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SELF-INTERSECTION OF THE RELATIVE DUALIZING SHEAF ON MODULAR

CURVES X(N)

MIGUEL GRADOS AND ANNA-MARIA VON PIPPICH

Abstract. Let N ≥ 3 be a composite, odd, and square-free integer and let Γ be the principalcongruence subgroup of level N . Let X(N) be the modular curve of genus gΓ associated to Γ. In thisarticle, we study the Arakelov invariant e(Γ) = ω2/ϕ(N), with ω2 denoting the self-intersection of therelative dualizing sheaf for the minimal regular model of X(N), equipped with the Arakelov metric, andwith Euler’s phi function ϕ(N). Our main result is the asymptotics e(Γ) = 2gΓ log(N) + o(gΓ log(N)),as the level N tends to infinity.

1. Introduction

1.1. Relevance of the known asymptotics. Arakelov invariants, such as the Faltings’s deltafunction and the self-intersection of the relative dualizing sheaf, play an important role in Arakelovgeometry. In particular, the self-intersection of the relative dualizing sheaf on an arithmetic surfaceis an essential contribution to the Faltings’s height of the Jacobian [24]. It is known that suitableupper bounds for this invariant lead to important applications in arithmetic geometry. For instance, ananalogue of the Bogomolov–Miyaoka–Yau inequality for arithmetic surfaces implies an effective versionof the Mordell conjecture [23]. Also, sufficiently good upper bounds for the self-intersection of therelative dualizing sheaf are crucial in the work of B. Edixhoven and his co-authors, when estimatingthe running time of his algorithm for determining Galois representations [9]. However, establishingsuch bounds turned out to be a complicated problem. The known bounds have been established sofar by the work of Abbes, Ullmo, Michel, Edixhoven, de Jong, and others, see, e.g., [3], [7], [9], [16],[17], [22]. They all deal with the calculation of the self-intersection of the relative dualizing sheaf forregular models of certain modular curves, Fermat curves, Belyi curves, or hyperelliptic curves, eitherestablishing bounds or computing explicit asymptotics for this numerical invariant.In their influential work [3], A. Abbes and E. Ullmo considered the modular curves X0(N)/Q. Denotingby X0(N)/Z a minimal regular model for X0(N)/Q, and by ωX0(N)/Z the relative dualizing sheaf onX0(N), equipped with the Arakelov metric, they established the asymptotics

ω2X0(N)/Z ∼ 3gΓ0(N) log(N),(1)

as N → ∞, where N is assumed to be square-free with 2, 3 - N and such that the genus gΓ0(N) ofX0(N) is greather than zero (see [3] and [22, Théorème 1.1]). As consequence of this result, P. Micheland E. Ullmo obtained in [22] an asymptotics for Zhang’s admissible self-intersection number [29],and with this they proved an effective version of the Bogomolov conjecture for the curve X0(N)/Q.Namely, if hNT denotes the Néron–Tate height on the Jacobian [4], then for all ε > 0 and sufficientlylarge N , the set

{x ∈ X0(N)(Q) | hNT(x) ≤ (2/3− ε) log(N)}is finite, whereas the set

{x ∈ X0(N)(Q) | hNT(x) ≤ (4/3 + ε) log(N)}is infinite.By its very definition, the self-intersection of the relative dualizing sheaf on modular curves is thesum of a geometric part that encodes the finite intersection of divisors coming from the cusps, and ananalytic part which is given in terms of the Arakelov Green’s function evaluated at these cusps. Theleading term 3gΓ0(N) log(N) in the asymptotics (1) is the sum of gΓ0(N) log(N) that comes from the

geometric part, and 2gΓ0(N) log(N) that comes from the analytic part. J. Kramer conjectured that this

result can be generalized to arbitrary modular curves, that is, that the main term in the asymptoticsof the self-intersection of the relative dualizing sheaf for a semi-stable model of modular curves of level

1

N and genus g is 3g log(N), as N tends to infinity. Following the lines of proof in [3], Kramer’s Ph.D.student H. Mayer [21] obtained a positive answer to this conjecture for the case of modular curvesX1(N)/Q. More precisely, for squarefree N of the form N = N ′ · q · r, where q, r > 4 are differentprimes, he proved the asymptotics

ω2X1(N)/Z ∼ 3gΓ1(N) log(N),

as N →∞; here, X1(N)/Z is a minimal regular model for X1(N)/Q. From this asymptotics he wasthen able to deduce the validity of the Bogomolov conjecture in this case.

1.2. Main result. In this article, we establish asymptotics for the self-intersection of the relativedualizing sheaf on modular curves X(N), as the level N tends to infinity. Following mostly the linesof proof in [3], we will first obtain a formula for this Arakelov invariant, in which the geometric andanalytic parts are explicitly given. Then, we proceed to compute the asymptotics for each of theseparts. Our main result is the following asymptotics (see Theorem 18)

1

ϕ(N)ω2X (N)/Z[ζN ] ∼ 2gΓ(N) log(N),

as N → ∞, where N ≥ 3 is a composite, odd, and square-free integer; here, Z[ζN ] denotes the ringof integers of the cyclotomic field Q(ζN ) with a primitive N -th root of unity ζN , and X (N)/Z(ζN )is a minimal regular model for X(N)/Q(ζN ). It turns out that the right hand side 2gΓ(N) log(N) inthe asymptotics comes from the analytic part, which partially confirms Kramer’s conjecture. Thedifference of our result with the previous cases lies in the underlying minimal regular model, whereall the computations take place, namely, our model is a Z[ζN ]-scheme which is not semi-stable. Theanalogue computations for a semi-stable model are a theme of our future research.

1.3. Sketch of proof. To prove the main result, we follow the strategy given in the work of Abbesand Ullmo [3]. The proof starts by observing that ω2X (N)/Z[ζN ]/ϕ(N) is the sum of an explicit geometric

contribution G(N) and an explicit analytic contribution A(N) (see Proposition 13). In particular,using results of Manin/Drinfeld and Faltings/Hriljac, we show that

G(N) =2gΓ(N)(V0, V∞)fin − (V0, V0)fin − (V∞, V∞)fin

2(gΓ(N) − 1).

We refer the reader to section 6.1 for the definitions of the vertical divisors V0 and V∞. From this, astraight-forward computation yields the asymptotics (see Proposition 14)

1

ϕ(N)G(N) = o(gΓ(N) log(N)),

as N →∞. Furthermore, the analytic contribution A(N is given in terms of the Arakelov (or canonical)Green’s function gAr(·, ·), evaluated at the corresponding cusps, and equals

A(N) = 4gΓ(N)(gΓ(N) − 1)∑

σ:Q(ζN )↪→C

gAr(0σ,∞σ).

To derive the desired asymptotics for A(N), one first uses a fundamental identity by Abbes/Ullmo,which expresses the Arakelov Green’s function gAr(q1, q2) at the cusps q1 and q2 of X(N) in termsthe scattering constant Cq1q2 at q1 and q2, the hyperbolic volume vΓ(N) of X(N), the constant term

in the Laurent expansion at s = 1 of the Rankin–Selberg transforms Rq1 resp. Rq2 of the Arakelovmetric at q1 and q2, respectively, and a contribution G involving the constant term of the automorphicGreen’s function Gs(z, w) at s = 1; we refer the reader to (5), (11), and (14) for precise definitions.More precisely, one has (see (13))

gAr(q1, q2) = −2π Cq1q2 −2π

vΓ(N)+ 2π(Rq1 + Rq2) + 2π G .

The relevant scattering constants Cq1q2 are computed, e.g., in [13], and the asymptotics for G followsfrom bounds proven in [18]. Furthermore, we show that to deal with the terms Rq1 + Rq2 for the cuspsin question one is reduced to only compute R∞ (see Lemma 15). To provide the explicit formula forR∞ given in Proposition 16, we apply methods from the spectral theory of automorphic functions,

2

namely, we represent R∞ in terms of a particular automorphic kernel. Decomposing this automorphickernel into terms involving hyperbolic and parabolic elements, R∞ is divided into hyperbolic andparabolic contributions Rhyp∞ and R

par∞ (see identity (12)). The computation of R

hyp∞ and R

par∞ are the

technical heart of the paper. Note that to determine the hyperbolic contribution Rhyp∞ , we proceeddifferently than Abbes–Ullmo and we reduce the calculation of the residue of the corresponding zetafunction to the well-known residue of the zeta function of a suitably generalized idèle class group ofthe quadratic extension, determined by the corresponding hyperbolic elements. From this, it is finallyshown that (see Proposition 17)

1

ϕ(N)A(N) = 2gΓ(N) log(N) + o(gΓ(N) log(N)),

as N →∞. Adding up the asymptotics for the geometric and for the analytic contribution then provesthe main result.

1.4. Outline of the article. The paper is organized as follows. In Section 2, we set our main notationand review basic facts on modular curves X(N), non-holomorphic Eisenstein series, the spectral theoryof automorphic forms, and the Arakelov metric. The core of this part is the identity (12), which statesthat the Rankin–Selberg constant of the Arakelov metric R∞ at the cusp ∞ is essentially the sumof two contributions Rhyp∞ and R

par∞ , associated to the hyperbolic and parabolic elements of Γ(N),

respectively. In Section 3, we turn our attention to the minimal regular model X/S of the curveX(N), where S = Spec(Z[ζN ]) with ζN an N -th root of unity. Here, in Proposition 1, we describethe good and bad fibers of X/S, and briefly recall the moduli interpretation of the cusps of X(N).Further, in Proposition 2, we describe how the cusps 0 and ∞ are mapped on X(N) under a givenembedding Q(ζN ) ↪−→ C. The importance of this result will be clear in the proof of Proposition 17. InSection 4, we begin with the study of the zeta function associated to an hyperbolic element of Γ(N). InProposition 4, we show that this zeta function is, in fact, equal to a partial zeta function of a suitablenarrow-ray ideal class, up to a holomorphic function. Using this fact, we then proceed to compute

an asymptotics for the hyperbolic contribution Rhyp∞ . This is done in Proposition 6. In Section 5, wedeal with the computation of the parabolic contribution Rpar∞ , and for this we proceed in the sameway as in [3], adapting the results therein to our case. Finally, in Section 6, we state our main resultin Theorem 18, and for the proof, we first provide a formula for the self-intersection of the relativedualizing sheaf in Proposition 13, where the geometric and analytic parts are explicitly given. Then,using the computations from previous sections, we determine the desired asymptotics.

