1 We say that a map of sets is finite if all its fibers are finite. (i) Suppose P ∈ X(S) and [E] = φ(P ). Then it holds N E ≤ 2 6d 3 5d N 2 T and h(P ) ≤ 6h F (E) + 3 log max(1, h F (E)) +42.
(ii) There is an elliptic curve E ′ over K that satisfies h F (E ′ ) = h F (E) and (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 ) ≥ √ 3/2. We write q = exp(2πiτ σ ) and ∆(τ σ ) = q ∞ n=1 (1 − q n ) 24 . From [Sil86, Proposition 1.1] we get with the sum taken over all embeddings σ : K ֒→ C. Here |·| denotes the complex absolute value. Further, on using the elementary inequalities log |∆(τ σ )/q| ≤ 24 |q| /(1 − |q|) and This together with the displayed formula for log ∆ E implies the statement.
We remark that the proof shows in addition that Faltings We shall need an estimate for the conductor. If v is a closed point of B and if f v denotes the conductor exponent at v of an elliptic curve over K (see Section 2.2), then This follows directly from the result of Brumer-Kramer which we stated in (2.4).
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 for λ an "indeterminate". Then we observe that defines an (universal) elliptic curve E over Y . We take P ∈ X(S). On using that Y T ∼ = X T , we obtain a morphism T → Y T induced by P . Let E be the fiber product of E Y T → Y T with this morphism T → Y T . Then E is an elliptic curve over T and therefore we see that defines a map φ : X(S) → M (T ). If P ′ ∈ X(S) satisfies φ(P ) = φ(P ′ ), then it follows that z(P ′ ) = (z 1 − z 2 )/(z 3 − z 2 ) with pairwise distinct z 1 , z 2 , z 3 ∈ {0, 1, z(P )}. Thus φ is finite.
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 that 2v(λ) = v(j) − 8v(2) for any finite place v of K with v(λ) ≤ −1 and that |σ(λ)| 2 ≤ |σ(j)| for any embedding σ : There are at most h K − 1 points in B − U , any v ∈ B − U satisfies N 2 v ≤ D K and the class group of U is trivial. Then we may and do take coprime elements l, m ∈ O U (U ) such that λ = l/m.
Let E ′ be an elliptic curve over K defined by the Weierstrass equation y 2 = x(x−l)(x−m).
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 ∆ and c 4 be the usual quantities associated to the above Weierstrass equation of E ′ , see [Sil09,p.42]. They take the form ∆ = 2 4 (lm(l − m)) 2 and c 4 = 2 4 ((l − m) 2 + lm).
If v(∆) ≥ 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 [Sil09,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 with the product taken over all v ∈ B such that v ∈ B − U or v | 2. Therefore, on using the properties of U , we see that the estimates in (3. It remains to prove (iii). We notice 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 v with the product taken over all v ∈ B with v | 2. 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|∆| − 2 log 2, and [Sil09, p.257] shows that |∆| ≤ 2 12 ∆ 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

Mordell equations
We continue the notation introduced above and we now give an explicit Paršin construction for Mordell equations. For any nonzero 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 (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 where c 4 and c 6 are the usual quantities of a defining Weierstrass equation of W , see [Sil09,p.42]. It turns out that the definition of h(W ) does not depend on the choice of the defining Weierstrass equation of W . We obtain the following lemma.
The proof shows in addition that one can take in Lemma 3.5 any Weierstrass model 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 write b = −a/1728. Let Y be the spectrum of for c 4 and c 6 "indeterminates". We observe that we obtain a morphism T → Y T induced by P . We denote by E the fiber product of It follows that E is an elliptic curve over T and 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 ∆, 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 proof of (i). On using that E is a Weierstrass model of E K over T , we see that there exists u ∈ K that satisfies Thus Lemma 3.5, (3.9) and Lemma 3.3 lead to an upper bound for h(P ) as stated in (ii).
To estimate the conductor N E of E we take a closed point v of B. Let f v be the conductor over T , and if v ∤ 6, then f v ≤ 2. Thus (3.5) implies an estimate for N E as claimed in (ii).
To prove (iii) we may and do assume that be the set of points v ∈ S − T with v ∤ 6 such that E K has (resp. 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 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 [∆ = b] on (Ell).
Here the moduli problem [∆ = b] is defined similarly as [∆ = 1] in [KM85,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, see [vK] 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-Hellegoarch 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-Hellegoarch curves".

Variation of Faltings heights under isogenies
In this section, we collect results which control the variation of Faltings heights under isogenies. These results are rather direct consequences of theorems in the literature.
Let K be a number field and let A be an abelian variety over K of dimension g ≥ 1.
We denote by h F ( Let N A be the conductor of A defined in Section 2.2, let D K be the absolute value of the discriminant of K over Q and let d = [K : Q] be the degree of K over Q.
Lemma 4.1. Suppose A ′ is an abelian variety defined over K which is K-isogenous to A.
Then the following statements hold.
(i) There exists an effective constant µ, depending only on g, N A , d and D K , such that (iii) Suppose K = Q and A is semi-stable. Then any abelian subvariety C of A satisfies The main ingredients for the proof of this lemma are as follows. Raynaud [Ray85] proved Lemma 4.1 (i) for semi-stable abelian varieties. His proof relies on refinements of certain arguments in Faltings [Fal83b]; these refinements are due to Paršin and Zarhin. To prove (i) we reduce the problem to the semi-stable case established in [Ray85]. For this reduction we use the semi-stability criterion of Grothendieck-Raynaud [GR72], the criterion of Néron-Ogg-Shafarevich [ that A L is semi-stable. Let D L be the absolute value of the discriminant of L over Q and let l be the relative degree of L over K. We denote by T the set of finite places of L where A L has bad reduction. Let ℓ and ℓ ′ be the smallest rational primes such that any place in T has residue characteristic different to ℓ and ℓ ′ . An application of [Ray85, Théorème 4.4.9] with the L-isogenous abelian varieties A L and A ′ L implies for µ ′ an effective constant depending only on D L , l, d, |T |, ℓ, ℓ ′ and g. We now estimate these quantities effectively in terms of g, N A , d and D K . The criterion of Néron-Ogg- places v of K such that v ∤ 15 and such that A has good reduction at v. Thus [vK13, Lemma 6.2], which is based on Dedekind's discriminant theorem, gives for t the number of finite places of K where A has bad reduction. It holds |T | ≤ lt, and it is known that l can be explicitly controlled in terms of g (see [GR72]). Further, the explicit prime number theorem in [RS62] gives effective upper bounds for t, ℓ and ℓ ′ in terms of N A . We conclude that µ ′ is bounded from above by an effective constant µ which depends only on g, N A , d and D K . Then (4.2) and the stability of h F prove (i).
To show (ii) we assume that K = Q and g = 1. Let ϕ : A → A ′ be a Q-isogeny of minimal degree among all Q-isogenies A → A ′ . This isogeny ϕ is cyclic, since otherwise it factors through multiplication by an integer which contradicts the minimality of deg(ϕ).
To prove (iii) we assume that K = Q and that A is semi-stable. Let C be an abelian subvariety of A. Then there exists a short exact sequence of abelian varieties over Q. The semi-stability of A provides that C and D are semi-stable as well, see for example [BLR90,p.182]. Therefore [Ull00, Proposition 3.3] implies that showed that these isogeny estimates are effective, and completely explicit constants are given by Gaudron-Rémond [GR12]. For example, if A ′ is an abelian variety over K which is K-isogenous to A, then [GR12, Théorème 1.4] combined with (4.1) gives We remark that on calculating the constant µ in Lemma 4.1 (i) explicitly, it turns out that Lemma 4.1 (i) improves (4.3) in some cases, and vice versa in other cases.

