{'fraction_non_alphanumeric': 0.061865767064776776, 'fraction_numerical': 0.01431019941779797, 'mean_word_length': 3.6499683744465528, 'pattern_counts': {'":': 0, '<': 7, '<?xml version=': 0, '>': 6, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 12, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
The singularities of the invariant metric on the line bundle of Jacobi forms

A theorem by Mumford implies that every automorphic line bundle on a pure open Shimura variety, equipped with an invariant smooth metric, can be uniquely extended as a line bundle on a toroidal compactification of the variety, in such a way that the metric acquires only logarithmic singularities. This result is the key of being able to compute arithmetic intersection numbers from these line bundles. Hence it is natural to ask whether Mumford's result remains valid for line bundles on mixed Shimura varieties. In this paper we examine the simplest case, namely the sheaf of Jacobi forms on the universal elliptic curve. We show that Mumford's result cannot be extended directly to this case and that a new interesting kind of singularities appears. By using the theory of b-divisors, we show that an analogue of Mumford's extension theorem can be obtained. We also show that this extension is meaningful because it satisfies Chern-Weil theory and a Hilbert-Samuel type of formula.


Introduction
In [10], [11], Faltings introduced the notion of logarithmically singular metrics on a projective variety defined over a number field and proved that they satisfy a Northcott type property, namely that the set of algebraic points not lying on the singular set of the metric with bounded height and degree, is finite. A prominent example of logarithmically singular metric is the Hodge bundle ω on a toroidal compactification of the moduli space of principally polarized abelian varieties of dimension g (with level structure if you do not want to work with stacks) A g equipped with the Petersson metric.
On the other hand, Mumford [22] introduced the concept of a good metric on a vector bundle, which is a class of singular metrics. He showed that, even being singular, Chern-Weil theory carries over to good metrics. He also proved that the invariant metric on a fully decomposable automorphic vector bundle on a toroidal compactification of the quotient of a hermitian symmetric domain by an arithmetic group is a good metric. This fact allowed him to extend Hirzebruch's proportionality principle to non-compact varieties.
The conclusion of the above facts is that the natural metrics that appear when studying vector bundles on toroidal compactifications of pure Shimura varieties are singular, but the singularities are mild enough so we can use the metrics to study geometric and arithmetic problems.
It is natural to ask whether this theory of logarithmically singular metrics can be extended to mixed Shimura varieties, to obtain geometric and arithmetic information of them.
In this paper we examine the first example of a mixed Shimura variety, namely the universal elliptic curve of full level N over the modular curve E 0 (N) → Y (N). On it we consider the line bundle of Jacobi forms equipped with the translation invariant metric.
It turns out that, in this case, a new kind of singularities appears. These new singularities are concentrated in codimension two. Therefore, if we remove a set of codimension two, we can extend the line bundle of Jacobi forms to a line bundle with a good hermitian metric on a partial compactification of E 0 (N). Since algebraic line bundles can be uniquely extended along codimension two subsets, we obtain a line with a singular metric on a compactification E(N) of the universal elliptic curve.
It turns out that this naive approach is not a good idea. First, it is not functorial. If we consider different toroidal compactifications of E 0 (N), then the resulting extensions are not compatible. Second, even if the characteristic forms associated with the metric are locally integrable and define cohomology classes, they fail to satisfy a Chern-Weil theory. The cohomology class of the the characteristic form does not agree with the characteristic class of the extended line bundle.
In this paper we propose a different approach to understand the extension of the line bundle of Jacobi forms to a compactification of the universal elliptic curve. The "right" extension is not a line bundle, but a b-Q-Cartier divisor. That is, a limit of different Cartier divisors with rational coefficients on all possible toroidal compactifications of E 0 (N). Defined in this way, the extension is obviously functorial because we are taking into account all possible toroidal compactifications. What is remarkable is that, with this interpretation, Chern-Weil theory allows us to interpret intersection products in terms of integrals of singular differential forms (see theorems 5.2 and 5.5). Moreover, there is a Hilbert-Samuel type formula relating the asymptotic of the dimension of the space of Jacobi forms with the self-intersection of the b-divisor (Theorem 5.1).
The non-functoriality of the naive extension is exactly the height jumping introduced by Hain (see [13] and [23]). Acknowledgments We have benefited from many discussions with colleagues on the subject of this paper. We want to thank S. Boucksom, R. de Jong, B. Edixhoven, D. Holmes, G. Freixas, A. von Pippich, and M. Sombra for many useful discussions. We thank specially A. von Pippich for pointing to us the Tornheim zeta function that is computed in [25] and R. de Jong that has computed independently the self-intersection product in Theorem 4.11, for sharing with us his work on the asymptotics of the Néron height pairing [14], that gives a complementary point of view on the results of this paper. This research has been conducted during visits of the authors to the Humboldt University of Berlin, the ICMAT at Madrid and the University of Barcelona. Our thanks go to these institutions for their hospitality.

