the jacquet-langlands correspondence and the …gerard.freixas/articles/...2 gerard freixas i...

The Jacquet-Langlands correspondence andthe arithmetic Riemann-Roch theorem for

pointed curves

Gerard Freixas i Montplet

Abstract.- We show how the Jacquet-Langlands correspondence and the arithmeticRiemann-Roch theorem for pointed curves, relate the arithmetic self-intersection numbersof the sheaves of modular forms –with their Petersson norms– on modular and Shimuracurves: these are equal modulo

∑l∈S Q log l, where S is a controlled set of primes. These

quantities were previously considered by Bost and Kuhn (modular curve case) and Kudla-Rapoport-Yang and Maillot-Roessler (Shimura curve case). By the work of Maillot andRoessler, our result settles a question raised by Soule.

Contents

1 Introduction 1

2 The Jacquet-Langlands correspondence for Maass forms 6

3 Selberg zeta functions of modular and Shimura curves 12

4 Faltings’ heights of jacobians of modular and Shimura curves 16

5 Proof of the main theorem 22

1 Introduction

This article concerns the sheaves of modular forms on modular and Shimuracurves, from the perspective of arithmetic intersection theory. In either case,

2000 Mathematics subject classification 14G40, 11G18 (primary), 11F12, 11F72 (sec-ondary).Key words: Arakelov geometry, arithmetic Riemann-Roch, automorphic forms, Jacquet-Langlands correspondance.

1

2 Gerard Freixas i Montplet

the line bundle of modular forms has a natural hermitian structure, the so-called L2 or Petersson metric. Its arithmetic self-intersection number is aninvariant of interest. In the case of modular curves, it has been computedindependently by Bost [1] and Kuhn [30, Cor. 6.2]. Their methods takeadvantage of the presence of cusps. For instance, both works appeal to theKronecker limit formula [30, Prop. 5.2]. Nevertheless, the L2 metric is singu-lar near the cusps: an extension of the arithmetic intersection theory of Gilletand Soule [16] is required [2], [30]. On the other hand, the analogous compu-tation has been accomplished for Shimura curves by Kudla-Rapoport-Yang[29, Thm. 0.5] and Maillot-Roessler [31, Conj. 2.1, Prop. 2.3, Thm. 4.3], fol-lowing different techniques. We emphasize that the formalism of Gillet-Souledoes apply to the Shimura curve situation: the L2 metric is smooth. Espe-cially, the arithmetic Lefschetz fixed point formula of Kohler and Roessler[27] applies. This is the main issue in the approach of Maillot-Roessler. Thetwo numbers, for both the modular and Shimura curves, essentially agree1

and involve the quantity ζ ′(−1)/ζ(−1), the logarithmic derivative of the Rie-mann zeta function evaluated at −1. However, the natures of the proofs aredifferent and the two results remained unrelated so far.

Our purpose is to provide an explanation for the coincidence of the alludedarithmetic self-intersection numbers, without evaluating them (Theorem 1.1).This enterprise is thus independent of the literature just reviewed. The firstobservation is that these numbers fit into our arithmetic Riemann-Roch typeformula for pointed stable curves [14, Thm. A]. This allows a reinterpreta-tion in terms of the Selberg zeta functions and the Faltings’ heights of thecurves. Then the Jacquet-Langlands correspondence for GL2 (Theorem 2.5)intervenes to relate the Selberg zeta functions of the modular and Shimuracurves (Theorem 3.7), as well as their Faltings’ heights (Theorem 4.1). Theobvious outcome is that the computations by Bost and Kuhn on the onehand, and Kudla-Rapoport-Yang and Maillot-Roessler on the other hand,are equivalent. It seems worth pointing out the following facts:

i. the work of Maillot-Roessler and Theorem 1.1 lead to reinterpret thetheorem of Bost and Kuhn solely in terms of the arithmetic Lefschetz fixedpoint formula and the arithmetic Riemann-Roch theorem for pointed stablecurves. This settles a question raised by Soule [37, Sec. 7].

ii. the formula of Bost and Kuhn and Theorem 1.1 give a direct proof ofthe result by Kudla-Rapoport-Yang and Maillot-Roessler, without appealingto the arithmetic Lefschetz fixed point formula. We also notice that themethod of Kudla-Rapoport-Yang is much more involved (but much more

1There is an equality modulo∑

l∈S Q log l, for a suitable set of primes S.

