Szpiro’s Small Points Conjecture for Cyclic Covers

Ariyan Javanpeykar and Rafael von Känel

Received: May 04, 2014

Communicated by Takeshi Saito

Abstract. Let X be a smooth, projective and geometrically connected curve of genus at least two, defined over a number field. In 1984, Szpiro conjectured that X has a “small point”. In this paper we prove that if X is a cyclic cover of prime degree of the projective line, then X has infinitely many “small points”. In particular, we establish the first cases of Szpiro’s small points conjecture, including the genus two case and the hyperelliptic case. The proofs use Arakelov theory for arithmetic surfaces and the theory of logarithmic forms.

2010 Mathematics Subject Classification: 14G05, 14G40, 11J86. Keywords and Phrases: Szpiro’s small points conjecture, cyclic covers, Arakelov theory, arithmetic surfaces, theory of logarithmic forms

1 Introduction

Let X be a smooth, projective and geometrically connected curve of genus at least two, defined over a number field. In 1984, Szpiro [Szp85a] conjectured that X has a “small point”, where a “small point” is an algebraic point of the curve X with “height” bounded from above in a certain way. We refer to Section 2 for a precise formulation of Szpiro’s small points conjecture. Szpiro proved that his conjecture implies an “effective Mordell conjecture”. In other words, Szpiro’s remarkable approach shows that to bound the height of all rational points of any curve X, it suffices to produce for any curve X at least one “small point”. The small points conjecture was studied in Szpiro’s influential seminars [Szp85b, Szp90a], see also Szpiro’s articles [Szp86, Szp90b]. The results of this paper are as follows. Let C be the set of curves X as above which are cyclic covers of prime degree of the projective line. For example, if X has genus two or is hyperelliptic, then X ∈ C. Our first result (see Theorem 3.1) gives that any X ∈ C has infinitely many “small points”. In particular, we establish the first cases of Szpiro’s small points conjecture. Furthermore, we improve Theorem 3.1 for hyperelliptic curves (see Theorem 3.2) and for genus two curves (see Theorem 3.3) in the sense that we produce “smaller points” on such curves. To give the reader a more concrete idea of our results we now state a special case of Theorem 3.3. If the curve X has genus two and is defined over ℚ, then X has infinitely many algebraic points x that satisfy

max(h_{NT} (x), h(x))≤ (10 ∏ p)^{10^{6}}

with the product taken over all bad reduction primes p of X. Here h_{NT} is the Néron-Tate height and h is the stable Arakelov height, see Section 2. We also give in Proposition 3.4 and Proposition 5.3 versions of the above theorems with “exponentially smaller points”. However these versions are either conditional on the abc conjecture, or they depend on lower bounds for Faltings’ delta invariant which are not known to be effective. We remark that “effective Siegel or Mordell” applications of our completely explicit results require hyperbolic curves which admit Kodaira-Paršin type constructions with all fibers in C. There exists no hyperbolic curve for which Kodaira’s construction (see [Szp85b, p.266]) is of this form and we are not aware of a hyperbolic curve for which Paršin’s construction (see Paršin [Par68], or [Szp85b, p.268]) has all fibers in C. Therefore our results have at the time of writing no “effective Siegel or Mordell” applications. However, there is the hope that new and more suitable Kodaira-Paršin type constructions will be discovered. For example, recently Levin [Lev12] gave a new Paršin type construction for integral points on hyperelliptic curves with all fibers in C. We next describe the main ingredients for our proofs. Let X be a smooth, projective and geometrically connected curve of genus g ≥ 2, defined over a number field K. On using fundamental results of Arakelov [Ara74], Faltings [Fal84] and Zhang [Zha92], we show that for any ϵ > 0 there exist infinitely many algebraic points x of X that satisfy

h_{NT} (x) ≤ 2g(g − 1)h(x) ≤ g · e(X) + ϵ.  (1)

Here e(X) is a stable Arakelov self-intersection number, see (7). Thus, to produce “small points” of X, it suffices to estimate e(X) effectively in terms of K, S and g, for S the set of finite places of K where X has bad reduction. The proof of Theorem 3.1 uses properties of the Belyi degree deg_{B}(X) of X, which is defined in (8). From [Jav14, Thm 1.1.1] we obtain

e(X) ≤ 10^{8}deg_{B}(X)^{5}g.  (2)

Szpiro’s Small Points Conjecture for Cyclic Covers 1087

To control deg_{B}(X) we use an effective version of Belyi’s theorem [Bel79], worked out by Khadjavi in [Kha02]. We deduce an explicit upper bound for deg_{B}(X) in terms of K, g and the heights h(λ) of the cross-ratios λ of four (geometric) branch points of a finite morphism X → ℙ_{K}^{1}. To estimate h(λ) we assume that X ∈ C and then we combine [dJR11, Prop 2.1] of de Jong-Rémond with [vK14, Prop 6.1 (ii)]; here we mention that the latter result is based on the theory of logarithmic forms and the former result generalizes ideas of Paršin [Par72] and Oort [Oor74]. This leads to an explicit upper bound, in terms of K, S and g, for deg_{B}(X), then for e(X) by using (2), and then finally for h_{NT} (x) and h(x) by applying (1). To discuss the proof of Theorem 3.3 we assume that g = 2. It was shown in [vK14, Prop 5.1 (v)] that Faltings’ delta invariant δ(X_{ℂ}) of a compact connected Riemann surface X_{ℂ} of genus two satisfies −186 ≤ δ(X_{ℂ}). Then the Noether formula for arithmetic surfaces, due to Faltings [Fal84] and Moret-Bailly [MB89], leads to the following explicit inequality

e(X) ≤ 12h_{F} (J) + 201.  (3)

Here h_{F} (J) is the stable Faltings height (see [Fal83, p.354]) of the Jacobian J = Pic^{0}(X) of X. The inequalities in [vK14, Prop 4.1 (i), Prop 6.1 (ii)] slightly refine a method developed by Paršin [Par72], Oort [Oor74] and de Jong-Rémond [dJR11]. On combining these inequalities, we deduce an explicit upper bound, in terms of K, S and g, for h_{F} (J), then for e(X) by using (3), and then finally for h_{NT} (x) and h(x) by applying (1). To prove Theorem 3.2 we use the strategy of proof of Theorem 3.3. In particular, we combine this strategy with a formula of de Jong [dJ09, Thm 4.3] and we use in addition the explicit estimate for the hyperelliptic discriminant modular form which was established in [vK14, Lem 5.4]. The plan of this paper is as follows. In Section 2 we discuss Szpiro’s small points conjecture and its variation which involves the Néron-Tate height. Then in Section 3 we state our theorems, and we give Proposition 3.4 which is conditional on the abc conjecture. In Section 4 we collect results from Arakelov theory for arithmetic surfaces. We also give an upper bound for the Belyi degree of X. In Section 5 we consider curves X ∈ C and we prove Theorem 3.1. We also establish Proposition 5.3. It refines our theorems, with the disadvantage inherent that it involves a lower bound for Faltings’ delta invariant which is not known to be effective. In Section 6 we first show Lemma 6.1 and Lemma 6.2. They give explicit results for certain (Arakelov) invariants of hyperelliptic curves which may be of independent interest. Then we use these lemmas to prove Theorem 3.2. Finally, in Section 7, we give a proof of Theorem 3.3.