Modular forms and modular curves
In the first part of this section, we collect results from the theory of cusp forms. In the second part, we work out an explicit upper bound for the modular degree of newforms with rational Fourier coefficients. In the third part, we give explicit upper bounds for the stable Faltings heights of the Jacobians of certain classical modular curves.

Cusp forms
We begin to collect results for cusp forms which are given, for example, in the books of Shimura [Shi71] or Diamond-Shurman [DS05]. 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 with the product taken over all rational primes p which divide N . Let f ∈ S 2 (Γ 0 (N )) be a nonzero cusp form. If div(f ) denotes the usual rational divisor on X 0 (N ) C of f , then deg(div(f )) = d/6 (5.2) for deg(div(f )) ∈ Q the degree of div(f ). For any integer n ≥ 1, we denote by a n (f ) the n-th Fourier coefficient of f . 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-Lehner in [AL70, 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 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 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 We denote by ι : X 0 (N ) ֒→ J 0 (N ) the usual embedding over Q, which maps the cusp ∞ to the zero element of J 0 (N ). On composing the embedding ι with the natural projection J 0 (N ) → J 0 (N )/I f J 0 (N ), we obtain a finite morphism 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 ≥ 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 first strategy to prove an upper bound for m f .
with all Fourier coefficients rational integers. Then the following statements hold.
(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 + 1⌋} of cardinality g, which is independent of f , such that m f ≤ g! 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 write l = ⌊d/6 + 1⌋ and we denote by B = {f i , i ∈ I} the Atkin-Lehner basis for . . , l}, we assume the contrary and deduce a contradiction. If . . , l}, then we obtain a nonzero f 0 ∈ S 2 (Γ 0 (N )) with Fourier expansion n≥l a n (f 0 )q n . Hence, f 0 vanishes at ∞ of order at least l > d/6 and this contradicts (5.2). We conclude that F (J) is surjective for J = {1, . . . , l}. Therefore we may and do take J ⊂ {1, . . . , l} such that F = F (J) is an isomorphism.
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 and (f c , x) = 0 by the first equality of (5.7). Hence, the second equality of (5.7) shows 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 Section 5.1, we see that any element σ of the absolute Galois group of Q "permutes" the rows of the matrix F . Hence, we get 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 imply that det(F ) ≤ g! j∈J τ (j)j 1/2 and then the above inequalities give (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 ≤ ⌊1+d/6⌋ = l and hence (ii) implies that m 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 [Fre97,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 [Fre89] and Mai-Murty [MM94] 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 [Mur99, 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 [ARS12, Theorem 2.1] that any rational prime for all rational prime numbers p.

Faltings heights of Jacobians of modular curves
In this section, we give explicit upper bounds for the stable Faltings heights of the Jacobians of certain classical modular curves in terms of their level. These upper bounds are based on a result of Javanpeykar given in [Jav13].
We begin to state the result of Javanpeykar. Let X be a smooth, projective and connected curve overQ of genus g, whereQ is an algebraic closure of Q. We denote by P 1 the projective line overQ and we let D be the set of degrees of finite morphisms X → P 1 which are unramified outside 0, 1, ∞. Belyi's theorem [Bel79] shows that D is non-empty.
The Belyi degree deg B (X) of X is defined by deg B (X) = min D. Let Pic 0 (X) be the Jacobian of X, and let h F be the stable Faltings height defined in Section 2. We recall that h F (0) = 0 and then Javanpeykar's inequality [Jav13, Theorem 1.1.1] gives h F (Pic 0 (X)) ≤ 13 · 10 6 deg B (X) 5 g.
We point out that h F (Pic 0 (X)) is well-defined, since the height h F is stable. Let Γ ⊂ SL 2 (Z) be a congruence subgroup. The associated modular curve has a smooth, projective and connected model X(Γ) overQ. Let g Γ be the genus of X(Γ), and let ǫ ∞ be the number of cusps of X(Γ). The inclusion Γ ⊂ SL 2 (Z) induces a natural projection X(Γ) → X(1)Q and the degree d Γ of this projection satisfies where [SL 2 (Z) : Γ] denotes the index of the subgroup Γ ⊂ SL 2 (Z) and id ∈ SL 2 (Z) denotes the identity. Furthermore, the projection X(Γ) → X(1)Q ramifies at most over the two elliptic points of X(1)Q or over the cusp of X(1)Q, and it holds that X(1)Q ∼ = P 1 . Therefore it follows that deg B (X(Γ)) ≤ d Γ and then the displayed estimate for h F (Pic 0 (X)) implies for J(Γ) = Pic 0 (X(Γ)) the Jacobian of X(Γ).
For any integer N ≥ 1, we consider the classical congruence subgroups Γ 1 (N ) ⊂ SL 2 (Z) and Γ(N ) ⊂ SL 2 (Z), and to ease notation we write J 1 (N ) = J(Γ 1 (N )) and J(N ) = J(Γ(N )). Further, we let J 0 (N ) be the modular Jacobian defined in Section 5.2. On combining the above results, we obtain the following lemma.
Proof. We recall that h F (0) = 0. Hence, to prove the claimed inequalities, we may and do assume that the Jacobians are non-trivial. On combining (5.1) and (5.11), we obtain an upper bound for h F (J 0 (N )) as stated. For Γ = Γ 1 (N ) or Γ = Γ(N ) there exist standard formulas which express [SL 2 (Z) : Γ] and ǫ ∞ in terms of N , see for example [Shi71] or [DS05]. These formulas together with (5.10) and (5.11) imply upper bounds for h F (J 1 (N )) and h F (J(N )) as claimed. This completes the proof of Lemma 5.2.
To conclude this section we discuss results in the literature which are related to Lemma 5.2. We begin with a theorem of Ullmo and we put g = g Γ 0 (N ) . If N ≥ 1 is a square-free integer, then [Ull00, Théorème 1.2] gives the asymptotic upper bound Further, if N ≥ 1 is a square-free integer, with 2 ∤ N and 3 ∤ N , then Jorgenson-Kramer provide in [JK09, Theorem 6.2] the asymptotic formula On combining (5.10) with the above displayed results, one can slightly improve the bounds of Lemma 5.2 for special integers N . However, our proofs of the Diophantine results in the following sections require bounds for all integers N ≥ 1 and thus the above discussed results of Ullmo and Jorgenson-Kramer are not sufficiently general for our purpose.
6 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 [Fre89,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 in [ Proof. This follows from Proposition 6.1, since log ∆ E ≤ 12h(E) + 16 by Lemma 3.3.
On combining Arakelov theory for arithmetic surfaces with the theory of logarithmic forms, we obtained in [vK13] 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 ∆ E ≤ (25N E ) 162 provided by [vK13, 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 (3.7). 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 3 . 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 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.  To prove Proposition 6.1 we use a strategy of Frey [Fre89]. In the first part, we apply Lemma 4.1 (ii) 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 [Edi91] 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 subvariety of J 0 (N ).
Thus, on using for example the standard argument via Poincaré's reducibility theorem, we obtain an elliptic curve E ′ over Q which is Q-isogenous to E and a surjective morphism it follows that E ′ has conductor N and Lemma 4.1 (ii) 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 Section 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 [Fre89,] 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 Section 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 [Fal83b, 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 Section 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 Section 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.
Further, Edixhoven showed in [Edi91, 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 [Ken82] 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 (3.5). 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 Qisogenous 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 [Ken82, Theorem 2] give that each Q-isogeny class of elliptic curves over Q contains at most 8 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 Proposition 6.1 or Proposition 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 discuss additional 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 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 Section 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. for certain modular curves which is based on the theory of logarithmic forms. Furthermore, if p ∈ T then we get 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 M (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 Theorem 7.1 (i): If Y = M (P) is a moduli scheme, defined over S, then |Y (T )| ≪ |P| T N 0.1689 T for N T the product of all rational primes p not in T . Furthermore, the discussion surrounding (6.1) shows that the "Birch and Swinnerton-Dyer conjecture" together with the "Generalized Riemann Hypothesis" implies that for all ǫ > 0 there exists a constant c(ǫ), depending only on ǫ, such that |Y (T )| ≤ c(ǫ)|P| T N ǫ T . We notice 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 (7.2-7.4) the symbol S will denote 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).