Jacquet-Langlands and arithmetic Riemann-Roch 3

far-reaching).2

From a conceptual point of view, the present report suggests the combi-nation of the theory of automorphic forms and the arithmetic intersectiontheory of Gillet-Soule [16]–[18] and Burgos-Kramer-Kuhn [3]–[4], to establisharakelovian variants of the Grothendieck-Riemann-Roch theorem for certainShimura varieties of non-compact type, via the Jacquet-Langlands correspon-dence. Also, the article indirectly indicates the interest of evaluating certainarithmetic intersection numbers in terms of the Selberg trace formula, as wellas exploring the interaction with other known cases of Langlands functorial-ity. The author hopes to develop these ideas in the future.

Before the statement of the main theorem, we briefly introduce the in-volved relevant objects.

Shimura curves. Let M ≥ 4 be an integer and D an indefinite quaternionalgebra over Q, whose discriminant d is relatively prime toM . Assume d = pqis the product of two different prime numbers. Fix a maximal order OD inD. Let XD

1 (M) → Spec Z[1/dM ] be the scheme representing the algebraicstack of abelian schemes (A/T, i, α) of relative dimension 2, together with aninjective unitary ring homomorphism i : OD → EndT (A) and a ΓD1 (M)-levelstructure α [6, Sec. 1, Sec. 2].3 The morphism XD

1 (M) → Spec Z[1/dM ]is proper, smooth, with geometrically irreducible fibers of dimension 1 [6,Thm. 2.1, Prop. 2.4]. Let πD : A → XD

1 (M) be the universal abelianscheme. If Ω1

A/XD1 (M)

is the sheaf of relative differentials, we define ωXD1 (M) =

detπD∗ Ω1A/XD

1 (M). The invertible sheaf ωXD

1 (M) comes equipped with a natural

L2 metric ‖ · ‖L2 . If x ∈ XD1 (M)(C) and θ ∈ detH0(Ax,Ω1

Ax), then

‖θ‖2L2,x =1

(2π)2

∫Ax(C)

|θ ∧ θ|.

The couple ωXD1 (M),L2 := (ωXD

1 (M), ‖ · ‖L2) is a smooth hermitian line bundlein the sense of Arakelov geometry. There is a canonical isomorphism [10,Lemma 7], [28, Prop. 3.2]

κD : ωXD1 (M)

∼−→ ωXD1 (M)/Z[1/dM ].

Let ‖ · ‖P be the hermitian metric on ωXD1 (M)/Z[1/dM ] built up from ‖ · ‖L2 and

κD. Consider a uniformization X1(M)(C) ' Γ\H, where H is the upper half-plane and Γ ⊂ PSL2(R) a torsion free fuchsian subgroup. The coordinate τ ∈

2Kudla-Rapoport-Yang actually deal with algebraic stacks and compute the exact con-tribution of all the finite places, including those of bad reduction.

3In loc. cit. the ΓD1 (M)-level structures (resp. ΓD

0 (M)) are called V1(M)-level struc-tures (resp. V0(M)). The objects (A/T, i, α) are commonly known as false elliptic curves.


H serves as a local analytic coordinate on X1(M)(C). Then the expressionof the metric ‖ · ‖P is ‖dτ‖P = 4π Im τ [28, Chap. III, Sec. 3, Eq. 3.6].4

At the intermediate steps we will need the Shimura curves XD0 (M) →

Spec Z[1/dM ]. These are coarse moduli schemes for the algebraic stacks offalse elliptic curves (A/T, i, α) with a level structure α of type ΓD0 (M) [6, Sec.2] (see also Proposition 5.4 below). The morphism XD

0 (M)→ Spec Z[1/dM ]is smooth, with geometrically connected fibers of dimension 1.