The universal elliptic surface
In the whole paper we fix an integer N ≥ 3. In this section we will revisit the definition of the universal elliptic surface of level N lying over the modular curve of level N. In particular, we will recall the construction of its smooth toroidal compactification. For further details and references the reader is referred [17].
The modular curve of level N. Let H denote the upper halfplane given by and H * := H ∪ P 1 (Q) the extended upper half-plane. The principal congruence subgroup of level N acts in the usual way by fractional linear transformations on H; this action naturally extends to H * . The quotient space X(N) := Γ(N)\H * is called the modular curve of level N; it is the compactification of Y (N) := Γ(N)\H by adding the so-called cusps. The modular curve X(N) is a compact Riemann surface of genus where the index of Γ(N) in SL 2 (Z) is given as The number p N of cusps of X(N) is given by we denote the cusps by P 1 := [∞], P 2 , . . . , P p N . We recall that Γ(N) is a normal subgroup of SL 2 (Z) and that the quotient group SL 2 (Z)/Γ(N) acts transitively on the set of cusps (with stabilizers of order N). Therefore, it suffices in the sequel to consider the cusp P 1 = [∞]. Since N ≥ 3 the group Γ(N) is torsion-free. Therefore, X(N) has no elliptic points. We recall that the modular curve X(N) is the moduli space of elliptic curves with a full level N-structure. A point [τ ] ∈ X(N) corresponds to the isomorphism class of elliptic curves determined by C/(Zτ ⊕ Z) with N-torsion given by (Z τ N ⊕ Z 1 N )/(Zτ ⊕ Z). The universal elliptic surface of level N. We consider the product H×C consisting of elements (τ, z) with τ ∈ H and z = x+iy ∈ C. On H×C the semi-direct product Γ(N)⋉Z 2 acts by the assignment where a b c d ∈ Γ(N) and (λ, µ) ∈ Z 2 . Since N ≥ 3, the group Γ(N) is torsion-free. Hence, the action of Γ(N) ⋉ Z 2 on H × C is free and the quotient space E 0 (N) := Γ(N) ⋉ Z 2 \H × C is a smooth complex surface together with a smooth surjective morphism The surface E 0 (N) is known to extend to a compact complex surface E(N) together with a surjective morphism π : E(N) −→ X(N), the so-called universal elliptic surface of level N. To describe this extension, it suffices to describe the fibers π −1 (P j ) above the cusps (i) The function f satisfies the functional equations , the function f has a Fourier expansion of the form and similar Fourier expansions at the other cusps. We denote the vector space of Jacobi forms of weight k, index m for Γ(N) by J k,m (Γ(N)). If condition (ii) on the Fourier expansions is restricted to the summation over n ∈ N >0 and r ∈ Z such that 4mn − Nr 2 > 0, the function f is called Jacobi cusp form of weight k, index m for Γ(N) and the span of these functions is denoted by J cusp k,m (Γ(N)). If condition (ii) on the Fourier expansions is dropped, the function f is called weak Jacobi form of weight k, index m for Γ(N). The span of these functions is denoted by J weak k,m (Γ(N)). Riemann theta functions. The Riemann theta function θ 1,1 : H × C → C is defined by the convergent power series and satisfies the functional equation for all a b c d , (λ, µ) ∈ SL 2 (Z) ⋉ Z 2 with a character χ(·), which is an 8-th root of unity. Therefore, θ 8 1,1 is a weak Jacobi form of weight 4, index 4 for Γ(1) = SL 2 (Z). Moreover, from the definition power series (2.5) it follows that θ 8 1,1 is a Jacobi form. 6 Dimension formulae. We recall the dimension formulae for the space of Jacobi forms. For simplicity, we restrict ourselves to the case m = k = 4ℓ. We denote by j : From [17], we cite the following result.
Proposition 2.6. There is a distinguished subsheaf F ℓ of the sheaf j * L 4ℓ,4ℓ,N such that we have an isomorphism where Q(n) denotes the largest integer whose square divides n and H(∆) is the Hurwitz class number.
Proof. The first statement is [ To prove the assymptotic estimate one uses that Q(n) is at most √ n, that the number of divisors of an integer n is o(n ε ) for any ε > 0 and that, by the Brauer-Siegel theorem, the Hurwitz class number H(∆) is o(|∆| 1/2+ε ) for any ε > 0.