Alternative methods
The first finiteness proof for S-unit equations (1.1) goes back to Mahler [Mah33]. He effective. The first effective finiteness proof of (1.1) was given 4 by Baker's method, using the theory of logarithmic forms; see for example Baker-Wüstholz [BW07]. Another effective finiteness proof of (1.1) is due to Bombieri-Cohen [BC97]. They generalized Bombieri's method in [Bom93], 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 [BC03].

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 [BG06,p.16]. We obtain the following corollary. 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 Section 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. 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 will improve in [vKM13] 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 (1.1). We notice 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. We observe that this inequality of Stewart-Yu is better than Corollary 7.2 for all sets S with sufficiently large N S . This concludes our comparison.

Number of solutions
To discuss explicit upper bounds for the number of solutions of S-unit equations (1.1), we recall that n S = 2 7 N S . In the special case of the moduli scheme P 1 Z[1/2] − {0, 1, ∞}, one can refine the proof of Theorem 7.1 and one obtains the following result. Proof. We use the terminology and notation introduced in Section 3. The discussion in Therefore, on replacing ν S by n S in the proof of Proposition 6.4, we deduce |φ(Y (T ))| ≤ 2 3 n S
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

Once punctured Mordell elliptic curves: Mordell equations
In the first part of this section, we briefly review alternative methods which give finiteness 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.

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 [Bak68b] for a discussion (of partial resolutions) of this classical problem, which goes back at least to Bachet 1621.
Mordell [Mor22,Mor23] 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 [Fal83b] and Kim [Kim05,Kim10] give finiteness of (1.2).
Siegel's method uses Diophantine approximations, and the methods of Faltings and Kim are briefly described in Section 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 [Bom93] and Kim's discussions in [Kim12]. The first effective finiteness result for solutions in Z × Z of Mordell's equation (1.2) was provided by Baker [Bak68b]. 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 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 Here N E denotes the conductor of E, and h(E) and h F (E) denote the relative and the stable Faltings height of E respectively; see Section 3.2 for the definitions. Proposition 6.1 provides that h(E) ≤ 1 4 N E (log N E ) 2 + 9 and it holds that h F (E) ≤ h(E). Therefore the displayed inequalities lead to the claimed estimate for h(x), and then for h(y) since

Comparison to known results
In what follows, we compare our Corollary 7.4 with the actual best effective height upper Over the last 45 years, many authors improved the explicit bound provided by Baker [Bak68b], using refinements of the theory of logarithmic forms; see Baker-Wüstholz [BW07] for an overview. The actual best explicit upper bound is due to Hajdu-Herendi [HH98], and due to Juricevic [Jur08] in the important special case O = Z.
It remains to discuss the case of arbitrary O. To state the rather complicated bound in [HH98], we have to introduce some notation. As in [HH98], we define 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.

A refinement of Stark's theorem
We now discuss a refinement of the following theorem of Stark On using (7.2) and the fact that h(a) = log|a| for a ∈ Z − {0}, we see that Corollary 7.5 generalizes and refines Stark's theorem [Sta73, Theorem 1] discussed above.
We remark that Stewart-Yu [SY01] obtained an exponential version of the abc- Corollary 7.6. The number of solutions of (1.2) is at most 2 3 a S p|a S (1 + 1/p) with the product taken over all rational primes p which divide a S .
Proof. In this proof, we use the terminology and notation which we introduced in Section 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 [Pou00]. 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 [HV06,EV07]. On combining their results (see [vK] for details), 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 [Pou00] and Helfgott-Venkatesh [HV06] are fundamentally different from the method of Corollary 7.6; see the end of Section 6.1 for a brief discussion of the Diophantine results used in the proofs of [Pou00] and [HV06]. To conclude our comparison, we mention that Evertse-Silverman [ES86] 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 can not 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.

Thue equations
Let m be an integer and let f ∈ Z[x, y] be an irreducible binary form of degree n ≥ 3, with discriminant ∆. We consider the classical Thue equation The famous result of Thue, based on Diophantine approximations, gives that (7.3) has only finitely many solutions. Moreover, Baker [Bak68a] used his theory of logarithmic forms to prove an effective finiteness result for Thue equations; see [BW07] 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 m∆ = 0.
Thus we get a smooth, projective and geometrically connected genus one curve On using classical invariant theory of cubic binary forms, one obtains a finite Q-morphism ϕ : X → Pic 0 (X) of degree 3 and one computes

Abelian varieties of product GL 2 -type
In the first part of this section, we define and discuss abelian varieties of product GL 2type. Then we state Theorem 8.1 and Proposition 8.2 which provide explicit inequalities relating the stable Faltings height and the conductor of abelian varieties over Q of product GL 2 -type. In the second part, we prove Theorem 8.1 and Proposition 8.2.
Let S be a connected Dedekind scheme, with field of fractions K a number field. Let A be an abelian scheme over S of relative dimension g ≥ 1. We say that A is of GL 2 -type if there exists a number field F of degree [F : Q] = g together with an embedding The terminology GL 2 -type comes from the following property: If A is of GL 2 -type and if Blumenthal type) of abelian varieties of GL 2 -type; see for instance [vdG88].
More generally, we say that A is of product GL 2 -type if A is isogenous to a product A 1 × S . . . × S A n of abelian schemes A 1 , . . . , A n over S which are all of GL 2 -type.

Height and conductor
Let A be an abelian variety over Q of dimension g ≥ 1. We denote by h F (A) the stable Faltings height of A, and we denote by N A the conductor of A. See Section 2 for the definitions of h F (A) and N A . We obtain the following result.
Theorem 8.1. If A is of product GL 2 -type, then the following statements hold.
(i) There is an effective constant k, depending only on g, N A , such that h F (A) ≤ k.
We point out that Theorem 8.1 generalizes Proposition 6.1 which holds for elliptic curves over Q, since any elliptic curve is an abelian variety of GL 2 -type.
Further, we mention that the proofs of Theorem 8.1 (i) and (ii) are in principle the same. The only difference is that in (i) we use isogeny estimates based on "essentially algebraic" methods, and in (ii) we apply isogeny estimates coming from transcendence.
On calculating explicitly the constant k in our proof of (i), it turns out that the resulting inequality is worse than (ii). However, for certain abelian varieties the method of (i) is capable to produce inequalities which are better than (ii). For example, on refining the proof of (i) for semi-stable elliptic curves over Q, we obtain the following asymptotic result in which γ = 0.5772 . . . denotes Euler's constant and f (x) = x log(x) log log x for x ≫ 1.
Proposition 8.2. If E is a semi-stable elliptic over Q, then We remark that if E is a semi-stable elliptic curve over Q with good reduction at the primes 2 and 3, then Proposition 8.2 can be slightly improved to Indeed, this inequality follows on replacing in the proof of Proposition 8.2 the asymptotic estimate (5.12) of Ullmo by the asymptotic formula (5.13) of Jorgenson-Kramer.

Proof of Theorem 8.1 and Proposition 8.2
In the first part of this section, we collect some useful properties of abelian varieties over Q of GL 2 -type. In the second part, we combine these properties with results obtained in Sections 2, 4 and 5.3 to prove Theorem 8.1 and Proposition 8.2.