Notation

Throughout this paper we shall use the following notations and conventions. Let K be a number field. We denote by K̄ a fixed algebraic closure of K. If L is a field extension of K, then we write [L : K] for the relative degree of L over K. By a curve X over K we mean a smooth, projective and geometrically connected curve X → Spec(K). For any finite place v of K, we write N_{v} for the number of elements in the residue field of v and we let v(𝔞) be the order of v in a fractional ideal 𝔞 of K. We denote by |S| the cardinality of an arbitrary set S. Finally, by log we mean the principal value of the natural logarithm and we define the product taken over the empty set as 1.

Acknowledgements

Most of the results were obtained (2012) at the IAS Princeton. Both authors would like to thank the IAS for the friendly environment and for the generous financial support. In addition, R.v.K. would like to thank Professor Szpiro for giving helpful explanations. R.v.K. was supported by the National Science Foundation under agreement No. DMS-0635607.

2 Small points conjectures

In this section we state and discuss Szpiro’s small points conjecture, and its variation which involves the Néron-Tate height. Let K be a number field, let g ≥ 2 be an integer and let X be a curve over K of genus g. We take an algebraic point x ∈ X(K̄). The classical result of Deligne-Mumford gives a finite extension L of K such that X_{L} has semi-stable reduction over B and such that x ∈ X(L), where B is the spectrum of the ring of integers of L. We denote by ω_{X /B} the relative dualizing sheaf of the minimal regular model X of X_{L} over B. Let ω = (ω_{X /B}, ∥·∥) with ∥·∥ the Arakelov metric, let (·, ·) be the intersection product of Arakelov divisors on X and identify ω and x with the corresponding Arakelov divisors on X, see [Fal84, Section 2] for definitions. Then the stable Arakelov height h(x) of x is the real number defined by

[L : ℚ]h(x) = (ω, x).  (4)

The factor [L : ℚ] and the semi-stability of X assure that the definition of h(x) does not depend on the choice of a field L with the above properties. Let S be a set of finite places of K. We say that a constant c, depending on some data (D), is effective if one can in principle explicitly determine the real number c provided that (D) is given. In 1984, Szpiro [Szp85a, p.101] formulated his small points conjecture in terms of the Arakelov self-intersection (x, x) of x. However, Arakelov’s adjunction formula (ω, x) = −(x, x) in [Ara74] shows that Szpiro’s small points conjecture coincides with the following conjecture.

Conjecture (sp). There exists an effective constant c, depending only on K, S and g, with the following property. Suppose that X is a curve over K of genus g, with set of bad reduction places S. If X has semi-stable reduction over the ring of integers of K, then there exists x ∈ X(K̄) that satisfies h(x) ≤ c.

It is known (see Szpiro [Szp85a, p.101]) that this conjecture implies an “effective Mordell conjecture”. Further, Szpiro established a rather strong geometric analogue of Conjecture (sp) in [Szp81]. We also mention that if Conjecture (sp) holds, then it holds without the semi-stable assumption. Indeed, on combining results of Grothendieck-Raynaud [GR72, Proposition 4.7] and Serre-Tate [ST68, Theorem 1] with Dedekind’s discriminant theorem, one obtains a finite field extension M of K such that X_{M} has semi-stable reduction over the ring of integers of M and such that [M : K] and the relative discriminant of M over K are effectively controlled in terms of K, S and g.

In the Grothendieck Festschrift, Szpiro [Szp90b] formulated another version of Conjecture (sp). This formulation involves the Néron-Tate height

h_{NT} (x)  (5)

of x ∈ X(K̄) which is defined as the Néron-Tate height of the divisor class (2g − 2)x − Ω^{1} in the Jacobian Pic^{0}(X_{L}) of X_{L} for L as above. Here Ω^{1} denotes the sheaf of differential one-forms of the curve X_{L} over L. We now give a version of Szpiro’s “Conjecture des deux petits points pour Néron-Tate” which was stated by Szpiro in [Szp90b, p.244].

Conjecture (sp)^{∗}. Any curve X over K of genus g has two distinct points x_{i} ∈ X(K̄), i = 1, 2, that satisfy h_{NT} (x_{i}) ≤ c, where c is an effective constant which depends only on K, g and the geometry of the bad reduction of X.

It follows for example from Lemma 4.5 below that Conjecture (sp) implies Conjecture (sp)^{∗}. An unconditional proof of the converse implication seems to be difficult. However, we point out that Szpiro’s arguments in [Szp90b, p.244] combined with Moret-Bailly’s proof of [MB90, Théorème 5.1] show that Conjecture (sp)^{∗} still implies an “effective Mordell conjecture”. We mention that the conjecture in [Szp90b, p.244] describes the constant c in Conjecture (sp)^{∗} more precisely. See also [JvK13, Section 2] for a discussion of the possible shape of the constants in Conjectures (sp) and (sp)^{∗}. For further discussions and conjectures related to the small points conjecture, we refer the reader to the works of Paršin [Par88] and Moret-Bailly [MB90].

3 Statements of results

In this section we state and discuss Theorems 3.1, 3.2 and 3.3. We also give Proposition 3.4 which relies on the abc conjecture. To state our results we need to introduce some notation. Let g ≥ 2 be an integer, let K be a number field and let S be a set of finite places of K. We denote by D_{K} the absolute value of the discriminant of K over ℚ and we write d = [K : ℚ] for the degree of K over ℚ. To measure K, S and g we use

D_{K}, d, N_{S} = ∏_{v∈S} N_{v}, g and ν = d(5g)^{5}.  (6)

We mention that the only purpose of introducing ν is to simplify the exposition. Let h be the stable Arakelov height and let h_{NT} be the Néron-Tate height. These heights are defined in (4) and (5) respectively. Let C = C(K) be the set of curves X over K of genus ≥ 2 such that there is a finite morphism X → ℙ_{K}^{1} of prime degree which is geometrically a cyclic cover. Our first theorem establishes in particular Conjectures (sp) and (sp)^{∗} for all curves X ∈ C.

Theorem 3.1. Let X be a curve over K of genus g, with set of bad reduction places S. If X ∈ C then there exist infinitely many x ∈ X(K̄) that satisfy

log max(h_{NT} (x), h(x))≤ ν^{dν}(N_{S}D_{K})^{ν}.