Modular curves. Let M ≥ 5 be an integer and X1(M) → Spec Z[1/M ]the scheme representing the algebraic stack of generalized elliptic curves(E/T, α), together with a Γ1(M)-level structure α [8, Def. 2.4.1, Rmk.4.2.2]. Let Y1(M) ⊂ X1(M) be the open subscheme parametrizing ellip-tic curves. The morphism X1(M) → Spec Z[1/M ] is proper, smooth withgeometrically connected fibers of dimension 1 [8, Thm. 3.2.7, Thm. 3.3.1,Thm. 4.2.1]. Let π : E → X1(M) be the universal generalized elliptic curve.Define ωX1(M) = π∗ωE/X1(M). For every x ∈ Y1(M)(C) and θ ∈ H0(Ex,Ω1

Ex),the rule

‖θ‖2L2,x =i

2π

∫Ex(C)

θ ∧ θ

defines a hermitian metric on ωX1(M),x. The collection of such norms consti-tute the L2 metric ‖·‖L2 on ωX1(M) |Y1(M). It is a smooth hermitian metric onY1(M)(C), and its asymptotic behavior near the cusps (X1(M) \ Y1(M))(C)can be determined. The metric ‖ · ‖L2 is pre-log-log in the sense of Burgos-Kramer-Kuhn [3, Sec. 7.3.2]–[4, Thm. 5.3], [30, Prop. 4.9, Par. 4.14]. Wewrite ωX1(M),L2 to refer to (ωX1(M), ‖ · ‖L2). There is a canonical Kodaira-Spencer isomorphism [26, Thm. 10.13.11]

κ : ω⊗2X1(M)

∼−→ ωX1(M)/Z[1/M ](cusps),

where cusps denotes the reduced divisor of cusps (X1(M) \ Y1(M))red [26,Chap. 8, Sec. 8.6, Eq. 8.6.3.2]. Let ‖ · ‖P denote the pre-log-log hermitianmetric on ωX1(M)/Z[1/M ](cusps) transported from ‖ · ‖L2 via κ. Uniformizethe curve Y1(M)(C) as Γ\H, for some torsion free fuchsian subgroup Γ ⊂PSL2(R). The coordinate τ ∈ H provides a local parameter for Y1(M)(C).The local expression of ‖ · ‖P is again ‖dτ‖P = 4π Im τ [30, Sec. 4.14].

We will also make use of the modular curves X0(M) → Spec Z[1/M ].These are coarse moduli schemes for generalized elliptic curves (E/T,C)with a cyclic subgroup C of order M , intersecting all the components of all

4The reader will note that in loc. cit. a factor 2 is missing.


the geometric fibers of Esm/T . The morphism X0(M) → Spec Z[1/M ] issmooth with geometrically connected fibers of dimension 1.

Theorem 1.1. Let M ≥ 5 be an integer and D an indefinite quaternionalgebra over Q of discriminant d prime to M . Assume d = pq is a productof two different prime numbers. Then the equality of normalized arithmeticself-intersection numbers

(ωXD1 (M),L2)2

degωXD1 (M)

=(ω⊗2

X1(M),L2)2

degω⊗2X1(M)

holds in the group R/∑

l|3p(p2−1)q(q2−1)M Q log l.

Remark 1.2. i. In the statement of the theorem, as well as throughout thisarticle, we write degL := degLC for a line bundle L over an arithmeticsurface X → Spec Z[1/M ].

ii. The restriction d = pq has been included for the sake of simplicity.The general case is in reach with the methods of this article, but it seemsa tedious task to work out the details (see Remark 3.8 and the proof ofProposition 4.3).

iii. The indeterminacy in Q log 3 is due to the usage of the curves XD0 (M),

X0(M) during the intermediate computations. They intervene to simplify themanipulations with scattering matrices and Selberg zeta functions (Section3), as well as in comparisons of Faltings’ heights (Section 4).

iv. The indeterminacy in Q log l, l | p(p2− 1)q(q2− 1), reflects the lack ofcontrol of the reduction of certain isogenies –between jacobians of modularand Shimura curves– at Eisenstein primes (Section 4).

The article is structured as follows. Section 2 reviews the Jacquet-Lang-lands correspondence for automorphic representations arising from Maasscusp forms. This is a well-known topic for the experts, but the author couldnot find a reference were all the required details are worked out. In viewof the importance of the correspondence for our purposes, it seems worthincluding a complete account. In Section 3, we combine the spectral corre-spondence derived from Jacquet-Langlands and the expression of the Selbergzeta function as a regularized determinant, proven by Efrat [11]–[12]. Theoutcome is a comparison of the Selberg zeta functions for modular curvesX0(M) and Shimura curves XD

0 (M). Section 4 carries out an analogousproject to compare the heights of Jacobians of modular and Shimura curves.The work of Prasanna [34] provided the base for these computations. Afterseveral reduction steps, Section 5 establishes Theorem 1.1.