2. Background material

2.1. The modular curve X(N). Let H := {z = x+ iy ∈ C | x, y ∈ R; y > 0} be the hyperbolic upperhalf-plane endowed with the hyperbolic volume form µhyp(z) := dxdy/y

2, and let H∗ := H tP1(R)be the union of H with its topological boundary. The group PSL2(R) acts on H

∗ by fractional lineartransformations. This action is transitive on H, since z = x+ iy = n(x)a(y)i with

n(x) :=

(1 x0 1

), a(y) :=

(y1/2 0

0 y−1/2

).

By abuse of notation, we represent an element of PSL2(R) by a matrix. We set I := ( 1 00 1 ) ∈ PSL2(R).Throughout the article, let N ≥ 3 be a composite, odd, and square-free integer and let Γ := Γ(N)denote the principal congruence subgroup of the modular group PSL2(Z). By Γz := {γ ∈ Γ | γz = z}we denote the stabilizer subgroup of a point z ∈ H∗ with respect to Γ.The quotient space Y (N) := Γ\H resp. X(N) := Γ\H∗ admits the structure of a Riemann surface anda compact Riemann surface of genus gΓ, respectively. The hyperbolic volume form µhyp(z) naturallydefines a (1, 1)-form on Y (N), which we still denote by µhyp(z), since it is PSL2(R)-invariant on H.Thus, the hyperbolic volume of Y (N) is given by vΓ :=

∫Y (N) µhyp(z). The genus gΓ and the hyperbolic

volume vΓ are related by the identity

gΓ = 1 +vΓ4π

(1− 6

N

).

3

By abuse of notation, we identify X(N) with a fundamental domain FΓ ⊂ H∗; thus, at times we willidentify points of X(N) with their pre-images in H∗.

A cusp of X(N) is the Γ-orbit of a parabolic fixed point of Γ in H∗. By PΓ ⊆ P1(Q) we denote acomplete set of representatives for the cusps of X(N) and we write pΓ := #PΓ. We will always identify acusp of X(N) with its representative in PΓ. Hereby, identifying P

1(Q) with Q∪{∞}, we write elementsof P1(Q) as α/β for α, β ∈ Z, not both equal to 0, and we always assume that gcd(α, β) = 1; we set1/0 :=∞. Given a cusp q = α/β ∈ PΓ, we choose the scaling matrix for q by σq := gqa(wq) ∈ PSL2(R),where gq :=

( α ∗β ∗)∈ PSL2(Z) and wq := [PSL2(Z)q : Γq]. Let U ⊂ Z be a set of representatives for

(Z/NZ)×/{±I} containing 1 ∈ Z. Given ξ ∈ U , the element 0ξ := n(N − 1)p(ξ)γN∞ ∈ P1(Q), wherep(ξ) :=

(ξ 1

rN ξ̃

)such that ξξ̃ − rN = 1 with ξ̃, r ∈ Z, and γN :=

(0 −11 0

)defines a cusp of Γ.

2.2. Eisenstein series, scattering constants, and the Rankin–Selberg transform. For a givencusp q ∈ PΓ with scaling matrix σq ∈ PSL2(R), the non-holomorphic Eisenstein series associated to qis defined by

Eq(z, s) :=∑

γ∈Γq\Γ

Im(σ−1q γz)s,

where z ∈ H and s ∈ C with Re(s) > 1. The Eisenstein series Eq(z, s) is Γ-invariant in z, andholomorphic in s for Re(s) > 1. Furthermore, Eq(z, s) admits a meromorphic continuation to the

complex s-plane with a simple pole at s = 1 with residue equal to v−1Γ . Now, for u ∈ U , consider thesubset M(u) ⊂ Z2 given by

M(u) := {(m,n) ∈ Z2 | (m,n) ≡ (0, u) modN},which is invariant under right multiplication by a matrix of Γ. Since we have w∞ = N , g∞ = I, andσ∞ = a(N), the Eisenstein series E∞(z, s) associated to ∞ can be conveniently written as follows

E∞(z, s) =1

N s

∑u∈U

Du(s)

( ∑(m,n)∈M(u)

ys

|mz + n|2s

),

where Du(s) is the Dirichlet series defined by

Du(s) :=

∞∑d=1

du≡1 modN

µ(d)

d2s,(2)

with µ(d) denoting the Möbius function. At s = 1, the series Du(s) admits the following Laurentexpansion ∑

u∈UDu(s) =

1

π

(vΓN3

)−1+O(s− 1).(3)

Let q1, q2 ∈ PΓ be two cusps, not necessarily distinct, with scaling matrices σq1 , σq2 ∈ PSL2(R),respectively. The scattering function ϕq1q2(s) at the cusps q1 and q2 is defined by

ϕq1q2(s) :=√π

Γ(s− 12

)Γ(s)

∞∑c=1

c−2s( ∑

dmod c( ∗ ∗c d )∈σ

−1q1

Γσq2

1

),

where s ∈ C with Re(s) > 1 and Γ(s) denotes the Gamma function. The Eisenstein series admits thefollowing Fourier expansion

Eq1(σq2z, s) = δq1q2ys + ϕq1q2(s)y

1−s +∑n 6=0

ϕq1q2(n; s)y1/2Ks−1/2(2π|n|y)e2πinx,(4)

where δq1q2 is the Dirac’s delta function, Kµ(Z) is the modified Bessel function of the second kind, and

ϕq1q2(n; s) :=2πs

Γ(s)|n|s−1/2

∑c>0

c−2s( ∑

dmod c( a ∗c d )∈σ

−1q1

Γσq2

e2πi(dm+an)/c).

4

The scattering function ϕq1q2(s) is holomorphic for s ∈ C with Re(s) > 1 and admits a meromorphiccontinuation to the complex s-plane. At s = 1, there is always a simple pole of ϕq1q2(s) with residue

equal to v−1Γ . The constant Cq1q2 given by

Cq1q2 := lims→1

(ϕq1q2(s)−

v−1Γs− 1

)(5)

is called the scattering constant at the cusps q1 and q2.

Let f : H −→ C be a Γ-invariant function which is of rapid decay at a cusp q ∈ PΓ, i.e., the 0-thcoefficient a0(y, q) in the Fourier expansion of f(σqz) satisfies a0(y, q) = O(y

−C) for all C > 0, asy →∞. For s ∈ C with Re(s) > 1, the Rankin–Selberg transform of f at q is given by the integral

Rq[f ](s) :=∫FΓf(z)Eq(z, s)µhyp(z).

The function Rq[f ](s) possesses a meromorphic continuation to the complex s-plane having a simplepole at s = 1, with residue equal to (1/vΓ)

∫FΓ f(z)µhyp(z). Furthermore, we have the following useful

identity

Rq[f ](s) =∫ ∞

0a0(y, q)y

s−2dy.

2.3. The spectral expansion and automorphic kernels. Let k be either 0 or 2 and fix it. Letz ∈ H and γ =

(a bc d

)∈ PSL2(R). Given a function f : H −→ C, we define f |[γ; k] by

(f |[γ; k])(z) := jγ(z; k)−1f(γz),where jγ(z; k) := ((cz + d)/|cz + d|)k is the weight-k automorphy factor. A function f : H −→ C is anautomorphic function of weight k with respect to Γ if the equality (f |[γ; k])(z) = f(z) holds for allγ ∈ Γ. Denote by L2(Y (N), k) the Hilbert space consisting of all automorphic functions of weight kwith respect to Γ that are measurable and square integrable, endowed with the inner product given by〈f, g〉 :=

∫FΓ f(z)g(z)µhyp(z). The hyperbolic Laplacian of weight k is defined by

∆hyp,k := −y2(∂2

∂x2+

∂2

∂y2

)+ iky

∂

∂x.

Considering the uppering and lowering Maass operators of weight k, namely

Uk :=k

2+ iy

∂

∂x+ y

∂

∂y, Lk :=

k

2+ iy

∂

∂x− y ∂

∂y,(6)

respectively, the following identities hold

∆hyp,k = Lk+2Uk −k

2

(1 +

k

2

)= Uk−2Lk +

k

2

(1− k

2

).

Since the hyperbolic Laplacian ∆hyp,k of weight k defines a symmetric and essentially self-adjointoperator, it extends to a unique self-adjoint operator ∆k on a suitable domain. Consequently, thereexists a countable orthonormal set {ψj}∞j=0 of eigenfunctions of ∆0, in fact eigenfunctions of ∆hyp,0,such that for all f ∈ L2(Y (N), k), we have the spectral expansion

f(z) =∞∑j=0

〈f, ψj〉ψj(z) +1

4π

∑q∈PΓ

∫ ∞−∞

〈f,Eq,k

(· , 1

2+ ir

)〉Eq,k

(· , 1

2+ ir

)dr,

which converges in the norm topology; moreover, if f is smooth and bounded, then the expansionconverges uniformly on compacta of H. Here, Eq,k(z, s) denotes the weight-k Eisenstein series of Γ atthe cusp q ∈ PΓ. For k = 0, this is the Eisenstein series Eq(z, s) from Section 2.2, but if k = 2, thenEq,2(z, s) is determined by Eq(z, s) via the uppering Maass operator U0 given by (6), namely

U0(Eq(z, s)) = sEq,2(z, s),(7)

where s is not a pole of Eq(z, s). In particular, using (4) and (7), the following Fourier expansion canbe deduced

E∞,2(σ∞z, s) = ys + ϕ∞∞,2(s)y

1−s

5

+y1/2

s

∑n6=0

((1

2− 2πny

)Ks−1/2(2π|n|y) + y

∂

∂yKs−1/2(2π|n|y)

)ϕ∞∞(n; s)e

2πinx,

where we have set

ϕ∞∞,2(s) :=1− ss

ϕ∞∞(s).

Next, we fix real numbers T > 0 and A > 1, once for all. Consider the holomorphic function hT (r)defined on the strip |Im(r)| < A/2 by

hT (r) := exp

(− T

(1

4+ r2

)).(8)

Let φk denote the inverse Selberg/Harish–Chandra transform of weight k of hT (r), namely, we have

φk(x) := −1

π

∫ ∞−∞

Q′(x+ t2)

(√x+ 1 + t2 − t√x+ 1 + t2 + t

)k/2dt

where x ≥ 0 and

Q

(1

4(eu + e−u − 2)

)=

1

2g(u), g(u) =

1

2π

∫ ∞−∞

hT (r)e−irudr.