Next, we present two results which improve Theorem 3.1 in certain cases. We say that a curve X over K of genus g is a hyperelliptic curve over K if there is a finite morphism X → ℙ_{K}^{1} of degree two. For example, any genus two curve over K is a hyperelliptic curve over K. We obtain in particular the following theorem for hyperelliptic curves over K.

Theorem 3.2. Suppose that X is a hyperelliptic curve over K of genus g, with set of bad reduction places S and set of Weierstrass points W. Then it holds

∑_{x∈W} h_{NT} (x) ≤ ν^{8^{g}dν}(N_{S}D_{K})^{ν}.

The Néron-Tate height h_{NT} is non-negative, and any hyperelliptic curve over K of genus g has exactly 2g + 2 Weierstrass points. Thus Theorem 3.2 gives another proof of Conjecture (sp)^{∗} for all hyperelliptic curves over K of genus g. Our next result refines Theorem 3.1 in the special case of genus two curves.

Theorem 3.3. Suppose that X is a curve over K of genus two, with set of bad reduction places S. Then there are infinitely many x ∈ X(K̄) that satisfy

max(h_{NT} (x), h(x))≤ ν^{2dν}(N_{S}D_{K})^{ν}, ν = 10^{5}d.

To state Proposition 3.4 we need to recall the abc-conjecture of Masser-Oesterlé [Mas02] over number fields. For any non-zero triple α, β, γ ∈ K we denote by H(α, β, γ) the usual absolute multiplicative Weil height of the corresponding point in ℙ^{2}(K), see [BG06, 1.5.4]. We define the height function H_{K} = H^{d} and we write S_{K}(α, β, γ) = ∏ N_{v}^{e_{v}} with the product extended over all finite places v of K such that v(α), v(β) and v(γ) are not all equal, where e_{v} = v(p) for p the residue characteristic of v. We mention that Masser [Mas02] added the ramification index e_{v} in the definition of the support S_{K} to obtain a natural behaviour of S_{K} with respect to finite field extensions.

Conjecture (abc). For any integer n ≥ 1, and any real r, ϵ > 1, there is a constant c, which depends only on n, r, ϵ, such that if K is a number field of degree [K : ℚ] ≤ n, and α, β, γ ∈ K are non-zero and satisfy α + β = γ, then H_{K}(α, β, γ) ≤ cS_{K}(α, β, γ)^{r}D_{K}^{ϵ}.

The following proposition is conditional on Conjecture (abc). It improves exponentially, in terms of N_{S} and D_{K}, the inequalities in our theorems. We put u(g) = 8^{(11g)^{3}8^{g}} and now we can state Proposition 3.4.

Proposition 3.4. Let r, ϵ > 1 be real numbers and write

Ω = (r + \frac{ϵ}{d}) log N_{S} + \frac{ϵ}{d} log D_{K}.

Suppose that Conjecture (abc) holds for n = 24g^{4}d, r, ϵ with the constant c. Then there exist effective constants c_{1}, c_{2}, c_{3}, depending only on c, r, ϵ, d and g, such that for any curve X over K of genus g, with set of bad reduction places S, the following statements hold.

(i) If X ∈ C, then there are infinitely many x ∈ X(K̄) that satisfy

log max(h_{NT} (x), h(x))≤ ν^{ν}Ω + c_{1}.

(ii) Suppose that X is a hyperelliptic curve over K, with set of Weierstrass points W. Then it holds

∑_{x∈W} h_{NT} (x) ≤ u(g)(3g − 1)(8g + 4)Ω + c_{2}.

(iii) If g = 2, then there are infinitely many x ∈ X(K̄) that satisfy

\frac{1}{4}h_{NT} (x) ≤ h(x) ≤ 6u(2)Ω + c_{3}.

We remark that the factor u(g) appearing in Proposition 3.4 is not optimal. Its origin shall be explained after Proposition 5.3. We also point out that Proposition 3.4 only requires the validity of (abc) for some fixed n, r, ϵ, instead of all n, r, ϵ, and (abc) with fixed n, r, ϵ is often called “weak abc conjecture”. Further we mention that Elkies [Elk91] used Belyi’s theorem to show that (effective) (abc) implies (effective) Mordell. However, it is not clear if Conjecture (sp) or Conjecture (sp)^{∗} follows from the effective version of the Mordell conjecture which Elkies deduces from an effective version of (abc). In general, we conducted some effort to obtain constants reasonably close to the best that can be acquired with the present method of proof. However, to simplify the form of our inequalities we freely rounded off several of the numbers appearing in our estimates.

4 Self-intersection, Belyi degree and heights

In this section we first give two lemmas which describe properties of the Belyi degree, and then we collect results from Arakelov theory for arithmetic surfaces. We also prove a lemma which was used in Section 2. Throughout this section we denote by X a curve of genus g ≥ 2, defined over a number field K. Suppose that L, ω and (·, ·) are as in Section 2. Then the stable self-intersection e(X) of ω is the real number defined by

[L : ℚ]e(X) = (ω, ω).  (7)

We observe that this definition does not depend on any choices. Let D be the set of degrees of finite morphisms XK̄ → ℙ1 ; K̄ which are unramified outside 0, 1, ∞. Belyi’s theorem [Bel79] shows that D is non-empty, and then the Belyi degree deg_{B}(X) of X is defined by

deg_{B}(X) = min D.  (8)

Our proof of Theorem 3.1 uses two fundamental properties of the Belyi degree. We now state the first of these properties in the following lemma.

Lemma 4.1. It holds e(X) ≤ 10^{8}deg_{B}(X)^{5}g.

Proof. The statement is proven in [Jav14, Theorem 1.1.1].

The next lemma gives the second property. It is a consequence of an effective version of Belyi’s theorem [Bel79] worked out by Khadjavi [Kha02]. We denote by H(α) the usual absolute multiplicative Weil height of α ∈ ℙ^{1}(K̄), defined in [BG06, 1.5.4]. Then we define the height H_{Λ} of a subset Λ of ℙ^{1}(K̄) by H_{Λ} = sup{H(λ), λ ∈ Λ}.

Lemma 4.2. If φ : X → ℙ_{K}^{1} is a finite morphism, with set of (geometric) branch points Λ ⊂ ℙ^{1}(K̄) and if m = 4[K : ℚ](deg(φ) + g − 1)^{2}, then

deg_{B}(X) ≤ (4mH_{Λ})^{9m^{3}2^{m−2}m!}deg(φ).