2 The Jacquet-Langlands correspondence for

Maass forms

2.1 Notations

The following notations will prevail throughout this section. Let M be apositive integer and B an indefinite quaternion algebra over Q of discriminantd prime to M . The case d = 1 is allowed, corresponding to B = M2 Q.Otherwise, d is a product of an even number of distinct primes [40, Thm.3.1]. Let A (resp. Af ) denote the ring of adeles of Q (resp. finite adeles).For every place v | d, Bv := B⊗Q Qv is a division algebra. If v - d (includingv =∞), Bv is isomorphic to M2(Qv). We then fix an algebra isomorphism5

γv : M2(Qv)∼−→ Bv.

Observe that γv induces a group isomorphism GL2(Qv) → B×v . By meansof γvv one can give a sense to the restricted products defining B(A) andB×(A). We introduce a compact open subgroup of B×(Af ). For every finiteplace v of Q set

K0(M)v = (a bc d

)∈ GL2(Zv) | c ≡ 0 mod M.

Then define

KB0 (M) =

∏v|d

K1v ×

∏v-∞v-d

γv(K0(M)v) ⊂ B×(Af ).

Here, for v | d, K1v denotes the group of invertible elements in the maximal

order of Bv:6

K1v = x ∈ B×v | | Nv(x) |v= 1 (Nv is the reduced norm of Bv).

Finally we put

ΓB0 (M) = γ−1∞ (B×(Q) ∩ (KB

0 (M)×B×(R)+)) ⊂ SL2(R).7

This is an arithmetic subgroup of SL2(R), acting on the upper half-planeH by Mobius transformations. Accordingly there is a notion of Maass cuspform for ΓB0 (M). Later we will focus on automorphic representations of theHecke algebra of B×(A) generated by Maass cusp forms of weight 0.

5The isomorphism is unique up to conjugation.6A quaternion algebra over a non-archimedian local field has a unique maximal order

[40, Lemma 1.5].7Notice that ΓB

0 (M) actually depends on the family of isomorphisms γv.


2.2 Ramification and the Jacquet-Langlands correspon-dence

Ramification. We proceed to introduce the notion of ramification andconductor for certain irreducible admissible representations of (the Heckealgebra of) B×(A).8 The reader is referred to [7] and [15, Chap 4., Sec. B,Par. 3, pp. 71–74] for an elaboration on these concepts in the local caseGL2(Qp) and for infinite dimensional irreducible admissible representations.

Lemma 2.1. Let F be a non-archimedian local field and D a non-split quater-nion algebra over F , with reduced norm N . Denote by K1 the subgroup ofinvertible elements in the maximal order of D. Let ρ : D× → GL(E) be anirreducible admissible representation.

i. Suppose the restriction of ρ to K1 is the trivial representation. Thenthere exists an unramified quasi-character χ : F× → C× such that ρ = χN .

ii. If moreover ρ has trivial central character, then χ is a quadratic char-acter, i.e. takes values in ±1.

Proof. Observe that E is finite dimensional because D× is compact moduloits center. Denote by D1 the commutator subgroup of D×. By [25, Lemma4.1], there is a commutative diagram of exact sequences of topological groups

1 // D1 // K1 N // _

O×F // _

1

1 // D1 // D×N // F× // 1.

Hence the reduced norm N : D× → F× induces an isomorphism K1\D× 'O×F \F×. Under i, it follows that ρ is one dimensional and of the form χ N ,for some unramified (i.e. trivial on O×F ) quasi-character χ of F×. Finally, ifρ has trivial central character, then χ is trivial on N(F×) = (F×)2. Hence χtakes values in ±1.

Definition 2.2. With the assumptions of Lemma 2.1 i, we say that ρ isunramified.9

Definition 2.3. Let π = ⊗vπv be an irreducible admissible representationof B×(A). Assume that for every v | d, πv is unramified and that for every

8See [25, Chap. 9, 14] and [15, Chap. 10] for an account of admissible and automorphicrepresentations of B×(A).

9This notion is stronger that the conventional one, namely that ρ |K1 contains theidentity representation. Nevertheless, we only deal with representations ρ for which ρ |K1

is the identity.


finite place v - d, πv is infinite dimensional. Then we define the conductor ofπ to be the positive integer

c(π) =∏v-∞v-d

c(πv).

