Polynomial bounds for Arakelov invariants of Belyi curves

Polynomial bounds for Arakelov invariants of Belyi curves

Abstract.

We explicitly bound the Faltings height of a curve over polynomially in its Belyi degree. Similar bounds are proven for three other Arakelov invariants: the discriminant, Faltings’ delta invariant and the self-intersection of the dualizing sheaf. Our results allow us to explicitly bound these Arakelov invariants for modular curves, Hurwitz curves and Fermat curves in terms of their genus. Moreover, as an application, we show that the Couveignes-Edixhoven-Bruin algorithm to compute coefficients of modular forms for congruence subgroups of runs in polynomial time under the Riemann hypothesis for -functions of number fields. This was known before only for certain congruence subgroups. Finally, we use our results to prove a conjecture of Edixhoven, de Jong and Schepers on the Faltings height of a cover of with fixed branch locus.

Key words and phrases:
Arakelov theory, Belyi degree, arithmetic surfaces, Riemann surfaces, Arakelov invariants, Faltings height, discriminant, Faltings’ delta invariant, self-intersection of the dualizing sheaf, branched covers
1991 Mathematics Subject Classification:
11G30, 11G32, 11G50, 14G40, 14H55, 37P30

1. Introduction and statement of results

We prove that stable Arakelov invariants of a curve over a number field are polynomial in the Belyi degree. We apply our results to give algorithmic, geometric and Diophantine applications.

1.1. Bounds for Arakelov invariants of three-point covers

Let be an algebraic closure of the field of rational numbers . Let be a smooth projective connected curve over of genus . Belyi [3] proved that there exists a finite morphism ramified over at most three points. Let denote the Belyi degree of , i.e., the minimal degree of a finite morphism unramified over . Since the topological fundamental group of the projective line minus three points is finitely generated, the set of -isomorphism classes of curves with bounded Belyi degree is finite.

We prove that, if , the Faltings height , the Faltings delta invariant , the discriminant and the self-intersection of the dualizing sheaf are bounded by a polynomial in ; the precise definitions of these Arakelov invariants of are given in Section 2.3.

Theorem 1.1.1.

For any smooth projective connected curve over of genus ,

The Arakelov invariants in Theorem 1.1.1 all have a different flavour to them. For example, the Faltings height plays a key role in Faltings’ proof of his finiteness theorem on abelian varieties; see [16]. On the other hand, the strict positivity of (when ) is related to the Bogomolov conjecture; see [45]. The discriminant “measures” the bad reduction of the curve , and appears in Szpiro’s discriminant conjecture for semi-stable elliptic curves; see [44]. Finally, as was remarked by Faltings in his introduction to [17], Faltings’ delta invariant can be viewed as the minus logarithm of a “distance” to the boundary of the moduli space of compact connected Riemann surfaces of genus .

We were first led to investigate this problem by work of Edixhoven, de Jong and Schepers on covers of complex algebraic surfaces with fixed branch locus; see [15]. They conjectured an arithmetic analogue ([15, Conjecture 5.1]) of their main theorem (Theorem 1.1 in loc. cit.). We use our results to prove this conjecture; see Section 6 for a more precise statement.

1.2. Outline of proof

To prove Theorem 1.1.1 we will use Arakelov theory for curves over a number field . To apply Arakelov theory in this context, we will work with arithmetic surfaces associated to such curves, i.e., regular projective models over the ring of integers of . We refer the reader to Section 2.2 for precise definitions and basic properties of Arakelov’s intersection pairing on an arithmetic surface. Then, for any smooth projective connected curve over of genus , we define the Faltings height , the discriminant , Faltings’ delta invariant and the self-intersection of the dualizing sheaf in Section 2.3. These are the four Arakelov invariants appearing in Theorem 1.1.1.

We introduce two functions on in Section 2.3: the canonical Arakelov height function and the Arakelov norm of the Wronskian differential. We show that, to prove Theorem 1.1.1, it suffices to bound the canonical height of some non-Weierstrass point and the Arakelov norm of the Wronskian differential at this point; see Theorem 2.4.1 for a precise statement.