Proof. The absolute Galois group Gal(ℚ̄/ℚ) of ℚ̄ over ℚ acts in the usual way on ℙ^{1}(ℚ̄) ≅ ℙ^{1}(K̄). Let Λ′ = Gal(ℚ̄/ℚ) · Λ be the image of Λ under this action. The classical result of Hurwitz [Liu02, Theorem 7.4.16] implies that [K(λ) : K] and |Λ| are at most 2g − 2 + 2deg(φ) for K(λ) the field of definition of λ ∈ Λ. This gives |Λ′| ≤ m, and the Galois invariance [BG06, 1.5.17] of H shows H_{Λ} = H_{Λ′}. Then an application of [Kha02, Theorem 1.1] with the Galois stable set Λ′ gives a finite morphism ψ : ℙ1 ; K̄ → ℙ1 ; K̄ with the following properties. The morphism ψ is unramified outside 0, 1, ∞, with ψ(Λ) ⊆ {0, 1, ∞} and

deg(ψ) ≤ (4mH_{Λ})^{9m^{3}2^{m−2}m!}.

We observe that the composition ψ∘φ : XK̄ → ℙ1 ; K̄ is unramified outside 0, 1, ∞. This shows deg_{B}(X) ≤ deg(ψ)deg(φ) and then the displayed inequality implies Lemma 4.2.

The bound in Khadjavi’s [Kha02, Theorem 1.1], and thus Lemma 4.2, can be improved in special cases. See for example Liţcanu [Liţ04]. Let h be the stable Arakelov height defined in (4). To obtain infinitely many small points we shall use the following lemma which relies on a fundamental result of Zhang [Zha92].

Lemma 4.3. Suppose that ϵ > 0 is a real number. Then there exist infinitely many points x ∈ X(K̄) that satisfy

2(g − 1)h(x) ≤ e(X) + ϵ.

Proof. It follows for example from Lemma 4.4 (i) below that any point x ∈ X(K̄) satisfies h(x) ≥ 0. Then we see that [Zha92, Theorem 6.3] implies the statement of Lemma 4.3.

We remark that Moret-Bailly showed that for any real number ϵ > 0, there exists an algebraic point x ∈ X(K̄) which satisfies 4(g − 1)h(x) ≤ e(X) + ϵ.

For details we refer the reader to the proof of [MB90, Proposition 3.4] which uses in particular Faltings’ result [Fal84, Corollary, p.406]. The following lemma is a direct consequence of classical results of Arakelov [Ara74], Faltings [Fal84] and Moret-Bailly [MB89]. To state this lemma we need to introduce more notation. For any embedding σ : K ↪ ℂ, we denote by X_{σ} the compact connected Riemann surface which corresponds to the base change of X to ℂ with respect to σ. Let δ(X_{σ}) be Faltings’ delta invariant of X_{σ}, defined in [Fal84, p.402]. We denote by M_{g}(ℂ) the moduli space of smooth, projective and connected curves over ℂ of genus g. Faltings’ delta invariant, viewed as a function M_{g}(ℂ) → ℝ, has a minimum which we denote by

c_{δ}(g).  (9)

If g ≥ 3, then effective lower bounds for c_{δ}(g) are not known. Let h_{F} (J) be the stable Faltings height [Fal83, p.354] of the Jacobian J = Pic^{0}(X) of X, and let h_{NT} be the Néron-Tate height defined in (5). We now can state the lemma.

Lemma 4.4. The following statements hold.

(i) Any x ∈ X(K̄) satisfies h_{NT} (x) ≤ 2g(g − 1)h(x).

(ii) It holds e(X) + c_{δ}(g) ≤ 12h_{F} (J) + 4g log(2π).

Proof. To show (i) we take x ∈ X(K̄). Let L, X → B, ω and (·, ·) be as in (4). We identify x with the corresponding Arakelov divisor on X. Let Φ be the (unique) vertical ℚ-Cartier divisor on X such that the supports of Φ and x(B) are disjoint and such that any irreducible component Γ of any fiber of X → B satisfies ((2g − 2)x − ω + Φ, Γ) = 0. Szpiro [Szp85c, p.276] observed that the adjunction formula in [Ara74] together with [Fal84, Theorem 4.c)] leads to

2h_{NT} (x) = −e(X) + 4g(g − 1)h(x) + \frac{1}{[L : ℚ]}(Φ, Φ).

Further, [Fal84, Theorem 5.a)] provides that −e(X) ≤ 0. Therefore the inequality (Φ, Φ) ≤ 0 implies assertion (i). We now prove (ii). If v ∈ B is closed, then δ_{v} denotes the number of singular points of the geometric special fiber of X over v. The (logarithmic) stable discriminant Δ(X) of X is the real number defined by

[L : ℚ]Δ(X) = ∑ δ_{v} log N_{v}  (10)

with the sum taken over all closed points v of B. Then we see that Moret-Bailly’s refinement [MB89, Théorème 2.5] of the Noether formula [Fal84, Theorem 6] implies the following formula

12h_{F} (J) = Δ(X) + e(X) − 4g log(2π) + \frac{1}{[L : ℚ]} ∑ δ(X_{σ})

with the sum taken over all embeddings σ : L ↪ ℂ. Therefore the estimates Δ(X) ≥ 0 and ∑ δ(X_{σ}) ≥ [L : ℚ]c_{δ}(g) prove assertion (ii) and this completes the proof of Lemma 4.4.

The above results lead to the following useful lemma.

Lemma 4.5. Suppose that ϵ > 0 is a real number and assume that x_{0} ∈ X(K̄). Then there exist infinitely many x ∈ X(K̄) that satisfy

h_{NT} (x) ≤ 4g^{2}(g − 1)h(x_{0}) + ϵ.

Proof. On combining Lemma 4.3 with Lemma 4.4 (i), we see that there exist infinitely many points x ∈ X(K̄) that satisfy

h_{NT} (x) ≤ g · e(X) + ϵ.  (11)

Further [Fal84, Theorem 5.b)] gives that any x_{0} ∈ X(K̄) satisfies the inequality e(X) ≤ 4g(g − 1)h(x_{0}) and thus (11) implies Lemma 4.5.

We conclude this section by the following remarks. Let S be the set of finite places of K where X has bad reduction. Suppose that there exists a finite morphism φ : X → ℙ_{K}^{1}, with deg(φ) and H_{Λ} effectively bounded in terms of K, S and g, where H_{Λ} is the height of the set Λ of (geometric) branch points of φ. Then Lemma 4.1, Lemma 4.2 and Lemma 4.3 show that X has infinitely many “small points”, and therefore X satisfies in particular Szpiro’s small points conjecture. Similarly, if deg_{B}(X) is effectively bounded in terms of K, S and g, then Lemma 4.1 and Lemma 4.3 show that X has infinitely many “small points”. For example, if the base change of X to ℂ, with respect to some embedding K ↪ ℂ, is a classical congruence modular curve, then a result of Zograf in [Zog91] gives deg_{B}(X) ≤ 128(g + 1).