Preliminaries
To prove Theorem 8.
of abelian varieties over Q. Here J 1 (N ) denotes the usual modular Jacobian, defined for example in Section 5.3. We note that Serre and Ribet used inter alia the Tate conjecture [Fal83b] to prove the implication "Serre's modularity conjecture ⇒ (8.1)".
We now collect additional results which shall be used in the proof of Theorem 8.1.
Assume that A is an abelian variety over Q of GL 2 -type. Then there exists a Q-simple abelian variety B over Q of GL 2 -type, an integer n ≥ 1 and a Q-isogeny gives for any positive integer M that any normalized newform f ∈ S 2 (Γ 1 (M )) satisfies . We now suppose that A is the Q-simple abelian variety over Q of GL 2type, which appears in (8.1). Then on combining the above observations, we see that one can choose the number N in (8.1) such that We are now ready to prove Theorem 8.1 and Proposition 8.2.

Proofs
We continue the notation of the previous section. For any abelian variety A over Q, we denote by h F (A) the stable Faltings height of A.
As already mentioned, our proofs of Theorem 8.1 (i) and (ii) are essentially the same.
We now describe the proof of (ii), which is divided into the following two parts. In the first part, we use (8.1) and (8.3) to show that any abelian variety A as in Theorem 8.1 is Q-isogenous to a product A e i i , where e i ≥ 1 is an integer and A i is an abelian subvariety of J 1 (N i ) for N i ≥ 1 an integer dividing N A i . In the second part, we combine results from Sections 2 and 4 to deduce then an upper bound for h F (A) in terms of h F (J 1 (N i )), dim(J 1 (N i )) and g = dim(A), and then in terms of N i and g by Section 5.3, and finally in terms of N A and g since each N i divides N A i and since N e i A i = N A .
Proof of Theorem 8.1. We take an abelian variety A over Q of dimension g ≥ 1 and we assume that A is of product GL 2 -type.
1. Poincaré's reducibility theorem gives positive integers e i together with Q-simple abelian varieties A i over Q such that A is Q-isogenous to the product A e i i . We write g i for the dimension of A i . The Q-simple "factors" A i of A are unique up to Q-isogeny, and by assumption A is Q-isogenous to a product of abelian varieties over Q of GL 2 -type. Therefore (8.2) implies that A i is Q-isogenous to an abelian variety over Q of GL 2 -type, and thus A i is of GL 2 -type as well. Then the results collected in (8.1) and (8.3) provide a positive integer N i with N g i i = N A i together with a surjective morphism of abelian varieties over Q. Let B i be the identity component of the kernel of J i → A i . It is an abelian subvariety of J i . Then Poincaré's reducibility theorem gives a complementary abelian subvariety A ′ i of J i together with a Q-isogeny A ′ i × Q B i → J i induced by addition. We next verify that A i and A ′ i are Q-isogenous. The kernel of the surjective morphism J i → A i is a Q-subgroup scheme of J i whose dimension coincides with dim(B i ). Hence, the dimension formula implies that the dimensions of A i and A ′ i coincide. Let A ′ i → A i be the morphism obtained by composing the natural inclusion and then with the morphism J i → A i . We recall that the surjective morphism A ′ i × Q B i → J i is induced by addition, and B i is the identity component of the kernel of J i → A i . Therefore we see that the morphism A ′ i → A i is surjective, and thus it is a Q-isogeny since dim(A ′ i ) = dim(A i ). Hence, after replacing A i by A ′ i , we may and do assume that A i is an abelian subvariety of J i and that there exists a Q-isogeny (8.5) 2. We now begin to estimate the heights. For any abelian variety B over Q, we denote by v B the maximal variation of the stable Faltings height h F in the Q-isogeny class of B; since A is an abelian variety over Q which is Q-isogenous to A ′ . Write n i = dim(J i ) and Here we used that the dimension of B i is at most dim(J ′ i ) = n i and that the lower bound for h F (B i ) in (2.2) holds in addition for B i = 0 since h F (0) = 0. To control n i in terms of N i , we consider the modular curve X 1 (N i ) = X(Γ 1 (N i )) over Q defined in Section 5.3. We recall that J i = J 1 (N i ) is the Jacobian of X 1 (N i ) and hence the genus of X 1 (N i ) coincides with the dimension n i of J i . Therefore (5.10) together with [DS05, p.107] implies To bound N J i in terms of N i , we use the classical result of Igusa which says that X 1 (N i ) has good reduction at all primes p ∤ N i . In particular, the Jacobian J i = Pic 0 (X 1 (N i )) of X 1 (N i ) has good reduction at all primes p ∤ N i . It follows that all prime factors of Then (2.4) gives an effective bound for N J i in terms of n i and N i , which together with (8.8) shows that there exists an effective constant c i , depending only on N i , such that We recall that A ′ = A e i i is an abelian variety over Q which is Q-isogenous to A. Hence, we get that N A = N A ′ and then the equality N A i = N g i i in statement (8.4) gives (8.10) In addition the dimension formula gives that g = e i g i . In particular we obtain e i ≤ g.
We now prove (i).
To simplify the bound we used here that one can assume N i ≥ 2 and g ≥ 2. Indeed, the equality N A i = N g i i in statement (8.4) together with Fontaine's result [Fon85] implies that N i ≥ 2, and if g = 1 then Proposition 6.1 combined with h F (A) ≤ h(A) gives an inequality which is even better than (8.11). Finally, it follows from (8.10) that N ′ A divides N A and then (8.11) implies (ii). This completes the proof of Theorem 8.1.
We now discuss a possible variation of the proof of Theorem 8.1. For any integer N ≥ 1, we denote by J(N ) the modular Jacobian defined in Section 5.3. In the proof of Theorem 8.1 it is possible to work with J(N ) instead of J 1 (N ) by using the canonical morphism J(N ) → J 1 (N ). However, the resulting inequality would not be as good as the inequalities provided by Theorem 8.1, since the bounds for dim(J(N )) and h F (J(N )) in terms of N (see Section 5.3) are worse than the corresponding estimates for J 1 (N ).
Proof of Proposition 8.2. We take a semi-stable elliptic curve E over Q as in the proposition. Write N = N E for the conductor of E, and let J 0 (N ) be as in Section 5.2. The Shimura-Taniyama conjecture gives a surjective morphism J 0 (N ) → E of abelian varieties over Q, see (6.3) for details. Then, as in the first part of the proof of Theorem 8.1, we find an abelian subvariety E ′ of J 0 (N ) which is Q-isogenous to E.
The conductor N of the semi-stable abelian variety E is square-free. Therefore we get that J 0 (N ) is semi-stable and then an application of Lemma 4.1 (iii) with the abelian subvariety E ′ of J 0 (N ) gives the inequality Here g denotes the dimension of J 0 (N ), and h(A) denotes the relative Faltings height of an arbitrary abelian variety A over Q. The dimension g coincides with the genus of the modular curve X 0 (N ) defined in Section 5.1. Therefore the upper bound for the genus of We point out that the above described proof of (8.12) for all elliptic curves over Q, which uses e.g. isogeny estimates and results from Arakelov theory, is very different to the proof of Proposition 6.1 which applies inter alia the theory of modular forms. In fact the only common tool is the "geometric" version (6.3) of the Shimura-Taniyama conjecture.

Effective Shafarevich conjecture
In the first part of this section, we discuss several aspects of the effective Shafarevich conjecture. In the second part, we give our explicit version of the effective Shafarevich conjecture for abelian varieties of product GL 2 -type and we deduce some applications. In 5 One uses in addition that h(E) ≤ hF (E) + 5 12 log NE + 2 log 2. This inequality follows by combining the Noether formula, [vK13, Proposition 5.2 (iv)] and the classification of Kodaira-Néron. the third part, we then prove the results of Section 9 and finally in the last part we give a generalization for Q-virtual abelian varieties of GL 2 -type. We mention that Conjecture (ES) would have striking applications to classical Diophantine problems. For example, the following proposition gives that Conjecture (ES) implies the effective Mordell conjecture for curves over number fields.