We estimate Arakelov-Green functions and Arakelov norms of Wronskian differentials on finite étale covers of the modular curve in Theorem 3.4.5 and Proposition 3.5.1, respectively. In our proof we use an explicit version of a result of Merkl on the Arakelov-Green function; see Theorem 3.1.2. This version of Merkl’s theorem was obtained by Peter Bruin in his master’s thesis. The proof of this version of Merkl’s theorem is reproduced in the appendix by Peter Bruin.

In Section 4 we prove the existence of a non-Weierstrass point on of bounded height; see Theorem 4.5.2. The proof of Theorem 4.5.2 relies on our bounds for Arakelov-Green functions (Theorem 3.4.5), the existence of a “wild” model (Theorem 4.3.2) and Lenstra’s generalization of Dedekind’s discriminant conjecture for discrete valuation rings of characteristic 0 (Proposition 4.1.1).

A precise combination of the above results constitutes the proof of Theorem 1.1.1 given in Section 4.6.

1.3. Arakelov invariants of covers of curves with fixed branch locus

We apply Theorem 1.1.1 to prove explicit bounds for the height of a cover of curves. Let us be more precise.

For any finite subset and integer , the set of smooth projective connected curves over such that there exists a finite morphism étale over of degree is finite. In particular, the Faltings height of is bounded by a real number depending only on and . In this section we give an explicit version of this statement. To state our result we need to define the height of .

For any finite set , define the (exponential) height as , where the height of an element in is defined as . Here is a number field containing and the product runs over the set of normalized valuations of . (As in [26, Section 2] we require our normalization to be such that the product formula holds.)

Theorem 1.3.1.

Let be a non-empty open subscheme in with complement . Let be the number of elements in the orbit of under the action of . Then, for any finite morphism étale over , where is a smooth projective connected curve over of genus ,

Theorem 1.3.1 is a consequence of Theorem 6.0.1. Note that in Theorem 6.0.1 we consider branched covers of any curve over (i.e., not only ). We use Theorem 1.3.1 to prove [15, Conjecture 5.1].

1.4. Diophantine application

Explicit bounds for Arakelov invariants of curves of genus over a number field and with bad reduction outside a finite set of finite places of imply famous conjectures in Diophantine geometry such as the effective Mordell conjecture and the effective Shafarevich conjecture; see [38] and [40]. We note that Theorem 1.1.1 shows that one “could” replace Arakelov invariants by the Belyi degree to prove these conjectures. We use this philosophy to deal with cyclic covers of prime degree. In fact, in [22], joint with von Känel, we utilize Theorem 1.1.1 and the theory of logarithmic forms to prove Szpiro’s small points conjecture ([42, p. 284] and [43]) for curves that are cyclic covers of the projective line of prime degree; see [22, Theorem 3.1] for a precise statement. In particular, we prove Szpiro’s small points conjecture for hyperelliptic curves.

1.5. Modular curves, Fermat curves, Hurwitz curves and Galois Belyi curves

Let be a smooth projective connected curve over of genus . We say that is a Fermat curve if there exists an integer such that is isomorphic to the planar curve . Moreover, we say that is a Hurwitz curve if . Also, we say that is a Galois Belyi curve if the quotient is isomorphic to and the morphism is ramified over exactly three points; see [8, Proposition 2.4], [47] or [48]. Note that Fermat curves and Hurwitz curves are Galois Belyi curves. Finally, we say that is a modular curve if is a classical congruence modular curve with respect to some (hence any) embedding .

If is a Galois Belyi curve, we have . In [49] Zograf proved that, if is a modular curve, then . Combining these bounds with Theorem 1.1.1 we obtain the following corollary.

Corollary 1.5.1.

Let be a smooth projective connected curve over of genus . Suppose that is a modular curve or Galois Belyi curve. Then

Remark 1.5.2.

Let be a finite index subgroup, and let be the compactification of obtained by adding the cusps, where acts on the complex upper half-plane via Möbius transformations. Let denote the compactification of . The inclusion induces a morphism . For an embedding, there is a unique finite morphism of smooth projective connected curves over corresponding to . The Belyi degree of is bounded from above by the index of in . In particular,

Remark 1.5.3.

Non-explicit versions of Corollary 1.5.1 were previously known for certain modular curves. Firstly, polynomial bounds for Arakelov invariants of with squarefree were previously known; see [46, Théorème 1.1], [46, Corollaire 1.3], [1], [34, Théorème 1.1] and [25]. The proofs of these results rely on the theory of modular curves. Also, similar results for Arakelov invariants of with squarefree were shown in [13] and [32]. Finally, bounds for the self-intersection of the dualizing sheaf of a Fermat curve of prime exponent are given in [9] and [27].