Remark 2.7. Since dim J 4ℓ,4ℓ Γ(N) − J cusp 4ℓ,4ℓ Γ(N) grows at most linearly with ℓ, we also have the asymptotic formula Translation invariant metric. Here we recall the translation invariant metric on the line bundle L k,m,N .
Lemma 2.9. For f ∈ J weak k,m (Γ(N)), we have for all a b c d , (λ, µ) ∈ Γ(N) ⋉ Z 2 . In particular, this shows that · induces a hermitian metric on the line bundle L k,m,N .
Proof. This is a straightforward calculation.
Lemma 2.10. Locally, in the affine chart W 0 ν over the cusp P 1 = [∞], the hermitian metric · is described by the formula Proof. Taking absolute values, we derive from (2.1) With these formulae we compute .
From this the proof follows immediately from the definition of the hermitian metric · .

Mumford-Lear extensions and b-divisors
In this section we will introduce Mumford-Lear extensions of a line bundle and relate them with b-divisors. We first recall the different notions of growth for metrics and differential forms that will be useful in the sequel.
Notations. Let X be a complex algebraic manifold of dimension d and D a normal crossing divisor of X. Write U = X \ D, and let j : U −→ X be the inclusion.
Let V be an open coordinate subset of X with coordinates z 1 , . . . , z d ; we put r i = |z i |. We say that V is adapted to D, if the divisor D is locally given by the equation z 1 · · · z k = 0. We assume that the coordinate neighborhood V is small enough; more precisely, we will assume that all the coordinates satisfy r i < 1/e e , which implies that log 1/r i > e and log(log 1/r i ) > 1.
If f and g are two functions with non-negative real values, we will for all x in the domain of definition under consideration.
log-log growth forms.
Definition 3.1. We say that a smooth complex function f on X \ D has log-log growth along D, if we have for any coordinate subset V adapted to D and some positive integer M. The sheaf of differential forms on X with log-log growth along D is the subalgebra of j * E * U generated, in each coordinate neighborhood V adapted to D, by the functions with log-log growth along D and the differentials d z i If D is clear form the context, a differential form with log-log growth along D will be called a log-log growth form.
Dolbeault algebra of pre-log-log forms. Clearly, the forms with log-log growth form an algebra but not a differential algebra. To remedy this we impose conditions on the derivatives as well.
Definition 3.3. A log-log growth form ω such that ∂ω,∂ω and ∂∂ω are also log-log growth forms is called a pre-log-log form (along D).
The sheaf of pre-log-log forms is the subalgebra of j * E * U generated by the pre-log-log forms. We will denote this complex by E * X D pre . The pre-log-log forms of degree zero are called pre-log-log functions.
The sheaf E * X D pre , together with its real structure, its bigrading, and the usual differential operators ∂,∂ is easily shown to be a sheaf of Dolbeault algebras. Moreover, it is the maximal subsheaf of Dolbeault algebras of the sheaf of differential forms with log-log growth.
Metrics with logarithmic growth and pre-log metrics. Let L be a line bundle on X and let · be a smooth hermitian metric on L| U . Definition 3.4. We will say that the metric · has logarithmic growth (along D) if, for every point x ∈ X, there is a coordinate neighbourhood V of x adapted to D, a nowhere zero regular section s of L on 9 V , and an integer M ≥ 0 such that Definition 3.6. We will say that the metric · is a pre-log metric (along D) if it has logarithmic growth and, for every rational section s of L, the function log s is a pre-log-log form along D \ div(s) on X \ div(s).
Mumford-Lear extensions. We are now able to define Mumford-Lear extensions. For the remainder of the section we fix a complex algebraic manifold X of dimension d, D and U as before, and we also fix a hermitian line bundle L = (L, · ) on U.
Definition 3.7. We say that L admits a Mumford-Lear extension to X if there is an integer e ≥ 1, a line bundle L on X, an algebraic subset S ⊂ X of codimension at least 2 that is contained in D, a smooth hermitian metric · on L| U that has logarithmic growth along D \ S and an isometry α : (L, · ) ⊗e → (L| U , · ). The 5-tuple (e, L, S, · , α) is called a Mumford-Lear extension of L. When the isomorphism α, the metric and the set S can be deduced by the context, we will denote the Mumford-Lear extension by (e, L). If e = 1, we will denote it by the line bundle L.
Remark 3.8. The name Mumford-Lear extension arises because they generalize (in the case of line bundles) the extensions of hermitian vector bundles considered by Mumford in [22] and the extensions of line bundles considered by Lear in his thesis [21].