Effective Shafarevich conjecture
Proposition 9.1. Suppose that Conjecture (ES) holds. If X is a smooth, projective and geometrically connected curve of genus at least 2, defined over an arbitrary number field, then one can determine in principle all rational points of X.
Let K be a number field. In what follows, by a curve over K we always mean a smooth, projective and geometrically connected curve over K. For any curve X over K, we denote by h F (X) the stable Faltings height of the Jacobian Pic 0 (X) of X. In the proof of Proposition 9.1 we will show that Conjecture (ES) implies in particular the following "classical" effective Shafarevich conjecture (ES) * for curves over K.
Conjecture (ES) * . Let T be a finite set of places of K. There exists an effective constant c, depending only on K, T and g, such that any curve X over K of genus g, with good reduction outside T , satisfies h F (X) ≤ c.
We now discuss several aspects of Conjectures (ES) and (ES) * . First, we mention that Conjecture (ES), which implies (ES) * , is a priori considerably stronger than (ES) * .
For example, if X is a curve over K, then Conjecture (ES) would allow in addition to control the finite places of K where the reductions of X and Pic 0 (X) are different; see the discussion at the end of Section 9 for more details. Furthermore, it is shown in [vKK] that already special cases of Conjecture (ES), as Theorem 9.2 below, have direct applications to the effective study of Diophantine equations. On the other hand, one needs to prove Conjecture (ES) * in quite general situations to get effective Diophantine applications.
We remark that de Jong-Rémond [dJR11] established Conjecture (ES) * for curves over K which are geometrically cyclic covers of prime degree of the projective line P 1 K . They combined the method introduced by Paršin [Par72] with the theory of logarithmic forms; see also the proof of [vK13, Theorem 3.2] for some refinements. The results in [dJR11] and [vK13] are not general enough to deduce effective applications for Diophantine equations via the known constructions of Kodaira or Paršin [Par68].
We point out that a geometric analogue of Conjecture (ES) was established by Faltings [Fal83a], see also Deligne [Del87,p.14] for some refinements. Finally, we mention that one can formulate Conjecture (ES) more classically in terms of Q-isomorphism classes of abelian varieties over Q of dimension g, with good reduction outside a finite set of rational prime numbers. However, our formulation of Conjecture (ES) in terms of abelian schemes is more compact and is more convenient for the effective study of integral points on moduli schemes.

Abelian schemes of product GL 2 -type
We continue the notation of the previous section. Let S be a non-empty open subscheme of Spec(Z) and let g ≥ 1 be an integer. We write N S = p with the product taken over all rational prime numbers p which are not in S. The following theorem establishes the effective Shafarevich conjecture (ES) for all abelian schemes of product GL 2 -type.
Theorem 9.2. Let A be an abelian scheme over S of relative dimension g. If A is of product GL 2 -type, then We point out that the bound in Theorem 9.2 is polynomial in terms of N S . In course of the proof of Theorem 9.2 we shall obtain the more precise inequality (9.7), which improves in particular the estimate of Theorem 9.2 and which is polynomial in terms of the relative dimension g of A. Moreover, it is possible to refine (9.7) in special cases. For example, we obtain the following result for semi-stable abelian varieties of product GL 2 -type.
Proposition 9.3. Let A be an abelian scheme over S of relative dimension g. If A is of product GL 2 -type and if the generic fiber of A is semi-stable, then Next, we deduce from Theorem 9.2 new cases of the "classical" effective Shafarevich conjecture (ES) * . We say that a curve X over Q is of product GL 2 -type if the Jacobian Pic 0 (X) of X is of product GL 2 -type. There exist many curves over Q of genus ≥ 2 which are of product GL 2 -type, see for example the articles in [CLQR04]. Let T be a finite set of rational prime numbers and write N T = p with the product taken over all p ∈ T .
Corollary 9.4. Let X be a curve over Q of genus g which is of product GL 2 -type. If Pic 0 (X) has good reduction outside T , then Proof. The Néron model of Pic 0 (X) over S = Spec(Z) − T is an abelian scheme, since Pic 0 (X) has good reduction outside T . Therefore Theorem 9.2 implies Corollary 9.4.
If X is a curve over Q with good reduction at a rational prime p, then Pic 0 (X) has good reduction at p. This shows that Corollary 9.4 establishes in particular the "classical" effective Shafarevich conjecture (ES) * for all curves over Q of product GL 2 -type.
We now derive new isogeny estimates for abelian varieties over Q of product GL 2 -type.
Masser-Wüstholz bounded in [MW93,MW95] the minimal degree of isogenies of abelian varieties. On combining Theorem 9.2 with the most recent version of the Masser-Wüstholz results, due to Gaudron-Rémond [GR12], we obtain the following corollary.
Corollary 9.5. Suppose that A and B are isogenous abelian schemes over S of relative dimension g. If A or B is of product GL 2 -type, then the following statements hold.
We point out that these isogeny estimates are independent of A and B. This is absolutely crucial for certain Diophantine applications such as for example [vKK] or Theorem 9.6 below. On calculating the constant of Lemma 4.1 (i) explicitly, we see that Corollary 9.5 (ii) is exponentially better in terms of N S and g than Lemma 4.1 (i). We also note that Corollary 9.5 (ii) holds with h F replaced by the relative Faltings height h.
We denote by M GL 2 ,g (S) the set of isomorphism classes of abelian schemes over S of relative dimension g which are of product GL 2 -type. Corollary 9.5 (i) is one of the main ingredients for the proof of the following quantitative finiteness result for M GL 2 ,g (S).
Theorem 9.6. The cardinality of We refer to Section 6 for a discussion of the important special case g = 1 of Theorem 9.6. To state some consequences of Theorem 9.6 for Q-isomorphism classes of abelian varieties over Q, we recall that T denotes a finite set of rational prime numbers and we let N T = p∈T p be as above. We obtain the following corollary.
Corollary 9.7. Let A be an abelian variety over Q of dimension g. We assume that A has the following properties: (a) A is of product GL 2 -type and (b) A has good reduction outside T . Then the following statements hold.
(i) Up to Q-isomorphisms, there exist at most (14g) (9g) 6 N (18g) 4 T abelian varieties over Q which are Q-isogenous to A.
(ii) Up to Q-isogenies, there exist at most (3g) 32g 2 N 4g T abelian varieties over Q of dimension g which have the properties (a) and (b). We remark that it is possible to prove a considerably more general version of Corollary 9.7 (ii) by refining Faltings' proof of [Fal83b,Satz 5] with an effectiveČebotarev density theorem; see for example Deligne [Del85]. However, the resulting unconditional bound for the number of isogeny classes would be worse than the estimate in Corollary 9.7 (ii).

Proof of the results of Section 9
In the first part of this section, we collect useful results for abelian schemes. In the second part, we first show Theorem 9.2, Proposition 9.3 and Corollary 9.5, then we prove Theorem 9.6 and Corollary 9.7, and finally we give the proof of Proposition 9.1.