For z, w ∈ H, the weight-k point pair invariant πk(z, w) associated to hT (r) is given by

πk(z, w) :=

(w − zz − w

)k/2φk(u(z, w)),

where u(z, w) := |z − w|2/4Im(z)Im(w). With this, we define the weight-k automorphic kernel by

Kk(z, w) :=∑γ∈Γ

jγ(z; k)πk(z, γw).

Now, let S2(Γ) be the C-vector space of dimension gΓ consisting of cusp forms of weight 2 with respectto Γ, endowed with the Petersson inner product. Once for all, we fix an orthonormal basis {f1, . . . , fgΓ}of S2(Γ) and write λj := 1/4 + r2j , with rj ∈ C, for the corresponding eigenvalue of ψj . Then, thefollowing spectral expansions hold

K0(z, w) =∞∑j=0

hT (rj)ψj(z)ψj(w) + S0(z, w),

K2(z, w) =

gΓ∑j=1

Im(z)Im(w)fj(z)fj(w) +∞∑j=1

hT (rj)

λj(U0ψj)(z)(U0ψj)(w) + S2(z, w),

(9)

where U0 is the uppering Maass operator of weight 0 given by (6), and

Sk(z, w) :=2− k

2v−1Γ +

1

4π

∑q∈PΓ

∫ ∞−∞

hT (r)Eq,k

(z,

1

2+ ir

)Eq,k

(w,

1

2+ ir

)dr.

In the sequel, we write Kk(z) for Kk(z, z) and Sk(z) for Sk(z, z). Letting

νk(γ; z) := jγ(z; k)πk(z, γz),

we have Kk(z) = Khypk (z) +K

park (z) with

Khypk (z) :=∑γ∈Γ

|tr(γ)|>2

νk(γ; z), Kpark (z) :=

∑γ∈Γ

|tr(γ)|=2

νk(γ; z).

Note that, by abuse of notation, γ = ±I are included in Kpark (z). With this, we define

H(z) := Khyp2 (z)−Khyp0 (z),

P(z) := (Kpar2 (z)− S2(z))− (Kpar0 (z)− S0(z)),

6

T (z) :=∞∑j=1

hT (rj)

λj|(U0ψj)(z)|2 −

∞∑j=0

hT (rj)|ψj(z)|2.

2.4. The Arakelov metric and the Arakelov Green’s function. For z ∈ H, the Arakelov metricF (z) is the Γ-invariant function defined on H given by

F (z) :=Im(z)2

gΓ

gΓ∑j=1

|fj(z)|2.

It can be verified that F (z) is of rapid decay at all cusps q ∈ PΓ, and that∫FΓ F (z)µhyp(z) = 1.

Moreover, if z ∈ H and α ∈ GL+2 (Q) such that α−1Γα = Γ, then the identity F (αz) = F (z) holds.Following the lines of Abbes–Ullmo, one obtains the following key identity for F (z). First, by definition,we have K2(z)−K0(z) = H(z) +Kpar2 (z)−K

par0 (z). Using the spectral decompositions given by (9),

we then obtain

K2(z)−K0(z) = gΓF (z) + T (z) + S2(z)− S0(z).

These two identities for the difference K2(z)−K0(z) therefore yield the identity

F (z) =1

gΓ

(− T (z) +H(z) + P(z)

).(10)

In the sequel, we refer to H(z) resp. P(z) as the hyperbolic contribution and parabolic contribution ofthe Arakelov metric, respectively.

The Rankin–Selberg constant of the Arakelov metric at q is defined by

Rq := lims→1

(Rq[F ](s)−

v−1Γs− 1

).(11)

For q = ∞, taking the Rankin–Selberg transform on both sides of (10) and grouping the constantterms of the Laurent expansion at s = 1, we obtain the following identity

R∞ =1

gΓ

(−v−1Γ2

∞∑j=1

hT (rj)

λj+ Rhyp∞ + R

par∞

).(12)

Here, Rhyp∞ resp. Rpar∞ denotes the constant term in the Laurent expansion at s = 1 of R∞[H](s) and

R∞[P](s), respectively, i.e., we have

Rhyp∞ := lims→1

(R∞[H](s)−

v−1Γs− 1

∫FΓH(z)µhyp(z)

),

Rpar∞ := lims→1

(R∞[P](s)−

v−1Γs− 1

∫FΓP(z)µhyp(z)

).

In the sequel, we refer to the constants Rhyp∞ and Rpar∞ as the hyperbolic and parabolic contributions

of R∞, respectively.

Next, the canonical volume form on H is given by µcan(z) := F (z)µhyp(z). Since F (z) and µhyp(z)are Γ-invariant on H, the canonical volume form µcan(z) is naturally defined on Y (N). Furthermore,µcan(z) extends to a (1, 1)-form on X(N), since it remains smooth at the cusps q ∈ PΓ. In addition,observe that

∫X(N) µcan(z) = 1.

Finally, the Arakelov Green’s function gAr is the function defined on X(N)×X(N) which is smoothoutside the diagonal and characterized by the following conditions

(i)1

iπ∂z∂zgAr(z, w) = µcan(z)− δw(z),

(ii)

∫X(N)

gAr(z, w)µcan(z) = 0, for all w ∈ X(N),

7

where δw denotes the Dirac delta distribution. Given two different cusps q1, q2 ∈ PΓ, we have thefollowing important identity (due to Abbes–Ullmo)

gAr(q1, q2) = −2πCq1q2 −2π

vΓ+ 2π(Rq1 + Rq2) + 2πG ,(13)

where Cq1q2 is defined in (5), Rqj is given in (11), and

G := −∫X(N)×X(N)

g(z, w)µcan(z)µcan(w).(14)

Here, the function g(z, w) is the constant term in the Laurent expansion of the automorphic Green’sfunction at s = 1. More precisely, let Gs(z, w) denote the automorphic Green’s function of Γ given, forz, w ∈ H, z 6≡ w mod Γ, and s ∈ C with Re(s) > 1, by the series

Gs(z, w) = −1

4π

∑γ∈Γ

Qs−1(1 + 2u(z, γw)),

where Qτ (ν) denotes the associated Legendre function of the second kind. The automorphic Green’sfunction Gs(z, w) admits a meromorphic continuation to the whole s-plane with a simple pole at s = 1.At s = 1, we have

Gs(z, w) = −v−1Γ

s(s− 1)− 1

4πg(z, w) +Oz,w(s− 1),

where g(z, w) depends only on z and w.

3. The minimal regular model of the modular curve X(N)

Let K be a number field with ring of integers OK . We set S := Spec(OK) and denote by η the genericpoint of S. An arithmetic surface X/S is an integral, regular, and 2-dimensional S-scheme with aprojective and flat structural morphism X −→ S, such that the generic fiber Xη is a geometricallyconnected curve over K.

Given a smooth projective curve X/K of genus g ≥ 1 defined over K, there exists a unique (minimal)arithmetic surface X/S whose generic fiber Xη is isomorphic to X/K [20, Proposition 10.1.8]. Thisarithmetic surface is called the minimal regular model of X/K. In particular, by the Riemann–Rochtheorem, the compact Riemann surface X(N) can be embedded in a projective space whose image is asmooth and projective algebraic curve X(N)/C. Since the algebraic curve X(N)/C is in fact definedover the cyclotomic number field Q(ζN ), there exists a minimal regular model of X(N)/Q(ζN ).

To simplify notation, we will write X/S for the minimal regular model of X(N)/Q(ζN ), where in thiscase S = Spec(Z[ζN ]). Also, given an embedding σ : Q(ζN ) ↪−→ C, we write Xη,σ for the base changeXη ⊗σ Spec(C).Next, we will provide an explicit description of the fibers of the arithmetic surface X/S. To do so, letus briefly introduce the moduli interpretation of X/S.

3.1. The moduli problem of canonical structures on elliptic curves. Let N ≥ 2 be a giveninteger and set ζN := e

2πi/N . Denote by µN (C) the group of the N -th roots of unity and by Y (N)/Q(ζN )the open subvariety of X(N)/Q(ζN ) given by the image of Γ(N)\H under the projective embeddingthat takes X(N) to X(N)/C.

An elliptic curve E/T over an arbitrary scheme T is a proper and smooth commutative group T -scheme with a given section, such that the geometric fibers are all connected and all have genus one.Given an elliptic curve E/T over T , the subscheme E[N ] of the N -torsion points of E/T is definedby E[N ] := E ×E T , which is obtained by base change using the N -fold morphism [N ] : E −→ Eand the given section of E/T . A canonical Γ(N)-structure on E/T is a homomorphism of groupsφ : (Z/NZ)2 −→ E[N ](T ) such that eN (φ(1, 0), φ(0, 1)) = ζN and the following identity of Cartierdivisors holds ∑

(a,b)∈(Z/NZ)2[φ(a, b)] = E[N ];

8

here, eN denotes the Weil pairing on E[N ] and [φ(a, b)] is the effective Cartier divisor induced by thesection φ(a, b) ∈ E[N ](T ).Let R be a noetherian regular and excellent ring, and denote by (Sch/R) the category of R-schemes.In addition, we let (Sets) be the category of sets. Let us consider the case when R = C. Given anelliptic curve E/C over C, we have E[N ] ' (Z/NZ)2, where the isomorphism is not canonical. Then,a canonical Γ(N)-structure on E/C is an isomorphism φ : (Z/NZ)2 −→ E[N ] which is compatiblewith the pairing (Z/NZ)2 × (Z/NZ)2 −→ µN (C) given by ((a, b), (c, d)) 7−→ ζad−bcN induced by theisomorphism φ and the Weil pairing on E[N ]. Then, it turns out that the variety Y (N)/Q(ζN )represents the functor F : (Sch/C) −→ (Sets) which takes T ∈ Ob(Sch/C) to the set of isomorphismclasses of pairs (E/T, φ), where E/T is an elliptic curve over T and φ is a canonical Γ(N)-structure onE/T .

In the general case, N. Katz and B. Mazur proved that an analogous functor F : (Sch/Z[ζN ]) −→ (Sets)is representable by a flat, regular, and 2-dimensional S-scheme Y/S, provided that N ≥ 3 is acomposite, odd, and square-free integer. Moreover, it turns out that the arithmetic surface X/S is thecompactification of Y/S.

Proposition 1 (Katz–Mazur). Let N ≥ 3 be a composite, odd, and square-free integer, and let X/Sbe the minimal regular model of X(N)/Q(ζN ). Then, the fiber Xp, for p ∈ S with p - N is a smoothcurve, whereas for p | p with p | N a prime number, the fiber Xp consists of p + 1 copies of smoothproper κ(p)-curves intersecting transversally at their supersingular points.