The Mumford-Lear extensions satisfy the following unicity property. Proposition 3.9. Assume that L admits a Mumford-Lear extension to X. Let (e 1 , L 1 , S 1 , · 1 , α 1 ) and (e 2 , L 2 , S 2 , · 2 , α 2 ) be two Mumford-Lear extensions of L to X. Then there is a unique isomorphism Proof. The composition α ⊗e 1 2 • (α −1 1 ) ⊗e 2 defines an isomophism between the line bundles L ⊗e 2 1 | U and L ⊗e 1 2 | U that is the only one that makes the diagram in the theorem commutative. Put S = S 1 ∪ S 2 . The proof of [22,Proposition 1.3] shows that this isomorphism extends uniquely to an isomorphism ϕ 1 : L ⊗e 2 1 | X\S → L ⊗e 1 2 | X\S . Since X is smooth and S has codimension 2, the isomorphism ϕ 1 extends to a unique isomorphism ϕ : L ⊗e 2 1 → L ⊗e 1 2 satisfying the condition of the proposition.
The next result is an immediate consequence of Proposition 3.9.

Proof.
This result allows us to associate to each rational section of L a Q-Cartier divisor on X. We will denote by Q-Ca(X) the group of Q-Cartier divisors of X.
Definition 3.11. Assume that L admits a Mumford-Lear extension to X and let (e, L, S, · , α) be one such extension. Let s be a rational section of L. Then we define the divisor of s on X as the Q-Cartier divisor div X (s) = 1 e div(α(s ⊗e )) ∈ Q-Ca(X), where div(α(s ⊗e )) is the divisor of α(s ⊗e ) viewed as a rational section of L on the whole X.
Mumford-Lear extensions and birational maps. We now consider Mumford-Lear extensions on different birational models of X.
Notation 3.12. Let C be the category whose objects are pairs (Y, π Y ), where Y is a smooth complex variety and π Y : Y → X is a proper birational map, and whose morphisms are maps ϕ : We denote by BIR(X) the set of isomorphism classes in C. Since the set of morphisms between two objects of C is either empty or contains a single element, the set BIR(X) is itself a small category equivalent to C. In fact BIR(X) is a directed set. As a shorthand, an element (Y, π Y ) of BIR(X) will be denoted by the variety Y , the map π Y being implicit. For an object Y of BIR(X) we will denote . By abuse of notation we will denote also by π Y the induced proper birational map from U Y to U. Finally, we will denote by BIR ′ (X) the subset consisting of the elements Y with D Y a normal crossing divisor. This is a cofinal subset. Definition 3.13. We say that L admits all Mumford-Lear extensions over X if, for every object Y of BIR ′ (X), the hermitian line bundle π * Y L on U Y admits a Mumford-Lear extension to Y . Definition 3.14. Assume that L admits all Mumford-Lear extensions over X.
. The Q-Cartier divisors of Definition 3.14 do not need to be compatible with inverse images. As we will see in concrete examples, it may happen that there are maps ϕ : This lack of compatibility with inverse images is related with the phenomenon of height jumping (see [13] and [23] for a discussion of height jumping).
In contrast, the divisors associated with Mumford-Lear extensions are compatible with direct images.
Proof. Let T be the subset of Z where ϕ −1 is not defined. Since Z is smooth, hence normal, by Zariski's main theorem, T has codimension at least 2. Write W = Z \ T and let U ′ = U Z ∩ W . Then L induces a line bundle on U ′ that admits a Mumford-Lear extension to W . Since T has codimension 2, the restriction map is an isomorphism. Moreover, using the definition is easy to see that Thus the proposition follows from the commutativity of the diagram .

B-Divisors.
Recall that the Zariski-Riemann space of X is the projective limit We are not going to use the structure of this space, which is introduced here merely in order to make later definitions more suggestive.
For the definition of b-divisors, we will follow the point of view of [3]. The groups Q-Ca(Y ), Y ∈ BIR(X), form a projective system with the push-forward morphisms and an inductive system with the pull-back morphisms. We define the group of Q-Cartier divisor on X as the inductive limit and the group of Q-Weil divisors on X as the projective limit Since, for any map ϕ in BIR(X) the composition ϕ * •ϕ * is the identity, it is easy to see that there is a map Q-Ca(X) → Q-We(X). Note also that, since BIR ′ (X) is cofinal, the above projective and inductive limit can be taken over BIR ′ (X).
The group of Q-Weil divisors of X is closely related to the group of b-divisors of X defined in [24]. Thus a Q-Weil divisors of X will be called a b-divisors of X.
The following definition makes sense thanks to Proposition 3.9.