Preliminaries
Let S be a connected Dedekind scheme, with field of fractions K. We begin to prove useful properties of morphisms of abelian schemes over S. Suppose that A and B are abelian schemes over S with generic fibers A K and B K respectively. Then base change from S to K induces an isomorphism of abelian groups Hom(A, B) ∼ = Hom(A K , B K ). (9.1) We now verify (9.1). Any abelian scheme over S is the Néron model of its generic fiber, see for example [BLR90,p.15]. Thus the Néron mapping property gives that any K-scheme morphism ϕ K : A K → B K extends to an unique S-scheme morphism ϕ : A → B. In addition, if ϕ K is a K-group scheme morphism, then ϕ is a S-group scheme morphism.
Therefore we see that base change from S to K induces a bijection of sets Hom(A, B) ∼ = Hom(A K , B K ). Finally, base change properties of group schemes show that this bijection is in fact a homomorphism of abelian groups and hence we conclude (9.1). Furthermore, base change from S to K induces an isomorphism of rings Indeed, if A = B then the isomorphism of abelian groups in (9.1) is an isomorphism of rings, since base change from S to K is a covariant functor from S-schemes to K-schemes.
We shall use the following property of semi-stable abelian varieties. Next, we give Lemma 9.8 which will allow us later to control the conductor of certain abelian varieties. In this lemma, we assume that K is a number field with ring of integers O K and we assume that S is a non-empty open subscheme of Spec(O K ). We write d = Lemma 9.8. Suppose that A is an abelian scheme over S of relative dimension g. Then the following statements hold.
(i) There exists a positive integer ν, depending only on K, S and g, such that N A | ν and such that ν ≤ (2g + 1) 6gdρ N 2g S .
(ii) If the generic fiber of A is semi-stable, then N A | N g S .
Proof. The generic fiber A K of A has good reduction at all closed points of S, since A is an abelian scheme over S. Therefore N A takes the form  [GR72,p.364]. Our additional assumption in (ii), that A K is semi-stable, implies that u v = 0 and δ v = 0. Therefore we deduce that Then the displayed formula for N A shows that N A | N g S which proves (ii). This completes the proof of Lemma 9.8.
We are now ready to prove the results of Section 9.