5 Cyclic covers of prime degree

In this section we prove Theorem 3.1 and Proposition 3.4 (i). We also give Proposition 5.3 which may be of independent interest. It improves certain aspects of the inequalities in Theorem 3.1 and Proposition 3.4 (i). However, Proposition 5.3 has the disadvantage inherent that it now involves a constant which is not known to be effective. Let X be a curve over a number field K of genus g ≥ 2. We denote by S the set of finite places of K where X has bad reduction. Further we write C = C(K) for the set of cyclic covers of prime degree which was introduced in Section 3. In this section we assume throughout that X ∈ C. We now give two lemmas which will be used in our proof of Theorem 3.1. To state and prove these lemmas we have to introduce some notation. Our assumption that X ∈ C provides a finite morphism φ : X → ℙ_{K}^{1} which is geometrically a cyclic cover of prime degree. Let q be the degree of φ and let L be a finite extension of K. We denote by U = S(L, q) the set of places of L which divide q or a place in S. Let O_{U}^{×} be the U-units in L and let h(α) be the usual absolute logarithmic Weil height of α ∈ L, defined in [BG06, 1.6.1]. We write μ_{U} = sup(h(λ), λ ∈ O_{U}^{×} and 1 − λ ∈ O_{U}^{×}). Let R be the set of field extensions L of K such that L is the compositum of the fields of definition of four distinct (geometric) ramification points of φ. We define

μ_{X} = sup(1, μ_{S(L,q)})  (12)

with the supremum taken over all fields L ∈ R. Let deg_{B}(X) be the Belyi degree of X, defined in (8), and let d be the degree of K over ℚ. We recall that ν = d(5g)^{5} and now we can state the following lemma.

Lemma 5.1. It holds log deg_{B}(X) ≤ ν^{ν/2}μ_{X}.

To prove this lemma we shall combine the estimate for deg_{B}(X) in Lemma 4.2 with the following observation of Paršin in [Par72]: The cross-ratios of the branch points, of a hyperelliptic map of a genus two curve X over K, are solutions of certain S(L, 2)-unit equations. Oort [Oor74, Lemma 2.1] and de Jong-Rémond [dJR11, Proposition 2.1] generalized Paršin’s idea to hyperelliptic curves and to cyclic covers of prime degree.

Proof of Lemma 5.1. We take three distinct (geometric) ramification points of φ. Let M be the compositum of their fields of definition. On composing φ with a suitable automorphism of ℙ_{M}^{1}, we get a finite morphism X_{M} → ℙ_{M}^{1} of degree q such that {0, 1, ∞} ⊂ Λ, where Λ is the set of (geometric) branch points of φ. Let H_{Λ} be the height of Λ, defined in Section 4. To prove the inequality

log H_{Λ} ≤ μ_{X},

we may and do take λ ∈ Λ with λ ≠ 0, 1, ∞. We write U = S(K(λ), q). From [dJR11, Proposition 2.1] we deduce that the cross-ratios λ = cr(∞, 0, 1, λ) and 1 − λ = cr(∞, 1, 0, λ) are U-units in K(λ). This implies that h(λ) ≤ μ_{X}, since K(λ) ⊆ L for some L ∈ R. Hence we obtain log H_{Λ} ≤ μ_{X} as desired. Next, we observe that the ramification indexes of φK̄ : XK̄ → ℙ1 ; K̄ are in {1, q}, and Gal(K̄/K) acts on the (geometric) ramification points of φ. Therefore Hurwitz leads to q ≤ 2g + 1 and [M : ℚ] ≤ 15dg^{3}. Then an application of Lemma 4.2 with X_{M} → ℙ_{M}^{1} gives an upper bound for deg_{B}(X) which together with the displayed inequality implies Lemma 5.1.

We remark that if L ∈ R, then Hurwitz (see the proof of Lemma 5.1) leads to q ≤ 2g + 1 and [L : K] ≤ 24g^{4}, and [dJR11, Lemme 2.1] of de Jong-Rémond implies that L is unramified outside S(K, q). Next, we go into number theory and we give an upper bound for μ_{X} in terms of the quantities N_{S}, ν, d and D_{K} which are defined in (6).

Lemma 5.2. The following statements hold.

(i) It holds μ_{X} ≤ ν^{dν/8}(N_{S}D_{K})^{ν}.

(ii) Let r, ϵ > 1 be real numbers. Suppose (abc) holds for n = 24g^{4}d, r, ϵ with the constant c. Then there exists an effective constant c^{∗}, depending only on c, r, ϵ, d and g, such that

dμ_{X} ≤ (dr + ϵ) log N_{S} + ϵ log D_{K} + c^{∗}.

To prove (i) we use [vK14, Proposition 6.1 (ii)]. It is based on the theory of logarithmic forms and we refer to the monograph of Baker-Wüstholz [BW07] in which the state of the art of this theory is exposed.

Proof of Lemma 5.2. We take L ∈ R, and we write U = S(L, q), T = S(K, q) and l = [L : K]. Then we observe that N_{T} = ∏_{v∈T} N_{v} satisfies N_{T} ≤ q^{d}N_{S}. Furthermore, the remark after the proof of Lemma 5.1 gives that q ≤ 2g + 1 and l ≤ 24g^{4}, and that L is unramified outside T. We now apply [vK14, Proposition 6.1] with U = U, T = T and S = T, where the symbols U, T, S on the left hand side of these equalities denote the sets in [vK14, Proposition 6.1]. In particular, [vK14, Proposition 6.1 (ii)] leads to μ_{U} ≤ ν^{dν/8}(D_{K}N_{S})^{ν} which implies statement (i). To show statement (ii) we take real numbers r, ϵ > 1. We may and do assume that Conjecture (abc) holds for n = 24g^{4}d, r, ϵ with the constant c. Then Conjecture (abc) holds in particular for n′ = ld, r, ϵ with the same constant c, since n′ ≤ n. Then [vK14, Proposition 6.1 (iii)] gives

dμ_{U} ≤ (dr + ϵ) log N_{T} + ϵ log D_{K} + ϵdt log l − \frac{ϵ}{l} log N_{T} + \frac{1}{l} log c

for t = |T |. From [vK14, Lemma 6.3] we get that ϵdt log l− \frac{ϵ}{l} log N_{T} is bounded from above by an effective constant, which depends only on ϵ, d and g. Hence we deduce statement (ii) and Lemma 5.2.

We recall that h denotes the stable Arakelov height and that h_{NT} denotes the Néron-Tate height, defined in (4) and (5) respectively. We now prove Theorem 3.1 and Proposition 3.4 (i) simultaneously.