Definition 3.16. Assume that L admits all Mumford-Lear extensions over X. Let s be rational section of L. Then the b-divisor associated to s is b-div(s) = (div Y (s)) Y ∈BIR ′ (X) ∈ Q-We(X). When it is needed to specify with respect to which metric we are compactifying the divisor, we will write b-div(s, · ).
Integrable b-divisors. From now on we restrict ourselves to the case when X is a surface. We want to extend the intersection product of divisors as much as possible to b-divisors.
It is clear that there is an intersection pairing Then, by the projection formula, the intersection product C Y · E Y does not depend on the choice of Y . Thus we define But, in general, we can not define the intersection product of two elements of Q-We(X). The following definition is the analogue for bdivisors of the concept of L 2 -function. Recall that, since BIR ′ (X) is a directed set, it is in particular a net. 13 Definition 3. 17.
exists and is finite.
exists and is finite.
where E is an exceptional divisor for the map ϕ. Hence, Thus, by the Hodge index theorem, Thus the convergence of (C 2 1,Y ) Y and (C 2 2,Y ) Y implies the convergence of (C 1,Y · C 2,Y ) Y .

The Mumford-Lear extension of the line bundle of Jacobi forms
In this section we will study the Mumford-Lear extensions of the line bundle of Jacobi forms.
The functions f n,m . We first study a family of functions that will be useful latter. Let (n, m) be a pair of coprime positive integers. Let u, v be coordinates of C 2 and denote U n,m ⊂ C 2 the open subset defined by |uv| < 1. Let D ⊂ U n,m be the normal crossing divisor of equation uv = 0. (ii) The equality ∂∂f n,m ∧ ∂∂f n,m = 0 holds. The differential forms f n,m , ∂f n,m , ∂f n,m , and ∂∂f n,m and all the products between them are locally integrable. Moreover, any product of ∂∂f n,m with a pre-log-log form along D is also locally integrable. (iii) Let π : U n,n+m → U n,m be the map given by (s, t) → (st, t). Note that U n,n+m is a chart of the blow-up of U n,m along (0, 0). Then π * f n,m (s, t) = 1 nm(n + m) log(tt) + f n,n+m (s, t).
Proof. Put P n,m (u, v) = n log(uu) + m log(vv) With these notations, we have  We now prove (i). Let p = (0, v 0 ) ∈ D \ {(0, 0)}. Let U be a neighborhood of p such that | log(vv)| ≤ K, uu < 1 and n| log(uu)| ≥ 2mK ≥ 2m| log(vv)|, for some positive constant K. Therefore, on all the points of U, the estimate |P n,m | ≥ n 2 | log(uu)| holds. Then, for (u, v) ∈ U, Similarly, if t 1 and t 2 are smooth tangent vectors on U with bounded coefficients, from (4.2) and (4.3), we derive for suitable positive constants C 1 and C 2 . The estimates (4.5), (4.6) and (4.7) show that f n,m is a pre-log-log function. Thus we have proved (i).
Since pre-log-log forms are always locally integrable (cf. [8, Proposition 7.6]), in order to check (ii), it is only necessary to study a neighborhood of the point (0, 0). Thus we restrict ourselves to the open V defined by |u| < 1/e and |v| < 1/e. We show the local integrability of a form of the type ∂∂f n,m ∧ ϕ for a pre-log-log form ϕ, being the other cases analogous.
By the definition of pre-log-log forms, equation (4.3) shows that ∂∂f n,m ∧ ϕ = g(u, v) d u ∧ d u ∧ d v ∧ d v, with g a function satisfying |g(u, v)| ≤ C 1 |log(log(uu)) log(log(vv))| M |P 3 n,m uuvv| . for certain positive constants C 1 and M. Using the geometric vs. arithmetic mean inequality, and the fact that the logarithm grows slower than any polynomial, we see that g can be bounded as with C 2 and ε positive. Since the differential form is locally integrable, we deduce that ∂∂f n,m ∧ ϕ is locally integrable.
Every product between a smooth form and any of the forms f n,m , ∂f n,m , ∂f n,m , and ∂∂f n,m , will satisfy growth estimates not worse than the one satisfied by ∂∂f n,m ∧ ϕ, except the product ∂∂f n,m ∧ ∂∂f n,m . Since this last product is zero we conclude (ii).