Here c(πv) denotes the conductor of πv [7, Thm. 1], [15, Thm. 4.24].

Remark 2.4. For an irreducible admissible representation π, a decompositionπ = ⊗vπv exists, where the local factors πv are unique up to equivalence andunramified for almost every finite place v [25, Prop. 9.1]. In particular, inDefinition 2.3, c(πv) = 1 for almost all v and c(π) is a well defined positiveinteger.

Jacquet-Langlands correspondence. We next state the Jacquet-Lang-lands correspondence. We do so under the assumptions to be encounteredlater. We assume d > 1 to avoid any trivial assertion.

Theorem 2.5 (Jacquet-Langlands). i. There is an injective correspondencewhich associates to an irreducible automorphic representation π′ of B×(A),an irreducible automorphic cuspidal representation π = π(π′) of GL2(A),provided π′ is not one dimensional. The image consists of those irreducibleautomorphic cuspidal representations π with square-integrable local factorsπv, v | d.

ii. For every place v - d, πv ' π′v.iii. If for some v | d, π′v is unramified and of the form χv Nv, for

some quasi-character χv : Q×v → C×, then πv is the special representation

σ(χv| · |1/2v , χv| · |−1/2v ).

iv. The correspondence is a bijection between the set of irreducible auto-morphic representations π′ of B×(A), with trivial central character, unrami-fied at primes v | d, infinite dimensional at finite places v - d, and conductorc prime to d, with the set of irreducible automorphic cuspidal representationsπ of GL2(A), with trivial central character, infinite dimensional at placesv - d and conductor cd.

Proof. The proof of i -iii is the content of [25, Thms. 4.2, 14.4, 15.1, 16.1]and [15, Thms. 7.6, 10.1, 10.2, 10.5]. We comment on iv.

Let π′ be an automorphic representation of B×(A) with trivial centralcharacter, unramified at primes v | d and conductor c. By Lemma 2.1, forevery v | d, π′v is of the form χv Nv, for some unramified quadratic character

of Q×v . According to iii, πv = σ(χv| · |1/2v , χv| · |−1/2v ). Since χ2

v = 1, πv


has trivial central character. By [15, Rmk. 4.25], πv has conductor p (thecharacteristic of the residue field of v), because χv is unramified. On theother hand, πv = π′v for v - d. We conclude that π has conductor cd.

Let π be an irreducible automorphic cuspidal representation of GL2(A),with trivial central character and conductor cd. We claim that for everyv | d, the local factor πv has the form σ(χv| · |1/2v , χv| · |−1/2

v ), where χv is anunramified quadratic character of Q×v . Recall that d is a square-free integer.Therefore, for v | d, πv has conductor p (the characteristic of the residue fieldof v). By [15, Rmk. 4.25], πv is either a principal series π(µ1, µ2) or a special

representation σ(χv| · |1/2v , χv| · |−1/2v ). We discard the first possibility. Observe

that the central character of π(µ1, µ2) is the product of quasi-characters µ1µ2

of Q×v . The assumption of trivial central character yields µ1 = µ−12 . Looking

at loc. cit. we find

p = c(π(µ1, µ2)) =(conductor of µ1)(conductor of µ2)

=(conductor of µ1)2.

This is inconsistent and we conclude πv = σ(χv| · |1/2v , χv| · |−1/2v ), thus proving

the claim. Again by [15, Rmk. 4.25], we see that χv has to be unramified.Finally, for πv to have trivial central character, it must happen χ2

v = 1.Namely, χv is a quadratic character. We finish by i -iii.

2.3 Maass cusp forms and passage from the classicalto the automorphic formalism

Let f be a Maass cusp form of weight 0 for ΓB0 (M). We build up a functionφf : B×(A) → C from f . By the strong approximation theorem [40, Chap.III, Thm. 4.3] we can write

(2.1) B×(A) = B×(Q) · (KB0 (M)×B×(R)+).

If g = g0kg∞ ∈ B×(A) is a decomposition according to (2.1), then we defineφf (g) = f(g∞i). Observe that g∞ acts on H via the identification γ∞. Thedefinition of φf (g) does not depend on the particular choice of decompositionof g. We summarize the features of φf :

i. φf ∈ L20(Z

×(A)B×(Q)\B×(A)), where Z denotes the center of B;

ii. for every g ∈ B×(A), r(θ) ∈ SO(2,R) and k ∈ KB0 (M), we have

φf (gr(θ)k) = φf (g);

iii. if ∆ = −y2( ∂2

∂x2 + ∂2

∂y2) and ∆f = λf , then φf is an eigenvector of

eigenvalue λ for the Casimir element of B×(R).