Proof of Theorem 3.1 and Proposition 3.4 (i). On combining Lemma 4.1, Lemma 4.3, Lemma 4.4 and Lemma 5.1, we obtain infinitely many x ∈ X(K̄) that satisfy log max(h_{NT} (x), h(x))≤ 6ν^{ν/2}μ_{X}. Therefore we see that Lemma 5.2 (i) and Lemma 5.2 (ii) imply Theorem 3.1 (i) and Proposition 3.4 (i) respectively.

The remaining of this section is devoted to the following Proposition 5.3. Let c_{δ}(g) be the minimum of Faltings’ delta invariant on M_{g}(ℂ), defined in (9). We recall that if g ≥ 3, then effective lower bounds for c_{δ}(g) in terms of g are not known. Put u(g) = 8^{(11g)^{3}8^{g}} and now we can state the following result.

Proposition 5.3. The following statements hold.

(i) There are infinitely many x ∈ X(K̄) that satisfy

h(x) ≤ ν^{8^{g}dν}(D_{K}N_{S})^{ν} − c_{δ}(g).

(ii) Let r, ϵ > 1 be real numbers. Suppose that Conjecture (abc) holds for n = 24g^{4}d, r, ϵ with the constant c. Then there exists an effective constant c′_{1}, depending only on c, r, ϵ, d and g, with the property that there are infinitely many points x ∈ X(K̄) which satisfy

h(x) ≤ 6\frac{u(g)}{g − 1}((r + \frac{ϵ}{d}) log N_{S} + \frac{ϵ}{d} log D_{K})+c′_{1} − \frac{c_{δ}(g)}{2g − 2}.

We observe that these (ineffective) inequalities improve exponentially, in terms of N_{S} and D_{K}, the estimates provided by Theorem 3.1 and Proposition 3.4 (i). On using Lemma 4.4 one can formulate Proposition 5.3 also in terms of h_{NT}. Further we remark that the factor u(g) comes from explicit height comparisons of Rémond [Rém10] which rely inter alia on results of Bost-David-Pazuki [Paz12]. In fact, Rémond’s explicit height comparisons hold for arbitrary curves over K, and in our special case where X ∈ C it seems possible to improve these height comparisons, and thus u(g), up to a certain extent. Our proof of Proposition 5.3 uses in particular the following tools. We combine Lemma 5.2 with [vK14, Proposition 4.1 (i)]. This slightly refines the method developed by Paršin [Par72], Oort [Oor74] and de Jong-Rémond [dJR11]. We also use Lemma 4.3 which is based on a theorem of Zhang [Zha92], and Lemma 4.4 (ii) which relies inter alia on Moret-Bailly’s refinement [MB89] of the Noether formula in Faltings’ article [Fal84].

Proof of Proposition 5.3. We denote by h_{F} (J) the stable Faltings height of the Jacobian J = Pic^{0}(X) of X. The remark given below [vK14, Proposition 4 (i)] provides the following explicit inequality

h_{F} (J) ≤ u(g)μ_{X}.  (13)

Then Lemma 4.4 (ii) shows e(X) ≤ 12u(g)μ_{X} − c_{δ}(g) + 4g log(2π) for e(X) as in (7). Hence Lemmas 4.3 and 5.2 imply Proposition 5.3.

We conclude this section by the following remarks. Let h_{F} (J) be as above, let Δ(X) be the stable discriminant of X defined in (10) and let e(X) be as in (7). Further, we define the quantity δ(X) = \frac{1}{d} ∑ δ(X_{σ}) with the sum taken over all embeddings σ : K ↪ ℂ, where δ(X_{σ}) is defined in Section 4. Then it holds

log max(e(X), δ(X), h_{F} (J), Δ(X)) ≤ ν^{dν}(D_{K}N_{S})^{ν}.  (14)

Indeed, [Jav14, Theorem 1.1.1] gives that e(X), δ(X), h_{F} (J) and Δ(X) are at most 10^{9}g^{2}deg_{B}(X)^{5}, and then Lemmas 5.1 and 5.2 prove the displayed inequality. We mention that de Jong-Rémond [dJR11, Theorem 1.2] provides an estimate for h_{F} (J) which is better than (14). Further [vK14, Theorem 3.2] gives an upper bound for Δ(X) which is exponentially better than (14). However, [vK14, Theorem 3.2] involves a constant, depending at most on g, which is only known to be effective for hyperelliptic curves X over K. We note that [dJR11, Theorem 1.2] and [vK14, Theorem 3.2] both depend inter alia on the above mentioned explicit height comparisons of Rémond in [Rém10], and such height comparisons are not used in our proof of (14).

Finally we point out that the method of this paper using the Belyi degree gives in addition the following generalizations: Theorem 3.1, Proposition 3.4 (i) and inequality (14) hold more generally for any curve Y over K which admits a finite étale morphism to some X ∈ C. Indeed on using that Y is an étale cover of X we deduce from Hurwitz that deg_{B}(Y) is explicitly bounded in terms of deg_{B}(X) and the genus of Y, and then the above arguments prove the claimed generalization. Here we applied in addition [LL99, Corollary 4.10] which gives that Y has bad reduction at a finite place v of K if X has bad reduction at v.

6 Hyperelliptic curves

In this section we prove Theorem 3.2. We also show two lemmas which may be of independent interest. They provide explicit results for certain (Arakelov) invariants of hyperelliptic curves. Throughout this section we denote by X a hyperelliptic curve of genus g ≥ 2, defined over a number field K. As before, we denote by X_{σ} the compact connected Riemann surface corresponding to the base change of X to ℂ with respect to an embedding σ : K ↪ ℂ. Let T (X_{σ}) be the invariant of de Jong. It is the norm of a canonical isomorphism between certain line bundles on X_{σ} and we refer to [dJ05, Definition 4.2] for a precise definition of T (X_{σ}).

Lemma 6.1. It holds −36g^{3} ≤ log T (X_{σ}).

To prove this lemma we use de Jong’s formula [dJ05, Theorem 4.7]. It expresses T (X_{σ}) in terms of a certain hyperelliptic discriminant modular form, which we then estimate by using the explicit inequality in [vK14, Lemma 5.4].