The statement (iii) follows from a direct computation. Consider the divisor on E(N) given by Choose a smooth hermitian metric · ′ on O(C) and let s be a section of O(C) with div s = C.
where ϕ 1 is a pre-log-log along D 0 . (iii) On the affine coordinate chart W 0 j,ν defined on Section 2, we can write where ϕ 2 is pre-log-log along D. In consequence, if we denote also by · the singular metric on O(C) induced by α and · , then the 5-tuple (1, O(C), Σ, · , α) is a Mumford-Lear extension of the hermitian line bundle L to E(N) and the divisor of θ 8 1,1 on the universal elliptic surface E(N) is given by div E(N ) (θ 8 1,1 ) = C. Proof. The metric · on L 4,4,N over the open subset E 0 (N) is induced by a smooth metric on the trivial line bundle over H × C. Since N ≥ 3, the map H × C → E 0 (N) isétale. Hence, the metric · is smooth on E 0 (N). Therefore, the components of div E(N ) (θ 8 1,1 ) that meet the open subset E 0 (N) come from the theta function. Is well known that, for fixed τ ∈ H, the zeros of the Riemann theta function θ 1,1 (τ, z) are located at z ∈ Zτ ⊕ Z; all the zeros are simple. This proves that the restriction of div E(N ) (θ 8 1,1 ) to E 0 (N) is given by 8H. This finishes the proof of (i).
By the normality of the group Γ(N) in SL 2 (Z), in order to prove that L admits a Mumford-Lear extension and compute the divisor div E(N ) (θ 8 1,1 ), it suffices to work over the cusp . Consider the open affine chart W = W 0 1,ν . By Lemma 2.10, We first study the term log |θ 8 1,1 | 2 . For this, we rewrite expression (2.5) defining θ 1,1 in terms of the local coordinates u ν , v ν . Using formulas (2.1), we obtain Since the vertical component Θ 1,ν is characterized by the equation v ν = 0, the multiplicity of θ 1,1 along Θ 1,ν is given by For a real number x we write ⌊x⌋ for the bigger integer smaller or equal to x and ǫ(x) = x − ⌊x⌋. Then one easily checks that Note that this quantity depends on the value of ν and not just on the residue class of ν modulo N. This is because θ 1,1 (τ, z) is a multi-valued function on E 0 (N). Similarly, the multiplicity of θ 1,1 along Θ 1,ν+1 is given by Therefore, on W \ H, we can write where ϕ 3 is a smooth function. We next consider the remaining terms of the expresion of log θ 8 is pre-log-log along D. The terms 4ν 2 N log(v νvν ) and 4(ν+1) 2 N log(u νūν ) add 4ν 2 /N and 4(ν + 1) 2 /N to the multiplicity of the components Θ 1,ν and Θ 1,ν+1 respectively. Summing up, we obtain that where ϕ 2 is pre-log-log along D.
In order to finish the proof of (iii), it only remains to observe that, for ν = 0, . . . , N, Statement (ii) follows from (iii) and the fact that, by Proposition 4.1 (i), the term The self-intersection of C. By Proposition 4.9, the Mumford-Lear extension of L to E(N) is isomorphic to O(C). We next compute the self-intersection C · C.
Proposition 4.10. The self intersection product C · C is given by In particular, for N = 4, we have p 4 = 6, hence C · C = 136.
Proof. Using the adjunction formula (see for instance [17] proof of Proposition 3.2), we obtain Moreover otherwise. From these intersection products and the explicit description of C in (4.8), we derive the result.
The b-divisor of the line bundle of Jacobi forms. Proof. Recall that Σ ⊂ D denotes the set of double points of D. By Proposition 4.9(ii), the restriction L| E(N )\Σ has a pre-log metric along D \ Σ. Therefore, if p ∈ Σ and π : X → E(N) is the blow-up of E(N) at p we deduce that π * O(C) is a Mumford-Lear extension of L and that div X (θ 8 1,1 ) = π * div E(N ) (θ 8 1,1 ). Assume now that p ∈ Σ and π : X → E(N) is the blow-up of E(N) at p. Write Σ X for the set of double points of the total transform of D.
Then #Σ X = #Σ + 1, because we can write Σ X = (Σ \ {p}) ∪ {p 1 , p 2 }, where {p 1 , p 2 } is the intersection of the exceptional divisor E of the blow-up with the strict transform of D.
Proposition 4.1(iii) and Proposition 4.9(iii) imply that is a Mumford-Lear extension of L to X (in this case the co-dimension two set is Σ X , and the isomorphism and the metric are the ones induced by α and · ). Moreover div X (θ 8 1,1 ) = π * div E(N ) (θ 8 1,1 ) −

N E.