Proofs
We continue the notation of the previous section and we assume that K = Q. Let S be a non-empty open subscheme of Spec(Z), and let N S and g ≥ 1 be as above. We assume that A is an abelian scheme over S of relative dimension g which is of product GL 2 -type.
Let h F (A) be the stable Faltings height of A.
The principal ideas of the proof of Theorem 9.2 are as follows. Theorem 8.1 together with Lemma 9.8 implies directly Conjecture (ES) for A, with an inequality of the form S for c(g) a constant depending only on g. However, to obtain the better bound h F (A) ≤ c(g)N 24 S and to improve the dependence on g of c(g), we go into the proof of Theorem 8.1 and we apply therein Lemma 9.8 with the simple "factors" of A.
Proof of Theorem 9.2. 1. By assumption A is isogenous to a product of abelian schemes over S which are all of GL 2 -type. Then (9.2) gives that the generic fibers of these abelian schemes are all of GL 2 -type as well. It follows that the generic fiber A Q of A is Q-isogenous to a product of abelian varieties over Q of GL 2 -type. In other words, the abelian variety A Q is of product GL 2 -type and thus satisfies all the assumptions of Theorem 8.1.
2. We now go into the proof of Theorem 8.1 (ii). Therein we showed the existence of positive integers N i and e i , together with Q-simple abelian varieties A i over Q of dimension g i , such that A Q is Q-isogenous to A e i i and such that Here N A i denotes the conductor of A i . Furthermore, on following the proof of Theorem 8.1 (ii) and on calculating the first term in the upper bound for h F (A) given in (8.11) more precisely, we obtain the sharper inequality h F (A) ≤ e i 18 · 10 3 N 12 i + (8g) 6 log N i . (9.5) 3. Next, we estimate the numbers N i in terms of g and S. There exists a surjective morphism A Q → A i of abelian varieties over Q, and the abelian variety A Q has good reduction at all closed points of S since it extends to an abelian scheme over S. Therefore [ST68, Corollary 2] provides that A i has good reduction at all closed points of S, and this shows that the Néron model A i of A i over S is an abelian scheme over S. Then an application of Lemma 9.8 with the abelian scheme A i over S of conductor N A i gives that where ρ i = ρ(S, g i ) denotes the number of rational primes p / ∈ S with p ≤ 2g i + 1. Further, since A Q is isogenous to A e i i , we obtain g = e i g i . (9.6) It follows that g i ≤ g and this leads to ρ i ≤ ρ = ρ(S, g). Then the above upper bound for N A i together with (9.4) proves that N i ≤ (2g + 1) 6ρ N 2 S . 4. We observe that ρ ≤ 2g and (9.6) implies that e i ≤ g. Therefore, on combining (9.5) with the above estimate for N i , we deduce an inequality as claimed by Theorem 9.2.
To simplify the form of the final result, we assumed here that g ≥ 2. In fact, in course of the proof of Theorem 7.1 we obtained an inequality which directly implies the remaining case g = 1. This completes the proof of Theorem 9.2.
We recall that ρ = ρ(S, g) denotes the number of rational primes p / ∈ S with p ≤ 2g +1.
The proof of Theorem 9.2 gives in addition the following more precise result: If A is an abelian scheme over S of relative dimension g and if A is of product GL 2 -type, then Let s be the number of rational primes which are not in S. It follows that ρ ≤ s < ∞ and then we see that (9.7) is polynomial in terms of g, since s depends only on S. Furthermore, on looking for example at products of elliptic curves over S, we see that any upper bound for h F (A) has to be at least linear in terms of g. This shows that the polynomial dependence on g of (9.7) is already "quite close" to the optimum. On the other hand, Lemma 9.8 implies that Frey's height conjecture [Fre89,p.39] would give an upper bound for h F (A) which is linear in terms of log N S , while (9.7) depends polynomially on N S . We remark that an (effective) estimate for h F (A) which is linear in terms of log N S would be very useful, since such an estimate would imply inter alia a (effective) version of the abc-conjecture.
In the following proof of Proposition 9.3, we use the arguments of Theorem 9.2 and we replace therein Lemma 9.8 (i) by Lemma 9.8 (ii).
Proof of Proposition 9.3. We freely use the notations and definitions of the proof of Theorem 9.2. In addition, we assume that A Q is semi-stable. Therefore (9.3) implies that A i is semi-stable, since there exists a surjective morphism A Q → A i of abelian varieties over Q. We showed that A i extends to an abelian scheme A i over S. Thus an application of Lemma 9.8 (ii) with the abelian scheme A i over S of conductor N A i and relative dimension g i gives that N A i | N g i S . Hence, the equality N g i i = N A i in (9.4) implies that N i ≤ N S and then (9.6) together with the upper bound for h F (A) in (9.5) leads to an inequality as claimed. This completes the proof of Proposition 9.3.
Proof of Corollary 9.5. We suppose that A and B are isogenous abelian schemes over S of relative dimension g. Let A Q and B Q be the generic fibers of A and B respectively. By assumption A or B is of product GL 2 -type. Thus both are of product GL 2 -type.
To show (i) we observe that the constant κ(A Q ) in [GR12] depends only on g and h F (A). Let κ be the constant which one obtains by replacing the number h F (A) with (3g) 144g N 24 S in the definition of κ(A Q ); notice that κ depends only on N S and g. An application of Theorem 9.2 with A shows that κ(A Q ) ≤ κ. The abelian varieties A Q and B Q are Q-isogenous. Therefore [GR12, Théorème 1.4] gives Q-isogenies ϕ Q : A Q → B Q and ψ Q : B Q → A Q of degree at most κ(A Q ) ≤ κ. As in the proof of (9.1) we see that ϕ Q and ψ Q extend to S-group scheme morphisms ϕ : A → B and ψ : B → A respectively.
Furthermore, it follows from [BLR90,p.180] that ϕ and ψ are isogenies since A and B are in particular semi-abelian schemes over S. Hence we conclude (i).
It remains to prove (ii). We showed in (i) that there is a Q-isogeny ϕ Q : for h the relative Faltings height. Hence we deduce a version of (ii) involving h. To prove the version involving h F we use [GR72]. It provides a number field L such that A L and B L are semi-stable, and thus h F (A) = h(A L ) and h F (B) = h(B L ). If ϕ L : A L → B L is the base change of ϕ Q , then (4.1) gives that |h(A L ) − h(B L )| ≤ 1 2 log deg(ϕ L ). Therefore deg(ϕ L ) = deg(ϕ Q ) ≤ κ leads to (ii). This completes the proof of Corollary 9.5.
We refer to the introduction for an outline of the following proof of Theorem 9.6.
Proof of Theorem 9.6. We recall that M GL 2 ,g (S) denotes the set of isomorphism classes of abelian schemes over S of relative dimension g which are of product GL 2 -type. To bound |M GL 2 ,g (S)| we may and do assume that M GL 2 ,g (S) is not empty. 1. We denote by M GL 2 ,g (S) Q the set of Q-isomorphism classes of abelian varieties over Q of dimension g which extend to an abelian scheme over S and which are of product GL 2 -type. Base change from S to Q induces a canonical bijection To verify this statement we observe that M GL 2 ,g (S) coincides with the set of S-scheme isomorphism classes generated by abelian schemes over S of relative dimension g which are of product GL 2 -type. Further, it follows from (9.2) that the generic fiber A Q of any [A] ∈ M GL 2 ,g (S) is of product GL 2 -type. Thus base change from S to Q induces a map M GL 2 ,g (S) → M GL 2 ,g (S) Q , which is surjective by (9.2) and [BLR90,p.180]. The abelian scheme A is the Néron model of A Q over S and then the Néron mapping property shows that M GL 2 ,g (S) → M GL 2 ,g (S) Q is injective. We conclude that M GL 2 ,g (S) ∼ = M GL 2 ,g (S) Q .
2. Next, we estimate the number of distinct Q-isogeny classes of abelian varieties over Q generated by M GL 2 ,g (S) Q . Let [A] ∈ M GL 2 ,g (S) Q . In the proof of Theorem 8.1 we constructed positive integers N i and e i , together with Q-simple abelian varieties A i over Q of dimension g i and of conductor N A i = N g i i , such that A is Q-isogenous to A e i i and such that A i is a Q-quotient of J 1 (N i ). Here J 1 (N ) denotes the usual modular Jacobian of level N ∈ Z ≥1 defined in Section 5.3. The abelian variety A extends to an abelian scheme over S, since [A] ∈ M GL 2 ,g (S) Q . Thus each A i extends to an abelian scheme over S and then the arguments of the proof of Lemma 9.8 together with g i ≤ g lead to N A i | ν g i for ν = N 2 S p cp , c p = 6 + 2⌊log(2g)/ log p⌋.
Here the product is taken over all rational primes p / ∈ S with p ≤ 2g + 1, and for any real number x we write ⌊x⌋ for the largest integer at most x. We warn the reader that the displayed number ν is related to the number appearing in Lemma 9.8 (i), but these numbers are not necessarily the same. It follows that N i | ν since N g i i = N A i , and this implies that J 1 (N i ) is a Q-quotient of J 1 (ν). On using that A i is a Q-quotient of J 1 (N i ), we then see that there exists a surjective morphism of abelian varieties over Q Hence Poincare's reducibility theorem shows that each A i is Q-isogenous to a Q-simple "factor" of J 1 (ν). Furthermore, the dimension of J 1 (ν) coincides with the genus g ν of the modular curve X 1 (ν) = X(Γ 1 (ν)) defined in Section 5.3, and the abelian variety J 1 (ν) (resp. A) has at most g ν (resp. g) Q-simple "factors" up to Q-isogenies. Therefore there exists a set of at most g·g g ν distinct abelian varieties over Q such that any [A] ∈ M GL 2 ,g (S) Q is Q-isogenous to some abelian variety in this set. In other words, the abelian varieties in M GL 2 ,g (S) Q generate at most g · g g ν distinct Q-isogeny classes of abelian varieties over Q. 3. To bound the size of each Q-isogeny class we take an arbitrary [A] ∈ M GL 2 ,g (S) Q .
We denote by C the set of Q-isomorphism classes of abelian varieties over Q which are Q-isogenous to A. Let κ be the constant which appears in the proof of Corollary 9.5. If [B] ∈ C then the proof of Corollary 9.5 provides a Q-isogeny ϕ : A → B of degree at most κ. Furthermore, the quotient of A by the kernel of ϕ is an abelian variety over Q which is Q-isomorphic to B. On combining the above observations, we see that |C| is bounded from above by the number of subgroups of A t of order at most κ, where A t is the group of torsion points of A. It holds that A t ∼ = (Q/Z) 2g , and [MW93, Lemma 6.1] gives that (Q/Z) 2g has at most κ 2g subgroups of order at most κ. Hence we deduce that |C| ≤ κ 2g .
The arguments used in the proof of Theorem 9.6 give in addition Corollary 9.7.
Proof of Corollary 9.7. We observe that part 3. of the proof of Theorem 9.6 implies (i), and we notice that (ii) follows from part 2. of the proof of Theorem 9.6.
We now prove Proposition 9.1. In the first part of the proof we show that Conjecture (ES) implies Conjecture (ES) * , and in the second part we use the effective version of the Kodaira construction due to Rémond [Rém99].
Proof of Proposition 9.1. We recall some notation. Let K be a number field of degree d = [K : Q], with ring of integers O K . We denote by D K the absolute value of the discriminant of K over Q. Let h F be the stable Faltings height and let T be a finite set of places of K. We write N T = N v with the product taken over all finite places v ∈ T . Let X be a smooth, projective and geometrically connected curve over K of genus g ≥ 1.
1. To prove that Conjecture (ES) implies (ES) * we assume that Conjecture (ES) holds. In addition, we suppose that the Jacobian J K = Pic 0 (X) of X has good reduction outside T . The Weil restriction A Q = Res K/Q (J K ) of J K is an abelian variety over Q of dimension n = dg, which is geometrically isomorphic to J σ K . Here the product is taken over all embeddings σ from K into an algebraic closure of K, and J σ K is the base change of J K with respect to σ. The Galois invariance h F (J K ) = h F (J σ K ) implies that h F (A Q ) = dh F (J K ). Let S be the open subscheme of Spec(Z) formed by the generic point together with the closed points where A Q has good reduction. The Néron model A of A Q over S is an abelian scheme. Therefore an application of Conjecture (ES) with A, S and n gives an effective constant c, depending only on S and n, such that dh F (J K ) = h F (A Q ) ≤ c. (9.8) We write D = {D K , d, g, N T } and we now construct an effective constant c ′ , depending only on D, such that c ≤ c ′ . The finite places in T form a closed subset of Spec(O K ), whose complement S ′ has the structure of an open subscheme of Spec(O K ). The Néron model J of J K over S ′ is an abelian scheme, since J K has good reduction outside T . We denote by N J and N A the conductors of J K and A Q respectively. A result of Milne [Mil72,Proposition 1] gives that N A = N J D 2g K , and an application of Lemma 9.8 (i) with the abelian scheme J over S ′ of relative dimension g implies that N J ≤ ΩD −2g K for Ω = (3g) 12g 2 d (N T D K ) 2g .
We deduce that N A ≤ Ω and this leads to U and n. We define c ′ = max c U with the maximum taken over all U ∈ U . If D is given, then the set U can be determined effectively. Thus we see that c ′ is an effective constant, depending only on D. On using that S ∈ U , we obtain that c ≤ c ′ and then (9.8) gives In other words, we proved that Conjecture (ES) would give an effective constant c ′ , depending only on D, with the following property: If J K has good reduction outside T , then h F (J K ) ≤ c ′ . Further, if X has good reduction at a finite place v of K, then J K has good reduction at v. Therefore we conclude that Conjecture (ES) implies (ES) * .
2. It follows from part 1. that Conjecture (ES) implies (ES) * . Furthermore, [Rém99] gives that Conjecture (ES) * implies that the set of rational points of X can be determined effectively if g ≥ 2. This completes the proof of Proposition 9.1.
We remark that the above proof of Proposition 9.1 assumes the validity of Conjecture (ES) in quite general situations. In particular, it is a priori not possible to use the above arguments in order to deduce special cases of the effective Mordell conjecture from special cases of Conjecture (ES) such as for example Theorem 9.2. To "transfer" special cases between these conjectures, an effective version of Paršin's construction [Par68] would be more useful than Kodaira's construction which is used in the proof of Proposition 9.1.
We mention that the implication (ES) * ⇒ (ES) remains an interesting open problem, which is non-trivial since (ES) is a priori considerably stronger than (ES) * . To discuss parts of the additional information contained in Conjecture (ES), we consider an arbitrary hyperelliptic curve X of genus g ≥ 2 over a number field K. Let T be the set of finite places of K where Pic 0 (X) has bad reduction. Suppose that v is a finite place of K where X has bad reduction but Pic 0 (X) has good reduction; the minimal regular model of X over Spec(O v ) is then automatically semi-stable for O v the local ring at v. Then on combining the arguments of [vK13, Proposition 5.1 (i)] with part 1. of the proof of Proposition 9.1, we see that already very special cases of Conjecture (ES) would give an effective estimate for N v in terms of K, g and T . We note that Levin [Lev12] proved that such an effective estimate for N v would solve the following classical problem: Give an effective version of Siegel's theorem for arbitrary hyperelliptic curves of genus g ≥ 2 defined over a number field K. In fact the latter problem is already open for g = 2 and K = Q.
We also point out that one can improve our inequalities for abelian varieties with "real multiplications": Let A be an abelian variety over Q of positive dimension g, with End(A) ⊗ Z Q a totally real number field of degree g over Q. Serre showed in [Ser87, Théorème 5] that Serre's modularity conjecture (see Section 8.2.1) gives that A is a Qquotient of J 0 (N ), where J 0 (N ) is defined in Section 5.2 and N is the positive integer whose g-th power equals the conductor of A. Then, on combining the bounds for h F (J 0 (N )) in Lemma 5.2 with the arguments of Theorem 8.1 and Theorem 9.2, we see that these results hold with better inequalities for abelian varieties such as A.