Proof of Lemma 6.1. We begin to state the formula for T (X_{σ}) in [dJ05, Theorem 4.7]. Let ℌ_{g} be the Siegel upper half plane of complex symmetric g × g matrices with positive definite imaginary part. We denote by Δ_{g} the hyperelliptic discriminant modular form on ℌ_{g}, defined in [vK14, Section 5]. Since X is hyperelliptic, there exists a finite morphism φ : X_{σ} → ℙ_{ℂ}^{1} of degree two. Let H_{1}(X_{σ}, ℤ) be the first homology group of X_{σ} with coefficients in ℤ. On following Mumford [Mum07, Chapter IIIa] we construct a canonical symplectic basis of H_{1}(X_{σ}, ℤ) with respect to a fixed ordering of the 2g + 2 branch points of φ. Then [dJ05, Theorem 4.7] provides a basis of the global sections of the sheaf of differentials on X_{σ} with the following property: Integration of this basis over the canonical symplectic basis of H_{1}(X_{σ}, ℤ) gives a period matrix τ_{σ} ∈ ℌ_{g} that satisfies

T (X_{σ}) = (2π)^{−2g} |Δ_{g}(τ_{σ})det(im(τ_{σ}))^{2a}|^{−(3g−1)/(8bg)} ,

where a = \binom{2g+1}{g+1} and b = \binom{2g}{g+1}. Furthermore, [vK14, Lemma 5.4] gives an effective constant k_{1}, depending only on g, such that

|Δ_{g}(τ_{σ})det(im(τ_{σ}))^{2a}| ≤ k_{1}.

The effective constant k_{1} is explicitly computed in [vK14, (15)], and then the displayed formula for T (X_{σ}) leads to an inequality as stated in Lemma 6.1.

Let h_{F} (J) be the stable Faltings height of the Jacobian J = Pic^{0}(X) of X and let h_{NT} be the Néron-Tate height which is defined in (5).

Lemma 6.2. If W denotes the set of Weierstrass points of X, then

∑_{x∈W} h_{NT} (x) ≤ (3g − 1)(8g + 4)h_{F} (J) + 293g^{5}.

To prove Lemma 6.2 we use de Jong’s formula [dJ09, Theorem 4.3]. This formula involves inter alia h_{F} (J) and ∑_{x∈W} h_{NT} (x), and an analytic term related to T (X_{σ}) which we control by Lemma 6.1.

Proof of Lemma 6.2. To state the formula in [dJ09, Theorem 4.3] we introduce quantities A_{1}, A_{2} and A_{3}. Let L be a finite field extension of K such that X_{L} has semi-stable reduction over the spectrum B of the ring of integers of L and such that all Weierstrass points of X are in X(L). We define

A_{1} = −4g(2g − 1)(g + 1) log(2π) + \frac{8g^{2}}{[L : ℚ]} ∑ log T (X_{σ})

with the sum taken over all embeddings σ : L ↪ ℂ. Let X → B, (·, ·) and ω be as in (4). We denote by E the residual divisor on X defined in [dJ09, p.286]. Let Δ(X) be the stable discriminant of X in (10). Then we take

A_{2} = (2g − 1)(g + 1)Δ(X) + \frac{4}{[L : ℚ]}(E, ω).

For any section x ∈ X (B) of X → B, we denote by Φ_{x} the (unique) vertical ℚ-Cartier divisor on X with the following properties: The supports of Φ_{x} and x(B) are disjoint, and any irreducible component Γ of any fiber of X → B satisfies ((2g − 2)x − ω + Φ_{x}, Γ) = 0. We write

A_{3} = \frac{1}{[L : ℚ]g(g − 1)} ∑_{x∈W} −(Φ_{x}, Φ_{x})n_{x}

with n_{x} the multiplicity of x in the Weierstrass divisor W on X, where W is defined in [dJ09, p.286]. Then [dJ09, Theorem 4.3] gives

(3g − 1)(8g + 4)h_{F} (J) = A_{1} + A_{2} + A_{3} + \frac{2}{g(g − 1)} ∑_{x∈W} h_{NT} (x)n_{x}.

We now estimate the quantities A_{1}, A_{2} and A_{3} from below. To deal with A_{1} we use Lemma 6.1. It gives −293g^{5} ≤ A_{1}. The divisor E on X is vertical and effective, and our minimal X does not contain any exceptional curves. This implies that (E, ω) ≥ 0. Then Δ(X) ≥ 0 and −(ϕ_{x}, ϕ_{x}) ≥ 0 show that A_{2} and A_{3} are both non-negative. Furthermore, [dJ09, Lemma 3.2] gives that n_{x} = g(g − 1)/2. Thus we see that the above displayed formula and the lower bounds for A_{1}, A_{2} and A_{3} imply an inequality as claimed in Lemma 6.2.

On using the inequality given in Lemma 6.2 we now prove Theorem 3.2 and Proposition 3.4 (ii) simultaneously.

Proof of Theorem 3.2 and Proposition 3.4 (ii). Since X is a hyperelliptic curve over K, there exists a finite morphism X → ℙ_{K}^{1} which is geometrically a cyclic cover of prime degree two. Then, on combining Lemma 6.2, Lemma 5.2 and inequality (13), we deduce Theorem 3.2 and Proposition 3.4 (ii).

Let x ∈ W. We remark that the arguments of Burnol [Bur92, Theorem B] imply an upper bound for h_{NT} (x) in terms of certain Arakelov invariants of X. However, it turns out that the bound for h_{NT} (x) in Lemma 6.2 leads to a better inequality in Theorem 3.2.

7 Genus two curves

In this section we prove Theorem 3.3 and Proposition 3.4 (iii). Let c_{δ}(2) be the minimum of Faltings’ delta invariant on M_{2}(ℂ), see (9). The following lemma was established in [vK14, Proposition 5.1 (v)].

Lemma 7.1. It holds −186 ≤ c_{δ}(2).

We now combine Lemma 7.1 with the arguments used in the proof of Proposition 5.3 in order to prove Theorem 3.3 and Proposition 3.4 (iii).

Proof of Theorem 3.3 and Proposition 3.4 (iii). Let X be a genus two curve which is defined over a number field K. It is a hyperelliptic curve over K by [Liu02, Proposition 7.4.9]. Thus there exists a finite morphism X → ℙ_{K}^{1} which is geometrically a cyclic cover of prime degree two. As before, we denote by h and h_{NT} the stable Arakelov height and the Néron-Tate height respectively, defined in (4) and (5). Then Lemma 4.3, Lemma 4.4, inequality (13) and Lemma 7.1 give infinitely many points x ∈ X(K̄) that satisfy

\frac{1}{4}h_{NT} (x) ≤ h(x) ≤ 6u(2)μ_{X} + 101

for μ_{X} as in (12) and u(2) as in Proposition 3.4. Therefore the upper bounds for μ_{X}, given in Lemma 5.2 (i) and Lemma 5.2 (ii), imply Theorem 3.3 and Proposition 3.4 (iii) respectively.

References

[Ara74] S. J. Arakelov, An intersection theory for divisors on an arithmetic surface, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 1179–1192.

[Bel79] G. V. Belyĭ, Galois extensions of a maximal cyclotomic field, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 2, 267–276, 479.

[BG06] E. Bombieri and W. Gubler, Heights in Diophantine geometry, New Mathematical Monographs, vol. 4, Cambridge University Press, Cambridge, 2006.