A similar phenomenon occurs on any smooth surface birational to E(N). To describe it we need a little of terminology. Let π : X → E(N) be a proper birational map with X smooth. A point p ∈ X will be called mild if the metric of π * L is smooth or pre-log-log in a neighborhood of p. Put Σ X ⊂ X for the set of non mild points. We will say that a point p has type (n, m) and multiplicity µ if there is a coordinate neighborhood centered at p, with coordinates (u, v) such that log θ 8 1,1 = log s ′ + ϕ − µ nm log(uū) log(vv) n log(uū) + m log(vv) .
Observe that E(N) has Np N non-mild points, all of type (1, 1) and multiplicity 4/N. Assume that Σ X is finite and that L admits a Mumford-Lear extension (e X , O(C X ), Σ X , · , α) to X. Let D X be the total transform of D to X. If π : X ′ → X is the blow-up at a mild point p ∈ Σ X , then Σ X ′ = π −1 Σ X is finite and (e X , π * O(C X )) is a Mumford-Lear extension of L to X ′ . In particular, div X ′ (θ 8 1,1 ) = π ′ * div X (θ 8 1,1 ). Let now π : X ′ → X be the blow-up of X at a point p ∈ Σ X , with type (n, m) and multiplicity a/b, with a, b integers. Then, by Proposition 4.1(iii), Σ X ′ = (Σ X \ {p}) ∪ {p 1 , p 2 }, where {p 1 , p 2 } is the intersection of the exceptional divisor E X ′ of the blow-up with the strict transform of D X . Moreover, (bnm(n + m)e X , π * O(bnm(n + m)C X ) ⊗ (−aE X ′ )) is a Mumford-Lear extension of L to X ′ . Hence div X ′ (θ 8 1,1 ) = π * div X (θ 8 1,1 ) − a bnm(n + m) E X ′ .
Note also that the singular point p gives rise to two points in Σ X ′ , both of multiplicity a/b, one of type (n + m, m) and the other of type (n, n + m). Since the self-intersection of the exceptional divisor E X ′ is −1, we deduce that Since the elements of BIR ′ (E(N)) can be obtained by successive blow-ups at points, for all X ∈ BIR ′ (E(N)), the set Σ X is finite and L admits a Mumford-Lear extensions to X. Hence L admits all Mumford-Lear extensions over E(N).
From the previous discussion, it is clear that, to study the b-divisor b-div(θ 8 1,1 ), we can forget the blow-ups at mild points and concentrate on blow-ups along non-mild points.
Consider the labeled binary tree with root labeled by (1, 1) and such that, if a node is labeled (n, m), the two child nodes are labeled (n + m, m) and (n, n + m). Then the labels of the tree are in bijection with the set of ordered pairs of co-prime positive integers. This tree also describes the type of the non mild points that appear by successive blow-ups starting with a point of type (1, 1).
Thus this degree can be interpreted as the product of the weight of the Jacobi form, its index, the index of the subgroup Γ(N) in PSL 2 (Z) and a multiple zeta value.

Interpretation and open questions
In the previous section we have seen that, when taking into account the invariant metric, the natural way to extend the Cartier divisor div(θ 8 1,1 ) associated to the line bundle of Jacobi forms, from the universal family of elliptic curves to a compactification of it, is not as a Cartier divisor, but as a Q-b-divisor. In particular, this implies that we can not restrict ourselves to a single toroidal compactification, but we 22 have to consider the whole tower of toroidal compactifications. Considering purely the arithmetic definition of Jacobi forms, this fact was already observed by the third author [19, Remark 2.19].
In this section we will give further evidence that b-div(θ 8 1,1 ) is the natural extension of div(θ 8 1,1 ) by showing that if satisfies direct generalizations of classical theorems on hermitial line bundles. We will also state some open problems and future lines of research.
Proof. By Remark 2.7 and Theorem 4.11 we have Chern-Weil theory. The second task is to show that the self intersection product in the sense of b-divisors is compatible with Chern-Weil theory. We write holds.
Proof. By propositions 4.1 and 4.9, we know that the integral in the right hand side exists and is finite. Let C be the divisor on Proposition 4.9 and choose a pre-log metric · ′ on O(C) such that, each double point p j,ν = Θ j,ν ∩ Θ j,ν+1 ∈ W 0 j,ν has a neighborhood in which Thus ω = ω ′ + 1 2πi ∂∂f.
Since Chern-Weil theory can be extended to pre-log singularities ( [22], [7]), the equality we are led to compute the second integral of the right hand side of the previous equation. Note that the minus sign in the above formula comes from the fact that d ∂ = −∂∂. Since pre-log-log forms have no residue, in order to compute this integral we can focus on the double points p j,ν , j = 1, . . . , p N , ν = 0, . . . , N − 1 of D. For each point p j,ν and 0 < ε < 1/e, let V j,ν,ε be the poly-cylinder Using that ω ′ is a pre-log-log form and equation (4.2), it is easy to see that lim ε→0 ∂V j,ν,ε 2 2πi ∂f ∧ ω ′ = 0.