Proof. See [19]. �

3.2. The moduli interpretation of the closed subscheme of cusps. The standard N -gon overa scheme S′ is the proper curve over S′ (meaning a morphism C −→ S′ which is separated, flat, finitelypresented, and of pure relative dimension 1, with non-empty fibers) obtained from P1S′ × Z/NZ bygluing the ∞-section of P1S′ × {j} to the 0-section of P1S′ × {j + 1}, for all j ∈ Z/NZ.In [6], B. Conrad extended the notion of canonical Γ(N)-structures to generalized elliptic curves.Moreover, he proved that, for N sufficiently large, the scheme X/S represents the functor F′ :(Sch/Z[ζN ]) −→ (Sets) which takes T ∈ Ob(Sch/Z[ζN ]) to the set of isomorphism classes of pairs(E/T, φ), where E/T is now a generalized elliptic curve over T and φ is an extended canonical Γ(N)-structure on E/T . The advantage of this approach for X/S lies in the moduli interpretation of theclosed subscheme of cusps

Cusps(N) := (X \Y)red,

namely, the closed subscheme Cusps(N) corresponds to the extended canonical Γ(N)-structures on the

Tate curve over Z[[q1/N ]]. In practice, this amounts to describe an extended canonical Γ(N)-structureson the standard N -gon.

Proposition 2. Let N ≥ 3 be a composite, odd, and square-free integer, and let 0,∞ ∈ PΓ becusps regarded as Q(ζN )-rational points of Xη. Given an embedding σ : Q(ζN ) ↪−→ C such thatσ(ζN ) = e

2πiv/N with v ∈ (Z/NZ)×/{±1}, let 0σ, ∞σ be the corresponding points in Xη,σ(C) of 0,∞under the embedding σ, respectively. Then, there exists an isomorphism

ισ : Xη,σ −→ X(N)/Csuch that, on complex points, we have ισ(0

σ) = 0ξ for some ξ ∈ U , and ισ(∞σ) =∞.

Proof. First of all, note that, a point in Xη,σ(C) corresponds to a triple (E;P,Q), where E is ageneralized elliptic curve over C and P,Q is a basis of Esm[N ] such that eN (P,Q) = e

2πiv/N . Then, ισis given by

(E;P,Q) 7−→ (E,P +Q, v′Q),on complex points.

Secondly, for a given(a bc d

)∈ SL2(Z/NZ), the map

ϕ( a bc d

) : (Z/NZ)2 −→ Z/NZ× µN (C),9

defined by (1, 0) 7−→ (a, ζbN ) and (0, 1) 7−→ (c, ζdN ), describes explicitly how cusps corresponds to aΓ(N)-structure on the Tate curve. Thus, the map ϕ( 1 00 1 )

takes (1, 0) 7−→ P0 and (0, 1) 7−→ Q0 withP0 = (1, 1) and Q0 = (0, 1), and the moduli interpretation of the cusp ∞ is (Tate0/Q(ζN );P0, Q0).Similarly, ϕ( 1 0

v′ 1

) takes (1, 0) 7−→ P0 and (0, 1) 7−→ (v′, ζN ), and the moduli interpretation of 1/v′ is(Tate0/Q(ζN );P0, (v

′, ζN )).

Finally, we have

ισ(∞σ) = (Tate0;P0 + (0, e2πiv/N ), v′(0, e2πiv/N ))

= (Tate0; (1, e2πiv/N ), (0, e2πi/N ))

= (Tate0; (1, 1), (0, e2πi/N )) = (Tate0;P0, Q0) =∞,

where in the third equality we used the automorphism (a, z) 7−→ (a, ze−2πiv/N ) of the Tate curve. Asimilar reasoning can applied for the cusp 0, using the other automorphism of the Tate curve. Thisconcludes the proof. �

4. The hyperbolic contribution of R∞

Recall from identity (12) that Rhyp∞ is the hyperbolic contribution of the Rankin–Selberg constant

R∞ at ∞. In this chapter, we establish an asymptotics for Rhyp∞ as T → ∞, for a given level N(see Proposition 6). To do so, in Section 4.1, we first determine the residue at s = 1 of certain zetafunctions associated to hyperbolic elements of Γ (see Proposition 4). Next, in Section 4.2, in the proofof Proposition 6, we obtain an explicit identity for R∞[H](s) in terms of the aforementioned zetafunctions; thus, we derive an identity for Rhyp∞ from the Laurent expansion of R∞[H](s) at s = 1 inwhich the residues at s = 1 of the previous zeta functions arise.

4.1. The zeta function associated to Γ. For this section, let us consider the following settings. Letl be a fixed integer such that |l| > 2 and set ∆ := l2−4. Denote by Q∆ the set of binary quadratic formsof discriminant ∆ whose coefficients lie in Z. Given f(x, y) ∈ Q∆, we will write f · (x, y) := f(y,−x).Consider the sets

spl := {γ ∈ PSL2(Z) | tr(γ) = l}

spl(Γ) := {γ ∈ Γ | tr(γ) = l} ⊂ spl.

Observe that an element γ ∈ spl(Γ) has the form γ = γl(a, b, c), where

γl(a, b, c) :=

(l−bN

2 −cNaN l+bN2

)(15)

with a, b, c ∈ Z, a > 0, gcd(a, b, c) = 1, (l ± bN)/2 ≡ 1 modN , and b2 − 4ac is not a square. Fromnow on, we assume that γ ∈ spl(Γ) has always the form given by (15). Thus, for γ ∈ spl(Γ), we setD := b2 − 4ac, and also

L := Q(√D), θ := −b+

√D

2a, b := Z+ Zθ.

Further, if OL is the ring of integers of L and f ⊂ OL is an integral ideal, we write U(f) := O×L ∩ (1 + f),and U+(f) for the totally positive elements of U(f), i.e., we have

U+(f) = {ε ∈ U(f) |N(ε) > 0},where N(·) denotes the norm of an element in L. For the sequel, we set f := NOL.It is well-known that the assignment(

a bc d

)7−→ cx2 + (d− a)xy − by2

defines a bijection βl : spl∼−→ Q∆. Then, for convenience, we will write Q∆(Γ) := βl(spl(Γ)), and

also fγ := βl(γ) and γf := β−1l (f), for γ ∈ spl(Γ) and f(x, y) ∈ Q∆(Γ), respectively. Furthermore, the

10

action of the modular group PSL2(Z) on Q∆, namely (f ◦ γ)(x, y) := f((x y)γt), descents to an actionof Γ on Q∆(Γ).Let γ = γl(a, b, c) ∈ spl(Γ). Denote by Γfγ the stabilizer of fγ(x, y) ∈ Q∆(Γ) under the action of Γ.Since fγ = Ngγ with gγ(x, y) = ax

2 + bxy + cy2, we have Γfγ = Γgγ . Therefore, it can be verified that

Γfγ =

{(t−bv

2 −cvav t+bv2

) ∣∣∣∣ t, v ∈ Z, t2 −Dv2 = 4, v ≡ 0 modN, t− v2 ≡ 1 modN}.

Now, given u ∈ U , defineM

fγu := {(m,n) ∈M(u) | fγ · (m,n) > 0}.

Then, the group Γfγ acts on Mfγu by right multiplication. Indeed, if (m′, n′) = (m,n)δ with (m,n) ∈

Mfγu and δ ∈ Γfγ , then (m′, n′) ∈M(u) holds, and since

fγ(n′,−m′) = fγ((n −m)(δ−1)t) = (fγ ◦ δ−1)(n,−m) = fγ(n,−m) > 0,

the claim follows. Let Mfγu be the set of Γfγ -orbits of M

fγu , i.e., we have M

fγu := M

fγu /Γfγ .

Lemma 3. Let u ∈ U and γ ∈ spl(Γ). Suppose that b = (u/N)a−1f, for some non-zero integral ideala ⊂ OL. Then, the assignment (u/N + b)+ −→ M

fγu given by u/N + n′ + m′θ 7−→ (Nm′, u + Nn′)

defines a bijection whose inverse is given by (m,n) 7−→ (1/N)(n + mθ). In particular, we have thebijection

Mfγu ' (u/N + b)+/U+(f).

Proof. For the well-definition, note that if ξ = u/N + n′ +m′θ, then

ξξ =1

aN3fγ(u+Nn

′,−Nm′),

which in turn implies that fγ(u + Nn′,−Nm′) > 0, since N(ξ) = ξξ > 0. Therefore, the pair

(Nm′, u+Nn′) ∈Mfγu . The injectivity and surjectivity are immediate.Next, for the second statement of the lemma, it turns out that the map(

t−bv2 −cvav t+bv2

)7−→ t+ v

√D

2

establishes the bijection

Γfγ ' U+(f)(see [12, Proposition 4.2.11] for further details). This concludes the proof of the lemma. �

Next, we proceed to define a key zeta function and study its properties. Let γ ∈ spl(Γ) and u ∈ U . Fors ∈ C with Re(s) > 1, the zeta funcion associated to the pair (γ, u) is defined by

ζγ,u(s) :=∑

(m,n)∈Mfγu

fγ(n,−m)−s.

We claim that the function ζγ,u(s) possesses a meromorphic continuation to the complex plane havinga pole at s = 1. In order to prove the claim, we will relate ζγ,u(s) to the zeta function associated to anarrow ideal class of L.

Let us consider the ray m := m∞f, where m∞ denotes the ray consisting of all embeddings L ↪→ R, anddenote by Clm(L) the narrow-ray ideal class group of L. Given C ∈ Clm(L) and s ∈ C with Re(s) > 1,the partial zeta function ζ(s,C) is defined by

ζ(s,C) :=∑c∈Cc6=0

1

(N c)s,

11

where the sum runs over the non-zero integral ideals c contained in C and N c denotes the norm of theideal c. Suppose that C ∈ Clm(L) and a ∈ C, a nonzero integral ideal are given. Let τ ∈ L× such thatN(τ) > 0 and set b′ = (τ)a−1f. Then, the partial zeta function ζ(s,C) satisfies the following identity

ζ(s,C) = (N f)−s∑

β∈(τ+b′)+/U+(f)

(N (β)Nb′

)−s(see, e.g., [26]), where N (β) denotes the norm of the principal ideal generated by the element β.