1.6. The Couveignes-Edixhoven-Bruin algorithm

Corollary 1.5.1 guarantees that, under the Riemann hypothesis for -functions of number fields, the Couveignes-Edixhoven-Bruin algorithm to compute coefficients of modular forms runs in polynomial time; see Theorem 5.0.1 for a more precise statement.

Conventions

By we mean the principal value of the natural logarithm. Finally, we define the maximum of the empty set and the product taken over the empty set as 1.

Acknowledgements

I would like to thank Peter Bruin, Bas Edixhoven and Robin de Jong. They introduced us to Arakelov theory and Merkl’s theorem, and I am grateful to them for many inspiring discussions and their help in writing this article. Also, I would like to thank Rafael von Känel and Jan Steffen Müller for motivating discussions about this article. I would like to thank Jean-Benoît Bost and Gerard Freixas for discussions on Arakelov geometry, Yuri Bilu for inspiring discussions, Jürg Kramer for discussions on Faltings’ delta invariant, Hendrik Lenstra and Bart de Smit for their help in proving Proposition 4.1.1, Qing Liu for answering our questions on models of finite morphisms of curves and Karl Schwede for helpful discussions about the geometry of surfaces.

2. Arakelov geometry of curves over number fields

We are going to apply Arakelov theory to smooth projective geometrically connected curves  over number fields . In [2] Arakelov defined an intersection theory on the arithmetic surfaces attached to such curves. In [17] Faltings extended Arakelov’s work. In this section we aim at giving the necessary definitions and results for what we need later (and we need at least to fix our notation).

We start with some preparations concerning Riemann surfaces and arithmetic surfaces. In Section 2.3 we define the (stable) Arakelov invariants of appearing in Theorem 1.1.1. Finally, we prove bounds for Arakelov invariants of in the height and the Arakelov norm of the Wronskian differential of a non-Weierstrass point; see Theorem 2.4.1.

2.1. Arakelov invariants of Riemann surfaces

Let be a compact connected Riemann surface of genus . The space of holomorphic differentials carries a natural hermitian inner product:

For any orthonormal basis with respect to this inner product, the Arakelov -form is the smooth positive real-valued -form on  given by . Note that is independent of the choice of orthonormal basis. Moreover, .

Let be the Arakelov-Green function on , where denotes the diagonal; see [2], [11], [14] or [17]. The Arakelov-Green functions determine certain metrics whose curvature forms are multiples of , called admissible metrics, on all line bundles , where is a divisor on , as well as on the holomorphic cotangent bundle . Explicitly: for a divisor on , the metric on satisfies for all away from the support of , where . Furthermore, for a local coordinate at a point in , the metric on the sheaf satisfies

We will work with these metrics on  and (as well as on tensor product combinations of them) and refer to them as Arakelov metrics. A metrised line bundle is called admissible if, up to a constant scaling factor, it is isomorphic to one of the admissible bundles . The line bundle endowed with the above metric is admissible; see [2].

For any admissible line bundle , we endow the determinant of cohomology

of the underlying line bundle with the Faltings metric; see [17, Theorem 1]. We normalize this metric so that the metric on is induced by the hermitian inner product on  given above.

Let be the Siegel upper half space of complex symmetric -by--matrices with positive definite imaginary part. Let in  be the period matrix attached to a symplectic basis of and consider the analytic Jacobian attached to . On one has a theta function , giving rise to a reduced effective divisor  and a line bundle on . The function is not well-defined on  . Instead, we consider the function

(1)

with . One can check that descends to a function on . Now consider on the other hand the set of divisor classes of degree on . It comes with a canonical subset given by the classes of effective divisors and a canonical bijection mapping onto . As a result, we can equip with the structure of a compact complex manifold, together with a divisor and a line bundle . Note that we obtain as a function on . It can be checked that this function is independent of the choice of . Furthermore, note that gives a canonical way to put a metric on the line bundle on .

For any line bundle of degree  there is a canonical isomorphism from to , the fibre of at the point in determined by . Faltings proves that when we give both sides the metrics discussed above, the norm of this isomorphism is a constant independent of ; see [17, Section 3]. We will write this norm as and refer to as Faltings’ delta invariant of .