Intersections with curves. Similarly, we also note that the intersection of b-div(θ 8 1,1 ) with a curve can also be computed using the differential form c 1 (L 4,4,N , · ).
To a curve C contained in E(N), we associate the b-divisor that, on each X ∈ BIR ′ (E(N)) consist on the strict transform of C on X. We will denote this divisor by b-div(C). Note that this b-divisor is not integrable because by taking successive blow-ups in points of C, the strict transform of C has self-intersection more and more negative. Assume that C is irrecucible and is not contained in D = E(N)\E 0 (N). Then the product b-div(θ 8 1,1 ) · b-div(C) is well defined because after a finite number of blow-ups on the double points of D and of its total transforms, the strict transform of C will not meet any double point of the total transform of D. holds.
Proof. Let X → E(N) be a birational map obtained by successive blow-ups on double points of D and of its total transforms and such that the strict transform of C in X, that we denote by C X , does not meet any double point of the total transform of D to X. Then b-div(θ 8 1,1 ) · b-div(C) = div X (θ 8 1,1 ) · C X . Let (e, L, S, α, · ) be a Mumford-Lear extension of L = (L 4,4,N , · ) to X. Denote by s = α(θ 8e 1,1 ) the rational section of L determined by θ 8 1,1 . Since the metric · is pre-log on X \ S, we deduce that A toric analogue of the singular metric. We now give an interpretation of the harmonic double value ζ(2, 2; 2)/ζ(6) in terms of toric varieties and the volume of a convex surface. Consider the projective plane P 2 with projective coordinates (x 0 : x 1 : x 2 ) and the canonical line bundle O(1). On this line bundle we can put the canonical metric given by This metric is continuous.
We have an open immersion (C * ) 2 ֒→ P 2 that sends the point (z 1 , z 2 ) to (1 : z 1 : z 2 ). We define the valuation map val : The function log( x 0 can ) is constant along the fibers of val. Thus there exist a function Ψ can : R 2 → R such that log x 0 (p) can = Ψ can (val(p)).
As before, the function log x 0 sing is constant along the fibers of val and defines a function Ψ sing : R 2 → R that is given explicitly by if v ≤ min(0, u).
The function Ψ sing is conic but is not piecewise linear. Assume that we can extend the theory of toric varieties to toric b-divisors. Then to Ψ sing we would associate the convex figure and we should obtain (5.6) b-div(x 0 , · sing ) 2 = 2 Vol(∆ sing ).
We see that this is indeed the case.
Open questions. In this paper we have examined a particular example and observed, just by comparing numbers, that several classical results should be extendable to b-divisors and singular metrics with a shape similar to the one of the translation invariant metrics. We are in the process of investigating the following questions.
(1) Theorem 5.1 shows that the translation invariant metric encodes the asymptotic behavior of the space of Jacobi forms. It is possible to define global sections of a b-divisor. We can ask what is the exact relationship between the space of Jacobi forms and the global sections of the b-divisor div(θ 8 1,1 ). Moreover, once this is settled, we can ask whether there is a Riemann-Roch theorem or a Hilbert-Samuel theorem for b-divisors that imply directly Theorem 5.1.
(2) By Theorem 5.7, it is clear that much of the theory of toric varieties could be extended to toric b-divisors and singular metrics on toric varieties. (3) Theorem 5.2 shows that Chern-Weil theory of singular metrics can be useful to study b-divisors. It would be interesting to generalize this theorem to higher dimensions. In this direction, with R. de Jong and D. Holmes, we have shown that the local integrability property extends, at least, to the case of toroidal compactifications of families of abelian varieties. (4) The original motivation of this paper is to be able to define and study the height of cycles on the universal elliptic curve with respect to the bundle of Jacobi forms equipped with the translation invariant metric, extending the work in [18]. First it is clear how to define the height of an algebraic point of E 0 (N) and one may wonder whether the new singularities are mild enough so that Northcott property is still true. We can also define the height of an algebraic curve not contained in the divisor D. But it is not clear how to define the height of E(N). The naive definition of that height would give the value −∞ but it should be possible to extract a meaningful finite number. To this end, the study of toric varieties might be useful, because the techniques developed in [9] can be extended to the singular metrics of Theorem 5.7. In this case, we obtain that the stability set of the function associated to the metric is no longer a polytope but a convex set. In this case the regularized height should be defined from the integral along this convex set of the roof function, in analogy with [9, Theorem 5.2.5].