We denote by V B(M,λ) the vector space of functions satisfying propertiesi -iii above. It is known to be a finite dimensional Hilbert space [20, Thm.4.7], with the hermitian structure induced from L2

0(Z×(A)B×(Q)\B×(A)).

Every φ ∈ V B(M,λ) uniquely arises as a φf , with f a Maass cusp form forΓB0 (M) and with eigenvalue λ for ∆.

If m | M and d ∈ B×(Af ) satisfies d−1KB0 (M)d ⊂ KB

0 (m), then there isa linear map

Am,d : V B(m,λ) −→ V B(M,λ)

φ 7−→ (φd : g 7→ φ(gd)).

The Maass cusp form f is a newform if φf is orthogonal to the space generatedby the images of the distinct operators Am,d, m |M , m 6= M . More generally,if M = M0M1, we say that f is M0-new if φf is orthogonal to the spacegenerated by the images of the operators Am,d with M1 | m |M and m 6= M .We define V B(M,λ)M0−new to be the subspace of V B(M,λ) generated by theφf , with f being M0-new. The orthogonal complement of V B(M,λ)M0−new

in V B(M,λ) is denoted by V B(M,λ)M0−old. The space V B(M,λ)M0−old isspanned by functions Am,dφ, where φ ∈ V B(m,λ)new and M1 | m |M .

Now suppose that f is a Maass cusp newform for ΓB0 (M), of eigenvalueλ for ∆, and a simultaneous eigenfunction of the Hecke operators Tp, p -dM .10 Then the right translates of φf in L2

0(Z×(A)B×(Q)\B×(A)) span an

irreducible automorphic cuspidal representation πf of the Hecke algebra ofB×(A). The proof goes as in [15, Chap. 5, Sec. C] and [5, Chap. 3, Par.3.6, Thm. 3.6.1]. The representation πf decomposes as a product of localirreducible admissible representations: πf = ⊗vπf,v [25, Prop. 9.1].

Lemma 2.6. Assume M is square-free. Let v denote a finite place of Q.i. If v | d, then πf,v is unramified.ii. If v - d, then πf,v is infinite dimensional. If v - dM , then c(πf,v) = 1.

If v |M , then c(πf,v) = p (the residual characteristic of v).As a result, the conductor of π is well defined and c(π) = M .

Proof. The first item is clear. Indeed, for every v | d, φf is right invariantunder KB

0 (M)v = K1v .

We claim that for v - dM , πf,v is a principal series representation andhence infinite dimensional. Indeed, this is seen combining that f is a simul-taneous eigenvector of the Hecke operators Tp, p - dM , applying [15, Thm.4.23] and [5, Thm. 4.6.3, Prop. 4.6.5 and proof of Thm. 3.6.1]. Moreoverc(πf,v) = 1 because φf is right invariant under KB

0 (M)v = γv(GL2(Zv)).

10Once the family γvv chosen, the Hecke operators are defined as in the modular case.See [15, Chap. 5, Sec. B, p. 88] for details.


Let v |M . Assume πf,v is finite dimensional. Then it is one dimensionaland of the form χdet for some quasi-character χ : Q×v → C× [25, Prop. 2.7].Because φf is invariant under KB

0 (M), πf,v is trivial on K0(M)v. Namely,χ |detK0(M)v= 1. We observe that detK0(M)v = det GL2(Zv) = Z×v . Henceπf,v is trivial on GL2(Zv) and φf is invariant under the action of γv(GL2(Zv)).This is in contradiction with the newform assumption for f . We infer thatπf,v is infinite dimensional. The conductor of πf,v is thus defined. It remainsto check that c(πf,v) = p (the residual characteristic of v). Because φf isinvariant under KB

0 (M)v and M is square-free, we see that c(πf,v) | p. By[15, Rmk. 4.25], πf,v is a principal series representation or a special represen-tation. As in the proof of Theorem 2.5 iv, the first option is excluded becauseπf,v has trivial central character. Therefore πf,v is a special representation ofconductor at most p. Again by [15, Rmk. 4.25], we see that the conductorc(πf,v) is exactly p. The proof is complete.