Effective Shafarevich for Q-virtual abelian varieties of GL 2 -type
In this section, we show that our method allows in addition to deal with certain more general abelian varieties over arbitrary number fields. This generalization is required for the effective study (see [vKK]) of those S-points on Y which correspond to abelian schemes that are not necessarily defined over S. Here S is a non-empty open subscheme of Spec(Z) and Y is a certain coarse moduli scheme over S (e.g. Hilbert modular variety).
Following Wu [Wu11], we now define Q-virtual abelian varieties of GL 2 -type. They generalize in particular the Q-Hilbert-Blumenthal abelian varieties of Ribet [Rib94]. Let Q be an algebraic closure of Q. Write G Q = Gal(Q/Q) for the absolute Galois group of Q.
Let g ≥ 1 be an integer and let A be an abelian variety overQ of dimension g. We assume that there is a number field F of degree [F : Q] = g together with an embedding For any σ ∈ G Q and for any ϕ ∈ End 0 (A), we denote by A σ and ϕ σ ∈ End 0 (A σ ) the by σ :Q →Q induced base changes of A and ϕ respectively. In addition we assume that for any σ ∈ G Q there exists an isogeny µ σ : A σ → A such that µ σ • ϕ σ = ϕ • µ σ for all ϕ ∈ End 0 (A). (9.10) For any abelian variety A overQ of dimension g, we say that A is a Q-virtual abelian variety of GL 2 -type if A satisfies (9.9) and (9.10) and we say that A is non-CM if End 0 (A) contains no commutative Q-algebra of degree 2g. For instance, if E is a non-CM elliptic curve over Q which is isogenous to all its G Q -conjugates E σ , then E satisfies (9.9) and (9.10). Such elliptic curves E were studied for example by Ribet [Rib92] and Elkies [CLQR04]. Proposition 9.9. There exists an effective constant c, depending only on d and g, with the following property. Let A be an abelian scheme over S of relative dimension g. If AQ is a simple Q-virtual abelian variety of GL 2 -type which is non-CM, then In the proof of Proposition 9.9, we use a result of Wu [Wu11] to reduce the problem to abelian varieties over Q of GL 2 -type. Then we combine the techniques of the previous sections with a result of Silverberg [Sil92] to deduce the statement.
Proof of Proposition 9.9. Let A be an abelian scheme over S of relative dimension g. We suppose that AQ is a simple Q-virtual abelian variety of GL 2 -type which is non-CM.
1. Let L be a finite field extension of K with the following three properties: (c) and for all σ ∈ G Q the isogenies µ σ : A σ Q → AQ in (9.10) are defined over L.
We denote by C Q = Res L/Q (A L ) the Weil restriction of A L . The proof of [Wu11, Theorem 2.1.13] shows in addition that C Q has a Q-quotient B Q which is of GL 2 -type.
2. We now show that A L is L-isogenous to some abelian subvariety of B L . The abelian variety C L is L-isomorphic to A σ L with the product taken over all σ ∈ Gal(L/Q), where A σ L denotes the base change of A L with respect to σ : L → L. Thus on using that B Q is a Q-quotient of C Q , we obtain a surjective morphism A σ L ∼ = C L → B L of abelian varieties over L. Further, our assumptions on AQ imply that each A σ L is L-simple. Therefore Poincaré's reducibility theorem shows that there exists σ ∈ Gal(L/Q) such that A σ L is L-isogenous to some abelian subvariety A ′ L of B L . The abelian varieties A L and A σ L are L-isogenous, since AQ is a Q-virtual abelian variety with isogenies µ σ : A σ Q → AQ defined over L by (c). Hence A L is L-isogenous to the abelian subvariety A ′ L of B L . 3. We begin to estimate the stable Faltings height h F . Let N A L and N C be the conductors of A L and C Q respectively. Milne [Mil72,Proposition 1] gives that N C = N A L D 2g L for D L the absolute value of the discriminant of L over Q. Hence, if a rational prime number p does not divide N S ′ = rad(N A L D L ), then C Q has good reduction at p. This shows that C Q extends to an abelian scheme C over S ′ = Spec(Z[1/N S ′ ]) and thus the Q-quotient B Q of C Q extends to an abelian scheme B over S ′ . Therefore an application of Theorem 9.2 with the abelian scheme B over S ′ of GL 2 -type gives that h F (B) ≤ (3n) 144n N 24 S ′ for n the relative dimension of B which satisfies n ≤ dim(C Q ) = lg. Then on using that A L is L-isogenous to the abelian subvariety A ′ L of B L , we see that the arguments of Theorem 8.1 (ii) lead to h F (A) ≤ c ′ N 24 S ′ for c ′ an effective constant depending only on l and g. 4. It remains to control the quantities N S ′ and l. We observe that N S ′ = rad(N A L D L ) divides rad(N S D L ) since A is an abelian scheme over S. To estimate l and rad(D L ) we use [Sil92, Theorem 4.2]. It implies the existence of a field extension L of K, with the properties (a), (b) and (c), such that rad(D L ) | rad(N S D K ) and such that l = [L : Q] is effectively bounded in terms of d and g. It follows that N S ′ | rad(N S D K ) and then the inequality h F (A) ≤ c ′ N 24 S ′ from 3. implies Proposition 9.9.
On computing explicitly the constant c of Proposition 9.9, one sees that c depends double exponentially on d and g. However, in certain cases of interest it is possible to obtain that c depends exponentially on d and g. Further, we mention that one can use the arguments of the proof of Theorem 8.1 to remove in Proposition 9.9 the assumption that AQ is simple. In fact one can generalize all results of Section 9.2 (except Proposition 9.3) by replacing Theorem 9.2 with Proposition 9.9 in the proofs of the previous section.