Proposition 4. Let u ∈ U and γ ∈ spl(Γ). Suppose that γ = γl(a, b, c) is given by (15). Then, fors ∈ C with Re(s) > 1, there exists a narrow ray ideal class C ∈ Clm(L) such that

ζγ,u(s) =

(1

N

)sζ(s,C).

Furthermore, ζγ,u(s) admits a meromorphic continuation to the complex plane having a pole at s = 1,whose pole is given by

ress=1(ζγ,u(s)

)=

log(εγ)

N3√D,

where εγ denotes a generator of U+(f).

Proof. We know that, if ξ := u/N + n+mθ ∈ (u/N + b)+, then

fγ(u+ nN,−Nm) = N3N(ξ)

Nb= N

(N fN (ξ)Nb

).

Therefore, we have

ζγ,u(s) =∑

(m,n)∈Mfγu

fγ(n,−m)−s

=

(1

N

)s(N f)−s

∑ξ∈( uN +b)+/U+(f)

(N (ξ)Nb

)−s

=

(1

N

)sζ(s,C),

where C is the narrow-ray ideal class of the fractional ideal fb−1(u/N). Consequently, the functionζγ,u(s) inherits the analytical properties of ζ(s,C), in particular, the meromorphic continuation to thecomplex plane. This concludes the proof. �

4.2. Computation of the constant Rhyp∞ . Given u ∈ U , let us consider the following subsets ofQ∆(Γ)×M(u)

S+Γ (u) := {(f ; (m,n)) | (m,n) ∈Mfu , f ∈ Q∆(Γ)},

S−Γ (u) := {(f ; (m,n)) | (m,n) ∈M−fu , f ∈ Q∆(Γ)}.

It can be verified that the sets S+Γ (u) and S−Γ (u) remain invariant under the diagonal action of Γ.

Therefore, by the “allgemeine Prinzip” from [28], we have

S+Γ (u)/Γ '⊔

γ∈spl(Γ)/Γ

Mfγu ,

S−Γ (u)/Γ '⊔

γ∈spl(Γ)/Γ

M(−fγ)u .

Further, we have Q∆(Γ)×M(u) = S+Γ (u) t S−Γ (u).

12

Now, we conveniently define the integrals

J+k :=

∫FΓ

( ∑(f ;(m,n))∈S+Γ (u)

νk(γf ; z)Im(z)s

|mz + n|2s

)µhyp(z),

J−k :=

∫FΓ

( ∑(f ;(m,n))∈S−Γ (u)

νk(γf ; z)Im(z)s

|mz + n|2s

)µhyp(z),

and the matrix λ(l) by

λ(l) :=

(l/2 l2/4− 11 l/2

).

Lemma 5. Let u ∈ U , l ∈ Z such that |l| > 2, and s ∈ C with Re(s) > 1. Then, the following identityholds

J+k =

(∫H

νk(λ(l); z)Im(z)sµhyp(z)

) ∑γ∈spl(Γ)/Γ

ζγ,u(s),

J−k =

(∫H

νk(λ(−l); z)Im(z)sµhyp(z)) ∑γ∈spl(Γ)/Γ

ζγ,u(s),

where νk(γ; z) is given in Section 2.4.

Proof. For the proof of the lemma, observe that it suffices to prove the statement for the integralJ+k , since we have −fγ = f−γ = fγ−1 . Let (f ; (m,n)) ∈ S

+Γ (u) with f(x, y) = cx

2 + (d− a)xy − by2,therefore γf =

(a bc d

). Put (mδ, nδ) = (mn)δ

t, for δ ∈ Γ, and set

M :=1

f(n,−m)

(n −bm− n(d− a)/2−m cn−m(d− a)/2

).

Using the identities

νk(δ−1γδ; z) = νk(γ; δz),

Im(z)

|mδz + nδ|2=

Im(δz)

|m(δz) + n|2,

which hold for δ ∈ PSL2(R), we obtain∑(f ;(m,n))∈S+Γ (u)

νk(γf ; z)Im(z)s

|mz + n|2s=

∑(f ;(m,n))∈S+Γ (u)/Γ

∑δ∈Γ

νk(δ−1γfδ; z)

Im(z)s

|mδz + nδ|2s

=∑

(f ;(m,n))∈S+Γ (u)/Γ

∑δ∈Γ

νk(γf ; δz)Im(δz)s

|m(δz) + n|2s.

Thus, we have

J+k =

∫FΓ

( ∑(f ;(m,n))∈S+Γ (u)

νk(γf ; z)Im(z)s

|mz + n|2s

)µhyp(z)

=∑

(f ;(m,n))∈S+Γ (u)/Γ

(∫FΓ

∑δ∈Γ

νk(γf ; δz)Im(δz)s

|m(δz) + n|2sµhyp(z)

)

=∑

(f ;(m,n))∈S+Γ (u)/Γ

(∫H

νk(γf ; z)Im(z)s

|mz + n|2sµhyp(z)

).

Finally, applying the change of variables z 7→Mz in the last integral, and using the identitiesIm(Mz)

|m(Mz) + n|2=

Im(z)

f(n,−m), M−1γfM = λ(l),

we deduce

J+k =

(∫H

νk(M−1γfM ; z)Im(z)

sµhyp(z)

) ∑(f ;(m,n))∈S+Γ (u)/Γ

f(n,−m)−s.

13

This concludes the proof of the lemma. �

Next, let us set

Ik(s; l) :=

∫H

(νk(λ(l); z) + νk(λ(−l); z)

)Im(z)sµhyp(z).

Proposition 6. Let N ≥ 3 be an odd square-free integer and s ∈ C with 1 < Re(s) < inf{A, 3/2}.Then the following identity holds

R∞[H](s) =1

N s

∑l∈Z|l|>2

(I2(s; l)− I0(s; l)

)∑u∈U

(Du(s)

∑γ∈spl(Γ)/Γ

ζγ,u(s)

),

where Du(s) is the Dirichlet series given by (2). Furthermore, we have

Rhyp∞ =v−1Γ2

(lims→1

(Z ′ΓZΓ

(s)− 1s− 1

)− T + 1 + o(1)

),

as T →∞; here ZΓ(s) stands for the Selberg zeta function of Γ.

Proof. For the proof, we write Khypk (z) as follows

Khypk (z) =∑γ∈Γ

|tr(γ)|>2

νk(γ; z) =∑l∈Z|l|>2

∑γ∈spl(Γ)

νk(γ; z).

Then, note that

Khypk (z)E∞(z, s) =1

N s

∑l∈Z|l|>2

∑u∈U

Du(s)∑

(γ;(m,n))

νk(γ; z)Im(z)s

|mz + n|2s,

where the innermost sum runs over the elements of spl(Γ)×M(u), and since spl(Γ) ' Q∆(Γ), we have

Khypk (z)E∞(z, s) =1

N s

∑l∈Z|l|>2

∑u∈U

Du(s)∑

(f ;(m,n))

νk(γf ; z)Im(z)s

|mz + n|2s.

Thus, the previous identity yields

R∞[Khypk ](s) =∫FΓKhypk (z)E∞(z, s)µhyp(z)

=1

N s

∑l∈Z|l|>2

∑u∈U

Du(s)

∫FΓ

( ∑(f ;(m,n))

νk(γf ; z)ys

|mz + n|2s

)µhyp(z).

Using the decomposition Q∆(Γ)×M(u) ' S+Γ (u) t S−Γ (u) and Lemma 5, we get

R∞[Khypk ](s) =1

N s

∑l∈Z|l|>2

∑u∈U

Du(s)(J+k + J

−k )

=1

N s

∑l∈Z|l|>2

Ik(s; l)∑u∈U

(Du(s)

∑γ∈spl(Γ)/Γ

ζγ,u(s)

).

This proves the first assertion.

For the second part of the proposition, observe that∑u∈U

Du(s)ζγ,u(s) =

(∑u∈U

Du(s)

)(log(εγ)

N3√D

1

s− 1+ C +O(s− 1)

).

Then, by [3, Proposition 3.3.2], namely

I2(s; l)− I0(s; l) = πAl(T )(s− 1) +O((s− 1)2),14

where

Al(T ) := −1

2ηl+

1

4π

∫ ∞−∞

hT (r)14 + r

2e−2ir log(ηl)dr

with l > 2 and ηl := (l +√l2 − 4)/2, and by virtue of identity (3), we deduce the following expansion

at s = 1

R∞[H](s) =∑l∈Z|l|>2

∑u∈U

∑γ∈spl(Γ)/Γ

1

N s(I2(s; l)− I0(s; l))Du(s)ζγ,u(s)

=∑l∈Z|l|>2

∑γ∈spl(Γ)/Γ

log(εγ)√l2 − 4

Al(T )

vΓ+O(s− 1).

Therefore, we have

Rhyp∞ = R∞[H](1)

=1

vΓ

∑l∈Z|l|>2

∑γ∈spl(Γ)/Γ

log(εγ)√l2 − 4

Al(T ).

Now, by [3, Proposition 3.3.3], namely

Al(T ) = −1

2

∫ T0g(t, 2 log(ηl))dt

for l > 2 with

g(t, u) :=1√4πt

e−t4−u

2

4t ,

we get

Rhyp∞ = −v−1Γ2

∑l∈Z|l|>2

∑γ∈spl(Γ)/Γ

log(εγ)√l2 − 4

∫ T0g(t, 2 log(ηl))dt

= −v−1Γ2

∫ T0

ΘΓ(t)dt,

where

ΘΓ(t) :=∑l∈Z|l|>2

∑γ∈spl(Γ)/Γ

log(εγ)√l2 − 4

g(t, 2 log(ηl)).

Finally, since ∫ T0

ΘΓ(t)dt = T − lims→1

(Z ′ΓZΓ

(s)− 1s− 1

)− 1 + o(1)

holds as T →∞, the assertion follows. �

5. The parabolic contribution of R∞

Recall from identity (12) that Rpar∞ is the parabolic contribution of the Rankin–Selberg constant R∞at ∞. In this chapter, we compute Rpar∞ explicitly (see Proposition 11). To do so, we first note that

R∞[P](s) =∫ ∞

0p(y)ys−2dy,

where p(y) is the 0-th coefficient in the Fourier expansion of P(σ∞z). Further, since

p(y) =

4∑j=1

(pj(y; 2)− pj(y; 0)),

15

where pj(y; k) is given by (18), it suffices to compute the integrals

Mj(s) :=∫ ∞

0(pj(y; 2)− pj(y; 0))ys−2dy,

which is done in Section 5.1. Then, in Section 5.2, we gather the identities for Mj(s) and conclude thecomputation of Rpar∞ in Proposition 11.