Let be the invariant of defined in [11, Definition 2.2]. More explicitly, by [11, Theorem 2.5],

(2)

where is any point on . It is related to Faltings’ delta invariant . In fact, let be an orthonormal basis of . Let be a point on and let be a local coordinate about . Write for . We have a holomorphic function

locally about from which we build the -fold holomorphic differential . It is readily checked that this holomorphic differential is independent of the choice of local coordinate and orthonormal basis. Thus, the holomorphic differential extends over to give a non-zero global section, denoted by , of the line bundle . The divisor of the non-zero global section , denoted by , is the divisor of Weierstrass points. This divisor is effective of degree . We follow [11, Definition 5.3] and denote the constant norm of the canonical isomorphism of (abstract) line bundles

by . Then,

(3)

Moreover, for any non-Weierstrass point in ,

(4)

2.2. Arakelov’s intersection pairing on an arithmetic surface

Let be a number field with ring of integers , and let . Let be an arithmetic surface, i.e., an integral regular flat projective -scheme of relative dimension 1 with geometrically connected fibres. For the sake of clarity, let us note that is a regular projective model of the generic fibre in the sense of [29, Definition 10.1.1].

In this section, we will assume the genus of the generic fibre to be positive. An Arakelov divisor on is a divisor on , plus a contribution running over the embeddings of into the complex numbers. Here the are real numbers and the are formally the “fibers at infinity”, corresponding to the Riemann surfaces associated to the algebraic curves . We let denote the group of Arakelov divisors on . To a non-zero rational function on , we associate an Arakelov divisor with the usual divisor associated to on , and , where . Here is the Arakelov -form on . We will say that two Arakelov divisors on are linearly equivalent if their difference is of the form for some non-zero rational function on . We let denote the group of Arakelov divisors modulo linear equivalence on .

In [2] Arakelov showed that there exists a unique symmetric bilinear map with the following properties:

  • if and are effective divisors on without common component, then

    where runs over the complex embeddings of . Here denotes the usual intersection number of and as in [29, Section 9.1], i.e.,

    where runs over the set of closed points of , is the intersection multiplicity of and at and denotes the residue field of . Note that if or is vertical, the sum is zero;

  • if is a horizontal divisor of generic degree over , then for every ;

  • if are complex embeddings, then .

An admissible line bundle on is the datum of a line bundle on , together with admissible metrics on the restrictions of to the . Let denote the group of isomorphism classes of admissible line bundles on . To any Arakelov divisor with , we can associate an admissible line bundle . In fact, for the underlying line bundle of we take . Then, we make this into an admissible line bundle by equipping the pull-back of to each with its Arakelov metric, multiplied by . This induces an isomorphism

In particular, the Arakelov intersection of two admissible line bundles on is well-defined.

Recall that a metrised line bundle on corresponds to an invertible -module, , say, with hermitian metrics on the . The Arakelov degree of  is the real number defined by:

where is any non-zero element of  (independence of the choice of  follows from the product formula).

Note that the relative dualizing sheaf of is an admissible line bundle on if we endow the restrictions of to the with their Arakelov metric. Furthermore, for any section , we have

where we endow the line bundle on with the pull-back metric.

Definition 2.2.1.

We say that is semi-stable (or nodal) over if every geometric fibre of over is reduced and has only ordinary double singularities; see [29, Definition 10.3.1]. We say that is (relatively) minimal if it does not contain any exceptional divisor; see [29, Definition 9.3.12].

Remark 2.2.2.

Suppose that is semi-stable over and minimal. The blowing-up along a smooth closed point on is semi-stable over , but no longer minimal.

2.3. Arakelov invariants of curves

Let be a smooth projective connected curve over of genus . Let be a number field such that has a semi-stable minimal regular model ; see Theorems 10.1.8, 10.3.34.a and 10.4.3 in [29]. (Note that we implicitly chose an embedding .)

The Faltings delta invariant of , denoted by , is defined as

where runs over the complex embeddings of into . Similarly, we define

Moreover, we define

The Faltings height of is defined by

where we endow the determinant of cohomology with the Faltings metric; see Section 2.1. Note that coincides with the stable Faltings height of the Jacobian of ; see [41, Lemme 3.2.1, Chapitre I]. Furthermore, we define the self-intersection of the dualizing sheaf of , denoted by , as

where we use Arakelov’s intersection pairing on the arithmetic surface . The discriminant of , denoted by , is defined as

where runs through the maximal ideals of and denotes the number of singularities in the geometric fibre of over . These invariants of are well-defined; see [36, Section 5.4].