[Bur92] J.-F. Burnol, Weierstrass points on arithmetic surfaces, Invent. Math. 107 (1992), no. 2, 421–432.

[BW07] A. Baker and G. Wüstholz, Logarithmic forms and Diophantine geometry, New Mathematical Monographs, vol. 9, Cambridge University Press, Cambridge, 2007.

[Elk91] N. D. Elkies, ABC implies Mordell, Internat. Math. Res. Notices (1991), no. 7, 99–109.

[Fal83] G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), no. 3, 349–366.

[Fal84] , Calculus on arithmetic surfaces, Ann. of Math. (2) 119 (1984), no. 2, 387–424.

[GR72] A. Grothendieck and M. Raynaud, Modéles de Néron et monodromie, Groupes de monodromie en Géometrie Algebrique, I, II, Springer, 1972, pp. 313–523. Lecture Notes in Math., Vol. 288.

[Jav14] A. Javanpeykar, Polynomial bounds for Arakelov invariants of Belyi curves, with an appendix by P. Bruin, Algebra Number Theory 8 (2014), no. 1, 89–140.

[JvK13] A. Javanpeykar and R. von Känel, Szpiro’s small points conjecture for cyclic covers, Preprint: arXiv:1311.0043 (2013), 22 pages.

[dJ05] R. de Jong, Arakelov invariants of Riemann surfaces, Doc. Math. 10 (2005), 311–329.

[dJ09] , Local invariants attached to Weierstrass points, Manuscripta Math. 129 (2009), no. 3, 273–292.

[dJR11] R. de Jong and G. Rémond, Conjecture de Shafarevitch effective pour les revêtements cycliques, Algebra Number Theory 5 (2011), no. 8, 1133–1143.

[vK14] R. von Känel, On Szpiro’s Discriminant Conjecture, Int. Math. Res. Notices (2014), no. 16, 4457-4491.

[Kha02] L. S. Khadjavi, An effective version of Belyi’s theorem, J. Number Theory 96 (2002), no. 1, 22–47.

[Lev12] A. Levin, Siegel’s theorem and the Shafarevich conjecture, J. Théor. Nombres Bordeaux 24 (2012), no. 3, 705–727.

[Liţ04] R. Liţcanu, Propriétés du degré des morphismes de Belyi, Monatsh. Math. 142 (2004), no. 4, 327–340.

[Liu02] Q. Liu, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, vol. 6, Oxford University Press, Oxford, 2002, Oxford Science Publications.

[LL99] Q. Liu and D. Lorenzini, Models of curves and finite covers, Compositio Math. 118 (1999), no. 1, 61–102.

[Mas02] D. W. Masser, On abc and discriminants, Proc. Amer. Math. Soc. 130 (2002), no. 11, 3141–3150.

[MB89] L. Moret-Bailly, La formule de Noether pour les surfaces arithmétiques, Invent. Math. 98 (1989), no. 3, 491–498.

[MB90] , Hauteurs et classes de Chern sur les surfaces arithmétiques, Astérisque (1990), no. 183, 37–58, Séminaire sur les Pinceaux de Courbes Elliptiques (Paris, 1988).

[Mum07] D. Mumford, Tata lectures on theta. II, Modern Birkhäuser Classics, Birkhäuser Boston Inc., Boston, MA, 2007, Jacobian theta functions and differential equations, With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman and H. Umemura, Reprint of the 1984 original.

[Oor74] F. Oort, Hyperelliptic curves over number fields, Classification of algebraic varieties and compact complex manifolds, Springer, Berlin, 1974, pp. 211–218. Lecture Notes in Math., Vol. 412.

[Par68] A. N. Paršin, Algebraic curves over function fields. I, Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 1191–1219.

[Par72] , Minimal models of curves of genus 2, and homomorphisms of abelian varieties defined over a field of finite characteristic, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), no. 1, 67–109.

[Par88] A. N. Parshin, The Bogomolov-Miyaoka-Yau inequality for the arithmetical surfaces and its applications, Séminaire de Théorie des Nombres, Paris 1986–87, Progr. Math., vol. 75, Birkhäuser Boston, Boston, MA, 1988, pp. 299–312.

[Paz12] F. Pazuki, Theta height and Faltings height, Bull. Soc. Math. France 140 (2012), no. 1, 19–49.

[Rém10] G. Rémond, Nombre de points rationnels des courbes, Proc. Lond. Math. Soc. (3) 101 (2010), no. 3, 759–794.

[ST68] J.-P. Serre and J. Tate, Good reduction of abelian varieties, Ann. of Math. (2) 88 (1968), 492–517.

[Szp81] L. Szpiro, Séminaire sur les Pinceaux de Courbes de Genre au Moins Deux, Astérisque, vol. 86, Société Mathématique de France, Paris, 1981.

[Szp85a] , La conjecture de Mordell (d’après G. Faltings), Astérisque (1985), no. 121-122, 83–103, Seminar Bourbaki, Vol. 1983/84.

[Szp85b] L. Szpiro (ed.), Séminaire sur les pinceaux arithmétiques: la conjecture de Mordell, Société Mathématique de France, Paris, 1985, Papers from the seminar held at the École Normale Supérieure, Paris, 1983–84, Astérisque No. 127 (1985).

[Szp85c] , Un peu d’effectivité, Astérisque (1985), no. 127, 275–287, Seminar on arithmetic bundles: the Mordell conjecture (Paris, 1983/84).

[Szp86] , Small points and torsion points, The Lefschetz centennial conference, Part I (Mexico City, 1984), Contemp. Math., vol. 58, Amer. Math. Soc., Providence, RI, 1986, pp. 251–260.

[Szp90a] L. Szpiro (ed.), Séminaire sur Les Pinceaux de Courbes Elliptiques, Société Mathématique de France, Paris, 1990, À la recherche de “Mordell effectif”. Papers from the seminar held in Paris, 1988, Astérisque No. 183 (1990).

[Szp90b] , Sur les propriétés numériques du dualisant relatif d’une surface arithmétique, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 229–246.

[Zha92] S. Zhang, Positive line bundles on arithmetic surfaces, Ann. of Math. (2) 136 (1992), no. 3, 569–587.

[Zog91] P. Zograf, A spectral proof of Rademacher’s conjecture for congruence subgroups of the modular group, J. Reine Angew. Math. 414 (1991), 113–116.

Ariyan Javanpeykar Mathematical Institute University of Leiden 2300RA Leiden The Netherlands Current address: Institut für Mathematik J.G.-Universität Mainz 55099 Mainz Germany peykar@uni-mainz.de Rafael von Känel IHÉS 35 Route de Chartres 91440 Bures-sur-Yvette France Current address: MPIM Bonn Vivatsgasse 7 53111 Bonn Germany rvk@mpim-bonn.mpg.de 1104

Documenta Mathematica 19 (2014)