In this section, we will consider the following settings. We define λ(2) := ( 1 01 1 ) and λ(−2) :=(−1 0

1 −1),

and let Ik(s; 2) be the integral given by

Ik(s; 2) :=

∫H

(νk(λ(2); z) + νk(λ(−2); z)

)Im(z)sµhyp(z).(16)

From [3, Proposition 3.3.4], we have the Laurent expansion at s = 1 of the difference I2(s; 2)− I0(s; 2),namely

I2(s; 2)− I0(s; 2) =√π

Γ(s− 12)Γ(s)

(A2(T )(s− 1) + C1(T )(s− 1)2 +O((s− 1)3)

),(17)

where

A2(T ) := −1

2+

1

4π

∫ ∞−∞

hT (r)14 + r

2dr.

This expansion will be useful later on. Next, recall that φk stands for the inverse Selberg/Harish–Chandra transform of weight k of the function hT (r) given by (8). Then, we set

Ψ±k (y) := Ψk(y) + Ψk(−y),

where Ψk(y) is the function given by

Ψk(y) :=

∫ ∞−∞

φk(u2)

(1− iu1 + iu

)k/2e2iπuydu.

We will write

Ẽ∞,k(σ∞z, s) := E∞,k(σ∞z, s)− ys − ϕ∞∞,k(s),

where E∞,0(z, s) and ϕ∞∞,0(s) stands for E∞(z, s) and ϕ∞∞(s), respectively. For convenience, wedefine the following integrals

p1(y; k) :=

∫ 1/2−1/2

( ∑γ∈Γ \Γ∞|tr(γ)|=2

νk(γ;σ∞z))

dx;

p2(y; k) :=

( ∑γ∈Γ∞

νk(γ;σ∞z)

)− y

2π

∫ ∞−∞

hT (r)dr;

p3(y; k) := −y

2π

∫ ∞−∞

hT (r)

( 12 + ir12 − ir

)k/2ϕ∞∞

(1

2− ir

)y2irdr − 2− k

2v−1Γ ;(18)

p4(y; k) := −∫ 1/2−1/2

(1

4π

∫ ∞−∞

hT (r)∑q∈PΓ

∣∣∣∣Ẽ∞,k(σ∞z, 12 + ir)∣∣∣∣2dr)dx;

p∗4(y) := −1

4π

∑q∈PΓ

∫ ∞−∞

(hT (r)14 + r

2R∞

[ ∣∣∣∣Ẽ∞,k(σ∞z, 12 + ir)∣∣∣∣2](s))dr;

where the sum in the integrand of the function p1(y; k) runs over all elements of Γ that are not in Γ∞.16

5.1. Preparatory lemmas.

Lemma 7. Let N ≥ 3 be an odd square-free integer and s ∈ C with 1 < Re(s) < A. Then, thefollowing identity holds

M1(s) =2ζ(s)ζ(2s− 1)ζ(2s)N2s−1

(I2(s; 2)− I0(s; 2)),

where ζ(s) denotes the Riemann zeta funcion and Ik(s; 2) is given by (16). Furthermore, the Laurentexpansion of M1(s) at s = 1 is given by

12A2(T )

πN

1

s− 1+

12

πN

(C1(T ) +A2(T )(2C + γEM − 2 log(N))

)+O(s− 1),

where γEM denotes the Euler–Mascheroni constant and C1(T ) is a constant that depends only on thefixed positive real T .

Proof. By definition, we have

M1(s) =∫ ∞

0(p1(y; 2)− p1(y; 0))ys−2dy.

In order to compute M1(s), let us consider the following integral∫ ∞0

p1(y; k)ys−2dy =

∫ ∞0

∫ 1/2−1/2

( ∑γ∈Γ \Γ∞|tr(γ)|=2

νk(γ;σ∞z)

)ys−2dxdy.

Put B := {n(b) | b ∈ Z}. Then, note that we have bijections

NZ \ {0} × (B\PSL2(Z))∼−→ {γ ∈ Γ \ Γ∞ | tr(γ) = 2},

NZ \ {0} × −(B\PSL2(Z))∼−→ {γ ∈ Γ \ Γ∞ | tr(γ) = −2},

given by (m,B

(∗ ∗c d

))7−→ γ+(m; c, d) :=

(1 +mcd md2

−mc2 1−mcd

),(

m,B

(∗ ∗c d

))7−→ γ−(m; c, d) :=

(−1 +mcd md2−mc2 −1−mcd

),

respectively. With these bijections, we obtain∫ ∞0

p1(y; k)ys−2dy =

∫ ∞0

∫ 1/2−1/2

( +∑(m,B( ∗ ∗c d ))

νk(γ+(m; c, d);σ∞z)

+−∑

(m,B( ∗ ∗c d ))

νk(γ−(m; c, d);σ∞z)

)ys−2dxdy,

where the first sum runs over all pairs (m,B( ∗ ∗c d )) in NZ \ {0} × (B\PSL2(Z)), and the second sumruns over all pairs (m,B( ∗ ∗c d )) in NZ \ {0} × −(B\PSL2(Z)). Then it suffices to compute

P+k :=

∫ ∞0

∫ 1/2−1/2

( +∑(m,B( ∗ ∗c d ))

νk(γ+(m; c, d);σ∞z)

)ys−2dxdy.

Observe that,

P+k =

∫ ∞0

∫ 1/2−1/2

( ∑′(m,B( ∗ ∗c d ))

∑n∈Z

νk(γ+(m; c, d);σ∞z)

)ys−2dxdy,

17

where now, the first sum runs over all pairs (m,B( ∗ ∗c d )) representing a class inNZ\{0}×(B\PSL2(Z)/Γ∞);further, by convergence, we can rewrite P+k as follows

P+k =∑′

(m,B( ∗ ∗c d ))

∫ ∞0

(∑n∈Z

∫ 1/2−1/2

νk(γ+(m; c, d);N(x+ n) + iNy)dx

)ys−2dy.

Now, by a suitable change of variables, we obtain

P+k =1

N

∑′(m,B( ∗ ∗c d ))

∫ ∞0

∫ ∞−∞

νk(γ+(m; c, d);x+ iNy)ys−2dxdy

=1

N s

∑′(m,B( ∗ ∗c d ))

∫H

νk(γ+(m; c, d); z)Im(z)sµhyp(z).

Then, by [3, Lemme 3.2.12], namely∫H

νk(γ+(m; c, d); z)Im(z)sµhyp(z) =

1

|mc2|s

∫H

νk(λ(2); z)Im(z)sµhyp(z),

we get

P+k =1

N s

(∫H

νk(λ(2); z)Im(z)sµhyp(z)

) ∑′(m,B( ∗ ∗c d ))

1

|mc2|s,

Similarly, if we set

P−k :=

∫ ∞0

∫ 1/2−1/2

( −∑(m,B( ∗ ∗c d ))

νk(γ+(m; c, d);σ∞z)

)ys−2dxdy,

we will obtain

P−k =1

N s

(∫H

νk(λ(−2); z)Im(z)sµhyp(z)) ∑′

(m,B( ∗ ∗c d ))

1

|mc2|s.

Therefore, ∫ ∞0

p1(y; k)ys−2dy =

1

N sIk(s; 2)

∑′(m,B( ∗ ∗c d ))

1

|mc2|s,

and since ∑′(m,B( ∗ ∗c d ))

1

|mc2|s=

∑m∈NZ\{0}

1

|m|s

( ∞∑c=1

1

c2s

∑dmod c

( ∗ ∗c d )∈B\PSL2(Z)/Γ∞

1

)

=2ζ(s)ζ(2s− 1)ζ(2s)N s−1

,

it follows that

M1(s) = 2N1−2sζ(s)ζ(2s− 1)

ζ(2s)(I2(s; 2)− I0(s; 2)).

This proves the first assertion of the lemma. For the second part of the lemma, use (17) together withthe well-known Laurent expansion

√π

Γ(s− 1/2)Γ(s)

ζ(2s− 1)ζ(2s)

=3/π

s− 1+

6

πC +O(s− 1)

at s = 1 with C := 1− log(4π) + ζ ′(−1)/ζ(−1). This concludes the proof. �18

Lemma 8. Let N ≥ 3 be an odd square-free integer and s ∈ C with 1 < Re(s) < A. Then, thefollowing identity holds

M2(s) = 2ζ(s)(∫ ∞

0Ψ±2 (2y)y

s−1dy −∫ ∞

0Ψ±0 (2y)y

s−1dy

).

Furthermore, the Laurent expansion of M2(s) at s = 1 is given byC2(T ) + (4π)

−1

s− 1+

(1− log(4π)

4π+ γEMC2(T ) + C3(T )

)+O(s− 1),

where C2(T ) and C3(T ) are constants that depend only on the fixed positive real T .

Proof. By definition, we have

M2(s) =∫ ∞

0(p2(y; 2)− p2(y; 0))ys−2dy.

In order to compute M2(s), let us consider the integral∫ ∞0

p2(y; k)ys−2dy =

∫ ∞0

( ∑γ∈Γ∞

νk(γ, σ∞z)−y

2π

∫ ∞−∞

hT (r)dr

)ys−2dy.

We claim that ∑γ∈Γ∞

νk(γ;σ∞z) =y

2π

∫ ∞−∞

hT (r)dr + 2y∑n∈Zn6=0

Ψk(2ny).

Indeed, since γ ∈ Γ∞ has the form γ =(

1 nN0 1

), for some n ∈ Z, we have jγ(z; k) = 1. This implies

νk(γ;σ∞z) = πk(Nz,N(z + b))

=

(1− i(n/2y)1 + i(n/2y)

)k/2φk

(n2

4y2

),

where for the second equality, we used the definition of πk(z, w) and u(z, w) given in Section 2.3.Therefore, we get the identities∑

γ∈Γ∞

νk(γ;σ∞z) =∑n∈Z

(1− i(n/2y)1 + i(n/2y)

)k/2φk

(n2

4y2

)

=∑n∈Z

∫ ∞−∞

φk

(v2

4y2

)(1− i(v/2y)1 + i(v/2y)

)k/2e2πinvdv,

where in the last equality we applied the Poisson summation formula. The claim follows by noting that∫ ∞−∞

φk(v2)

(1 + iv

1− iv

)k/2dv =

1

4π

∫ ∞−∞

hT (r)dr.