Remark 2.7. The lemma suggests we apply Theorem 2.5 iv to the represen-tation πf .

Theorem 2.8. Let M be a square-free integer and D an indefinite quaternionalgebra over Q, of discriminant d > 1 prime to M . Let λ be a positive realnumber. Then there is a Hecke equivariant isomorphism

V D(M,λ)∼−→ V M2 Q(dM, λ)d−new.

As a consequence, dimV D(M,λ) = dimV M2 Q(dM, λ)d−new.

Proof. To begin with, observe that the Hecke operators preserve new and oldspaces. To prove the proposition, it will be enough to show that for everym |M , there is a Hecke equivariant isomorphism

V D(m,λ)new∼−→ V M2 Q(dm, λ)new.

The Hecke operators on Maass cusp forms commute and are normal. Thisis seen as in [15, Chap. 5, Sec. B]. Moreover they commute with the hyper-bolic Laplacian. Therefore V D(m,λ)new and V M2 Q(dm, λ)new admit bases ofnewforms which are simultaneous eigenfunctions of the Hecke operators andthe Casimir element.

First of all we notice that for B = D or M2 Q and φf ∈ V B(m,λ)new, theattached representation πf determines f up to a non-zero scalar. We adoptthe convention d = 1 in the matrix algebra case. We may assume f 6= 0.Let H ⊂ L2

0(Z×(A)B×(Q)\B×(A)) be the space on which πf acts. Let us

consider the space E = H ∩ V B(m,λ). We observe that φf ∈ E. We claimE is one dimensional. Indeed, by Lemma 2.6, πf,v is unramified at every


place v | d, hence one dimensional. At every other finite place, πf,v is infinitedimensional. The conductor of πf is well defined and equals m. Finally, thetrivial representation of SO(2,R) occurs with multiplicity one in πf,∞ [15,Chap. 4, Sec. A]. These facts together with the definitions of conductor andV B(m,λ) yield the claim.

Suppose φf ∈ V D(m,λ)new \ 0 and f is a simultaneous eigenfunctionof the Hecke operators. Consider πf the attached representation of D×(A).According to Lemma 2.6 and Theorem 2.5 iv, to πf corresponds an irre-ducible automorphic cuspidal representation πf of GL2(A), with trivial cen-tral character and well defined conductor dm. Also, remark that πf,∞ ' πf,∞and in particular the trivial representation of SO(2,R) appears with multi-plicity one in πf,∞. Let H ⊆ L2

0(Z×(A) GL2(Q)\GL2(A)) be the space on

which πf acts. By the preceding remarks and the definition of conductor,

the space of functions in H right invariant under KM2 Q0 (dm) × SO(2,R) is

one dimensional. Let φf be a basis of this space. The right translates ofφf must generate πf (irreducibility). Because πf has conductor dm and

πf,∞ ' πf,∞, we see that f ∈ V M2 Q(dm, λ)new. Moreover f is an eigen-function of the Hecke operators with the same eigenvalues as f , becauseπf,v ' πf,v for every finite place v - dm. The association φf 7→ φf sends a

basis of V D(m,λ)new to linearly independent functions in V M2 Q(m,λ)new, bythe strong multiplicity one theorem in the quaternion algebra case.11 Thisway we construct a Hecke equivariant monomorphism of finite dimensionalvector spaces V D(m,λ)new → V M2 Q(m,λ)new.

Analogously, one constructs a reverse Hecke equivariant monomorphismV M2 Q(m,λ)new → V D(m,λ)new. This ends the proof of the theorem.

3 Selberg zeta functions of modular and Shi-

mura curves

3.1 Notations

In this section, we fix a square-free positive integer M with a prime factorN ≡ 11 mod 12, and D an indefinite quaternion algebra over Q of discrim-inant d > 1 prime to M . Let also d′ denote a divisor of d. Construct groupsΓD0 (M) and Γ0(d

′M) = ΓM2 Q0 (d′M) as in §2.1. One can identify ΓD0 (M)

11Strong multiplicity one for irreducible automorphic representations of D×(A) is itself aconsequence of strong multiplicity one for irreducible automorphic cuspidal representationsof GL2(A) [15, Thm. 5.14], [5, Thm. 3.3.6] and the Jacquet-Langlands correspondence,Theorem 2.5 above.