To bound the above Arakelov invariants, we introduce two functions on : the height and the Arakelov norm of the Wronskian differential. More precisely, let and suppose that induces a section of over . Then we define the height of , denoted by , to be

Note that the height of is the stable canonical height of a point, in the Arakelov-theoretic sense, with respect to the admissible line bundle . We define the Arakelov norm of the Wronskian differential at as

These functions on are well-defined; see [36, Section 5.4].

Changing the model for might change the height of a point. Let us show that the height of a point does not become smaller if we take another regular model over .

Lemma 2.3.1.

Let be an arithmetic surface. Assume that is a model for . If denotes the section of over induced by , then

Proof.

By the minimality of , there is a unique birational morphism ; see [29, Corollary 9.3.24]. By the factorization theorem, this morphism is made up of a finite sequence

of blowing-ups along closed points; see [29, Theorem 9.2.2]. For , let denote the exceptional divisor of . Since the line bundles and agree on , there is an integer such that

Applying the adjunction formula, we see that . Since restricts to the identity morphism on the generic fibre, we have a canonical isomorphism of admissible line bundles

Let denote the section of over induced by . Then

where we used the projection formula in the last equality. Therefore, we conclude that

2.4. Bounding Arakelov invariants in the height of a non-Weierstrass point

In this section we prove bounds for Arakelov invariants of curves in the height of a non-Weierstrass point and the Arakelov norm of the Wronskian differential in this point.

Theorem 2.4.1.

Let be a smooth projective connected curve over of genus . Let . Then

Suppose that is not a Weierstrass point. Then

This theorem is essential to the proof of Theorem 1.1.1 given in Section 4.5. We give a proof of Theorem 2.4.1 at the end of this section.

Lemma 2.4.2.

For a smooth projective connected curve over of genus ,

Proof.

We kindly thank R. de Jong for sharing this proof with us. We follow the idea of [19, Section 2.3.2], see also [10, Appendice]. Let be the Siegel fundamental domain of dimension in the Siegel upper half-space , i.e., the space of complex -matrices in such that the following properties are satisfied. Firstly, for every element of , we have . Secondly, for every in , we have , and finally, is Minkowski-reduced, i.e., for all and for all such that are non-zero, we have and, for all we have . One can show that contains a representative of each -orbit in .

Let be a number field such that has a model over . For every embedding , let be an element of such that as principally polarized abelian varieties, the matrix of the Riemann form induced by the polarization of being on the canonical basis of . By a result of Bost (see [19, Lemme 2.12] or [37]), we have

(5)

Here we used that the Faltings height of equals the Faltings height of its Jacobian. Now, let be the Riemann theta function as in Section 2.1, where is in and is in with . Combining (5) with the upper bound

(6)

implies the result. Let us prove (6). Note that, if we write for in ,

Since is Minkowski reduced, we have for all in . Here . Also, for all (see [21, Chapter V.4] for these facts). We deduce that

This proves (6). ∎

Lemma 2.4.3.

Let and . Then, for all real numbers ,

Proof.

It suffices to prove that for all . To prove this, let . Then, if , we have . (To prove that , we may assume that . It is easy to show that is a non-decreasing function for . Therefore, for all , we conclude that .) If , the function attains its minimum value at on the interval . ∎

Lemma 2.4.4.

(Bost) Let be a smooth projective connected curve over of genus . Then

Proof.

See [18, Corollaire 8.4]. (Note that the Faltings height utilized by Bost, Gaudron and Rémond is bigger than due to a difference in normalization. In fact, we have . In particular, the slightly stronger lower bound holds.) ∎

Lemma 2.4.5.

Let be a smooth projective connected curve over of genus . Then

Proof.

By the explicit formula (2) for in Section 2.1 and our bounds on theta functions (Lemma 2.4.2),

Since , the statement follows from Lemma 2.4.3 (with , and ). ∎

Lemma 2.4.6.

Let be a smooth projective connected curve of genus over . Then

Proof.

By [11, Proposition 5.6],