Consequently, we have∫ ∞0

p2(y; k)ys−2dy = 2

∑n∈Zn6=0

∫ ∞0

Ψk(2ny)ys−1dy

= 2

∞∑n=1

∫ ∞0

(Ψk(2ny) + Ψk(−2ny))ys−1dy

= 2ζ(s)

∫ ∞0

Ψ±k (2y)ys−1dy,

and the first assertion of the lemma follows. For the second assertion, note that Ψ±k (2y) is one-halfthe function ψ(y) + ψ(−y) given by [3, p. 42] when k = 2, and given by [27, p. 326–329] when k = 0.Finally, the assertion of the lemma follows using [27, p. 389] and [3, Proposition 3.2.6]. This concludesthe proof. �

19

Lemma 9. Let N ≥ 3 be an odd square-free integer and s ∈ C with 1 < Re(s) < A. Then, thefollowing identity holds

M3(s) =(

s

1 + s

)hT

(is

2

)ϕ∞∞

(1 + s

2

).

Furthermore, the Laurent expansion of M3(s) at s = 1 is given by

v−1Γs− 1

+

(C∞∞

2+v−1Γ2

(T + 1)

)+O(s− 1).

Proof. The proof of the first identity follows from an immediate extension of [3, Lemme 3.2.17, p. 44]and [21, Lemma 5.1.1., p. 137] to subgroups of the modular group of finite index. This is possible sinceit only uses general analytic properties of the scattering function. Next, for the Laurent expansion ats = 1, we just multiply the following Laurent expansions at s = 1

1

2hT

(is

2

)=

1

2+T

4(s− 1) +O

((s− 1)2

);

ϕ∞∞

(1 + s

2

)=

2v−1Γs− 1

+ C∞∞ +O(s− 1);

2s

s+ 1= 1 +

1

2(s− 1) +O

((s− 1)2

).

This concludes the proof. �

Lemma 10. Let N ≥ 3 be an odd square-free integer and s ∈ C with 1 < Re(s) < A. Then, thefollowing identity holds

M4(s) = s(s− 1

2

)∫ ∞0

p∗4(y)ys−2dy,

where p∗4(y) is given by (18). Furthermore, the Laurent expansion of M4(s) at s = 1 is given by

C4(T ) +O(s− 1),

where C4(T ) is a constant that depends only on the fixed positive real T .

Proof. The lemma follows from an immediate extension of [21, Proposition 5.2.3, p. 141] to subgroupsof the modular group of finite index. �

5.2. Computation of the constant Rpar∞ .

Proposition 11. Let N ≥ 3 be an odd square-free integer and s ∈ C with 1 < Re(s) < inf{A, 3/2}.Then the following identity holds

R∞[P](s) =2ζ(s)ζ(2s− 1)ζ(2s)N2s−1

(I2(s; 2)− I0(s; 2)) +s

s+ 1hT

(is

2

)ϕ∞∞

(1 + s

2

)

+ ζ(s)

(∫ ∞0

Ψ±2 (y)ys−1dy −

∫ ∞0

Ψ±0 (y)ys−1dy

)+ s

(s− 1

2

)∫ ∞0

p∗4(y)ys−2dy.

Furthermore, the constant in the Laurent expansion at s = 1 of the previous expression is

Rpar∞ =12

πN

(C1(T ) +A2(T )

(2C + γEM − 2 log(N)

))+

1− log(4π)4π

+ γEMC2(T ) + C3(T ) +C∞∞

2+v−1Γ2

(T + 1) + C4(T ).

Proof. By summing up the Laurent expansions for eachMj(s) at s = 1 proved in the previous lemmas,we can immediately deduce the constant term Rpar∞ , namely we have

Rpar∞ =12

πN

(C1(T ) +A2(T )

(2C + γEM − 2 log(N)

))+

1− log(4π)4π

20

+ γEMC2(T ) + C3(T ) +C∞∞

2+v−1Γ2

(T + 1) + C4(T ).

The result follows after noting that

Nϕ(N)

vΓ

∏p|N

(1 +

1

p

)=

6

πN

and reordering terms. This concludes the proof. �

6. The self-intersection of the relative dualizing sheaf

Let ω := ωX/S be the relative dualizing sheaf of X/S, write ω for the relative dualizing sheaf ω,equipped with the Arakelov metric, and let ω2 := ω2X/S denote its self-intersection. In this chapter, we

establish the main result of our article, namely an asymptotics for the invariant

e(Γ) :=1

ϕ(N)ω2,

as N → ∞. To prove this asymptotics, we proceed in two steps. In section 6.1, we first obtain anexplicit formula for ω2 in terms of an geometric contribution G(N) and an analytic contribution A(N)(see Proposition 13). In section 6.2, we then establish asymptotics for the geometric and analyticcontribution (see Propositions 14 and 17), and we conclude with the main result in Theorem 18.

6.1. Explicit formula for the self-intersection. Let H0 resp. H∞ be the horizontal divisor onX/S defined by the cusps 0 and ∞, respectively. For a prime ideal p ∈ S such that p|p, where p|N is aprime number, we set

rp := p+ 1,

sp :=p− 1

24[PSL2(Z) : Γ(N/p)].

Note that sp is the number of supersingular points on the fiber Xp. Let us write C1,p, . . . , Crp,p forthe rp irreducible components of the fiber Xp stated in Proposition 1, and C0,p resp. C∞,p for theirreducible component of Xp intersected by H0 and H∞, respectively.Consider the following divisors on X with rational coefficients

V0 := −∑p|N

2(gΓ − 1)rpsp

C0,p, V∞ := −∑p|N

2(gΓ − 1)rpsp

C∞,p,

where both sums run over all prime ideals p ∈ S satisfying p|p for some prime number p|N . Now, forq ∈ {0,∞}, we define the admissible line bundle Lq on X as follows

Lq := ω ⊗O(Hq)⊗−(2gΓ−2) ⊗O(Vq).

In addition, we define the divisor M on X by

M := H∞ −H0 +1

2gΓ − 2(V0 − V∞).

Lemma 12. Let N ≥ 3 be a composite, odd, and square-free integer. Then, the identities

(Lq,O(V ))Ar = 0,(M, V )Ar = 0,

(19)

hold for all vertical divisors V of X and q ∈ {0,∞}. Furthermore, we have

(V∞, V∞)fin = (V0, V0)fin = −4(gΓ − 1)ϕ(N)(

1− 6N

)∑p|N

p2 log(p)

p2 − 1,

(V0, V∞)fin = 4(gΓ − 1)ϕ(N)(

1− 6N

)∑p|N

p log(p)

p2 − 1.

21

Proof. Throughout the proof, we let p, p̃ ∈ S be prime ideals satisfying p|N and p̃ - N , respectively.First of all, we claim that the identity

(Lq,O(V ))Ar = 0holds for V = Xp̃ and for V = Cq,p. Indeed, note that, from the definition of Lq, we have

(Lq,O(V ))Ar = (ω,O(V ))fin − (2gΓ − 2)(O(Hq),O(V ))fin + (O(Vq),O(V ))fin.Then, the claim follows by a straighforward computation using the following identities

(ω,O(V ))fin =

(2gΓ − 2) log(#k(p̃)), V = Xp̃;2gΓ − 2rp

log(#k(p)), V = Cq,p;

(O(Hq),O(V ))fin =

log(#k(p̃)), V = Xp̃;log(#k(p)), V = Cq,p;(O(Vq),O(V ))fin =

0, V = Xp̃;2(gΓ − 1)(rp − 1)

rplog(#k(p)), V = Cq,p;

(see, e.g., [20, Chapter 9]). Now, since every vertical divisor V of X can be written as a linearcombination of Xp̃ and Cq,p, with coefficients in Q, the first identity of (19) follows. Similarly, notethat (M, V )Ar = 0 holds for V = Xp̃ and V = Cq,p. Furthermore, if V = Cq′,p with q′ ∈ {0,∞} andq 6= q′, then we have

(O(Hq),O(V ))fin = 0,

(O(Vq),O(V ))fin = −2(gΓ − 1)

rplog(#k(p)).

This yield the first assertion of the lemma.

Next, to prove the second part of the lemma, observe that

(V0, V∞)fin =∑p|N

∑p′|N

4(gΓ − 1)2

rprp′spsp′(C0,p, C∞,p′)fin.

Since we have (C0,p, C∞,p′)fin = 0 provided that p 6= p′, and

(Cq,p, Cq′,p)fin =

sp log(#k(p)), q 6= q′;

−(rp − 1)sp log(#k(p)), q = q′;

where q′ ∈ {0,∞} with q 6= q′, we get

(V0, V∞)fin =∑p|N

4(gΓ − 1)2

r2psplog(#k(p)).

Further, if p|p for some prime number p|N , then we have

sp =N

24p(p+ 1)

∏p′|N

((p′)2 − 1);

therefore, we obtain

(V0, V∞)fin =4(gΓ − 1)

N

24(gΓ − 1)∏p′|N ((p

′)2 − 1)∑p|N

(p

p+ 1

∑p|p

log(#k(p))

)

= 4(gΓ − 1)ϕ(N)(

1− 6N

)∑p|N

p log(p)

p2 − 1.

22

The other identities can be proved in a similar way. This concludes the proof. �

Proposition 13. Let N ≥ 3 be a composite, odd, and square-free integer. Then, the following identityholds

ω2 = G(N) +A(N),where

G(N) = 2gΓ(V0, V∞)fin − (V0, V0)fin − (V∞, V∞)fin2(gΓ − 1)

,

A(N) = 4gΓ(gΓ − 1)∑

σ:Q(ζN )↪→C

gAr(0σ,∞σ).

Here, the sum runs over all embeddings of Q(ζN ) into C, and 0σ resp. ∞σ denote the image of 0,∞

in Xη(Q(ζN ))σ under the embedding σ, respectively.

Proof. Throughout the proof, we will write E2 := (E , E)Ar. Let Lq := ω ⊗O(Hq)⊗−(2gΓ−2) ⊗O(Vq).First of all, note that the pullback Lq := Lq ⊗Z[ζN ] Q(ζN ) of Lq to the generic fiber Xη defines aline bundle on X(N)/Q(ζN ), that is supported on the cusps (see [3, Lemme 4.1.1]). By the theoremof Manin–Drinfeld, Lq defines a torsion point of Jac(X(N))(Q(ζN )). Furthermore, the theorem ofFaltings–Hriljac implies

L2q = −2ϕ(N)hNT(Lq),

where hNT(·) denotes the Néron–Tate height. Since the latter vanishes on torsion points of the Jacobian,we obtain that

L2q = 0.Similarly, the restriction of the divisor M to the generic fiber Xη is supported on the cusps ofX(N)/Q(ζN ); therefore, by using the same arguments of the previous paragraph, we obtain M2 = 0.

Secondly, if we expand the left hand side of L2q = 0 using the definition of Lq, and apply the identities(19), then we obtain

ω2 = 2(2gΓ − 2)(ω,O(Hq))Ar − (2gΓ − 2)2O(Hq)2 − (ω ⊗O(Hq)

⊗−(2gΓ−2),O(Vq))Ar,

where q ∈ {0,∞}. By virtue of the equalities

(ω ⊗O(Hq)⊗−(2gΓ−2)

,O(Vq))Ar = −O(Vq)2,

(ω ⊗O(Hq),O(Hq))Ar = 0,we now have

ω2 = −4gΓ(gΓ − 1)O(Hq)2

+O(Vq)2,

for each q ∈ {0,∞}. Thus, by adding the resulting identities for q = 0 and q =∞, we get

ω2 = −2gΓ(gΓ − 1)(O(H0)2

+O(H∞)2) +

1

2(O(V0)

2+O(V∞)

2).

Thirdly, putting D := H∞ −H0 and E := (1/(2gΓ − 2))(V0 − V∞), we have

D2 =M2 − 2(M, E)Ar + E2 = E2,which in turn implies

O(H∞)2

+O(H0)2

=1

(2gΓ − 2)2

(O(V0)

2 − 2(O(V0),O(V∞))Ar +O(V∞)2)

+ 2(O(H∞),O(H0))Ar.

Therefore, we have

ω2 = −4gΓ(gΓ − 1)(O(H∞),O(H0))Ar +2gΓ(V0, V∞)fin − (V0, V0)fin − (V∞, V∞)fin

2(gΓ − 1).

23

Finally, since

(O(H∞),O(H0))Ar = −∑

σ:Q(ζN )↪→C

gAr(0σ,∞σ),

the result follows. This concludes the proof of the proposition. �

6.2. Asymptotics for the self-intersection.

Proposition 14. Let N ≥ 3 be a composite, odd, and square-free integer. Then, the followingasymptotics holds

1

ϕ(N)G(N) = o(gΓ log(N)),

as N →∞.

Proof. Indeed, using Lemma 12 and Proposition 13, we have

1

ϕ(N)G(N) = 4

(1− 6

N

)(log(N) + gΓ

∑p|N

p log(p)

p2 − 1+∑p|N

log(p)

p2 − 1

).

Using the fact that ∑p|N

log(p)

p= O(log log(N)),

as N →∞ (see, e.g., [5]), we can deduce the following asymptotics∑p|N

p log(p)

p2 − 1= O(log log(N)),

∑p|N

log(p)

p2 − 1= O(log log(N)),

as N →∞. This yields

1

ϕ(N)

G(N)gΓ log(N)

=4

gΓ log(N)

(1− 6

N

)(log(N) +O(gΓ log log(N))

),

and the result follows, since the right hand side of the previous identity converges to zero as N →∞.This concludes the proof. �

Lemma 15. Let N ≥ 3 be a composite, odd, and square-free integer. Then, the following identity holds

R0ξ = R∞,

for all ξ ∈ U .

Proof. By taking α =(−mN ξ−ξ̃ 1

)with det(α) = 1, one can verify that 0ξ = α

−1∞ and α−1Γα = Γ;therefore, from Section 2.4, we have that F (αz) = F (z) with z ∈ H. Thus, since the identityE∞(αz, s) = E0ξ(z, s) holds, we obtain R0ξ [F ](s) = R∞[F ](s). Consequently, by comparing coefficientsin the Laurent expansions of R0ξ [F ](s) and R∞[F ](s) at s = 1, we get R0ξ = R∞. This concludes theproof. �

Proposition 16. Let N ≥ 3 be a composite, odd, and square-free integer. Then, the following identityholds

R∞ =1

gΓ

(v−1Γ2

lims→1

(Z ′ΓZΓ

(s)− 1s− 1

)+

12

πN

(C1 − C −

γEM2

+ log(N)

)

+ v−1Γ

(C + 1− 2 log(N)−

∑p|N

log(p)

p2 − 1

)+

1− log(4π)4π

),

where C1 = limT→∞C1(T ).24

Proof. From identity (12), we know that

R∞ =1

gΓ

(−v−1Γ2

∞∑j=1

hT (rj)

λj+ Rhyp∞ + R

par∞

),

where now, by virtue of Proposition 6 and Proposition 11, we have

Rhyp∞ =v−1Γ2

(lims→1

(Z ′ΓZΓ

(s)− 1s− 1

)− T + 1 + o(1)

),

as T →∞, and

Rpar∞ =12

πN

(C1(T ) +A2(T )

(2C + γEM − 2 log(N)

))+

1− log(4π)4π

+v−1Γ2

(T + 1) +C∞∞

2+ γEMC2(T ) + C3(T ) + C4(T ).

Since

C∞∞ = 2v−1Γ

(C − 2 log(N)−

∑p|N

log(p)

p2 − 1

)(see, e.g., [13]), one gets

R∞ =1

gΓ

(v−1Γ2

lims→1

(Z ′ΓZΓ

(s)− 1s− 1

)+

12

πN

(C1(T ) +A2(T )(2C + γEM − 2 log(N))

)

+ v−1Γ

(C + 1− 2 log(N)−

∑p|N

log(p)

p2 − 1

)+

1− log(4π)4π

+ ϑ(T )

),

where

ϑ(T ) := −v−1Γ2

∞∑j=1

hT (rj)

λj+ γEMC2(T ) + C3(T ) + C4(T ) +

v−1Γ2oT (1),

as T →∞.Now, since hT (rj), C2(T ), C3(T ), and C4(T ) tend to zero as T →∞ (see [3]), we have that ϑ(T )→ 0.Consequently, we get

R∞ =1

gΓ

(v−1Γ2

lims→1

(Z ′ΓZΓ

(s)− 1s− 1

)+

12

πN

(C1 − C −

γEM2

+ log(N)

)

+ v−1Γ

(C + 1− 2 log(N)−

∑p|N

log(p)

p2 − 1

)+

1− log(4π)4π

).

This concludes the proof. �

Proposition 17. Let N ≥ 3 be a composite, odd, and square-free integer. Then, the followingasymptotics holds

1

ϕ(N)A(N) = 2gΓ log(N) + o(gΓ log(N)),

as N →∞.

Proof. By Proposition 13, we have

1

ϕ(N)A(N) = 4gΓ(gΓ − 1)

ϕ(N)

∑σ:Q(ζN )↪→C

gAr(0σ,∞σ)

=4gΓ(gΓ − 1)

ϕ(N)

∑ξ∈U

gAr(0ξ,∞),

where in the second identity we identified each embedding σ : Q(ζN ) ↪→ C with a unique elementξ ∈ U via the assignment σ(ζN ) = e2πiξ/N , and used the isomorphism ισ from Proposition 2.

25

Next, from the identity (13) and Lemma 15, we write gAr(0ξ,∞) as follows

gAr(0ξ,∞) = −2πC0ξ∞ −2π

vΓ+ 4πR∞ + G ,

where G is given by (14), and C0ξ∞ is given by

C0ξ∞ = 2v−1Γ

(C − log(N) +

∑p|N

p log(p)

p2 − 1

)+ κ(ξ̃; 1, N);

we refer to [13] for a proof of the latter identity and for the definition of κ(ξ̃; 1, N). Letting C :=4π(gΓ − 1)v−1Γ , we obtain

1

ϕ(N)A(N) = 2CgΓ log(N)− 2gΓC

(C +

1

2+∑p|N

p log(p)

p2 − 1

)− 8πgΓ(gΓ − 1)

ϕ(N)

∑ξ∈U

κ(ξ; 1, N)

+ 8πgΓ(gΓ − 1)R∞ + 2G gΓ(gΓ − 1).

Observe that, from the definition of κ(ξ; 1, N) and the orthogonality relations of Dirichlet characters,we obtain ∑

ξ∈Uκ(ξ; 1, N) = 0.

Therefore, since ∑p|N

p log(p)

p2 − 1= O(log log(N)),

as N →∞ (see the proof of Proposition 14), we get the following asymptotics1

ϕ(N)A(N) = 2CgΓ log(N) + 8πgΓ(gΓ − 1)R∞ + 2G gΓ(gΓ − 1) +O(gΓ log log(N)),

as N →∞.Now, we claim that

8πgΓ(gΓ − 1)R∞ + 2G gΓ(gΓ − 1) = o(gΓ log(N)),as N →∞. Indeed, using Proposition 16, we have

8πgΓ(gΓ − 1)R∞ = C lims→1

(Z ′ΓZΓ

(s)− 1s− 1

)+ 2C

(C + 1− log(N)−

∑p|N

log(p)

p2 − 1

)

+96(gΓ − 1)

N

(C1 −

(C +

γEM2− log(N)

))+ 2(gΓ − 1)(1− log(4π)),

and by virtue of [18, p. 27], namely

lims→1

(Z ′ΓZΓ

(s)− 1s− 1

)= O(N ε),

for ε > 0, as N →∞, the claim follows for a sufficiently small ε.Finally, note that

2CgΓ log(N) = 2gΓ log(N) + o(gΓ log(N)).

This concludes the proof of the proposition. �

Finally, we can prove the main result of this paper.

Theorem 18. Let N ≥ 3 be a composite, odd, and square-free integer. Then, the following asymptoticsholds

e(Γ) = 2gΓ log(N) + o(gΓ log(N)),

as N →∞.26

Proof. Recalling that e(Γ) = ω2/ϕ(N) and using Proposition 13, we get

e(Γ) =1

ϕ(N)(G(N) +A(N)) .

Substituting therein the asymptotics proven in Propositions 14 and 17, and adding up, yields theasserted asymptotics. �

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E-mail address: [email protected]

Fachbereich Mathematik, Technische Universität Darmstadt, Schloßgartenstr. 7, 64289 Darmstadt, GermanyE-mail address: [email protected]

27

1. Introduction2. Background material3. The minimal regular model of the modular curve X(N)4. The hyperbolic contribution of R5. The parabolic contribution of R6. The self-intersection of the relative dualizing sheafReferences