and Γ0(d′M) with their images in PSL2(R).12 These are fuchsian groups

and thus act discontinuously on the upper half plane H. Because N ≡ 11mod 12, they act without fixed points on H and are torsion free [40, Chap.IV, Sec. 3, Par. A]. Therefore the quotient spaces XD

0 (M)an = ΓD0 (M)\Hand Y0(d

′M)an = Γ0(d′M)\H have a structure of Riemann surface, such that

the respective coverings by H possess no branch points. While XD0 (M)an is

compact, Y0(d′M)an is not. Y0(d

′M)an is compactified into a Riemann surfaceX0(d

′M)an by adding the orbits of P1(Q), i.e. the cusps.Equip the upper half plane H with its unique complete hyperbolic rie-

mannian metric of constant curvature −1, whose riemannian tensor is ds2 =(dx2 + dy2)/y2. The scalar laplacian is ∆ = −y2( ∂

∂x2 + ∂∂y2

). These objects

are invariant under the action of PSL2(R), so that they descend to XD0 (M)an

and Y0(d′M)an. We use the same notations ds2 and ∆ to refer to the de-

scended objects. Because the coverings by H have no branch points, ds2 and∆ agree with the intrinsic complete hyperbolic metric of curvature −1 andthe associated scalar laplacian on XD

0 (M)an or Y0(d′M)an.

The Selberg zeta functions of XD0 (M)an and Y0(d

′M)an are defined interms of the length spectrum of ds2. If R denotes either of these surfaces,then

Z(R, s) =∏γ

∞∏k=0

(1− e−(s+k)l(γ))2, Re s > 1.

The outer product runs over the closed primitive non oriented geodesics ofR, and l(γ) stands for the length of the geodesic γ. The product is absolutelyconvergent for Re s > 1, and admits a meromorphic continuation to the wholecomplex plane, with a simple zero at s = 1 [22, Chap. 10, Sec. 5], [20, Sec.10.8].

3.2 Selberg zeta functions and determinants of lapla-cians

For the sake of simplicity, we now assume d = pq, where p, q are differentprime numbers, so that d′ ∈ 1, p, q, pq. Let λ ≥ 0 be any real number.Consider the spaces V D(M,λ) and V M2 Q(d′M,λ) introduced in §2.3. Recallthese are finite dimensional complex vector spaces.

Notation 3.1. We define

mD(0) = md′(0) = 1

12If I is the identity matrix, we observe that −I 6∈ ΓD0 (M),Γ0(d′M). This is clear

because for v - d, γv : M2(Qv)→ D(Qv) is an algebra isomorphism, hence γv(−I) = −1.


and

mD(λ) = dimV D(M,λ) ≥ 0, md′(λ) = dimV M2 Q(d′M,λ) ≥ 0 if λ > 0.

Remark 3.2. i. It is known that mD(λ) and md′(λ) vanish except on a discretesubset of R+: the discrete spectrum of the laplacian ∆ on XD

0 (M)an andY0(d

′M)an, respectively [20, Thm. 4.7]. Then mD(λ) (resp. md′(λ)) is themultiplicity of λ as an eigenvalue of ∆ on XD

0 (M)an (resp. Y0(d′M)an).

ii. In the Shimura curve case, λ = 0 is an eigenvalue of multiplicity oneby compactness and connectedness. In the modular curve case, λ = 0 is inthe residual spectrum, known to be trivial and simple –i.e. multiplicity one–for congruence subgroups [20, Sec. 11.2, Thm 11.3].

Lemma 3.3. For every real number λ ≥ 0, we have the relation

mD(λ) = mpq(λ) + 4m1(λ)− 2mp(λ)− 2mq(λ).

Proof. The relation is clear for λ = 0. For λ > 0 the lemma is a formalapplication of Lemma 2.6, [7, Cor. to the proof, p. 306], Theorem 2.8 andthe definition of pq-new forms §2.3.

Notation 3.4 ([11]–[12]). Let s, w ∈ C with Re s > 1 and Rew > 1. Wedefine the spectral zeta functions

ζD(w, s) =∑λ≥0

mD(λ)∑σ∈C

λ=σ(1−σ)

(λ− s(1− s))−w

and

ζd′(w, s) =∑λ≥0

md′(λ)∑σ∈C

λ=σ(1−σ)

(λ− s(1− s))−w

− 1

2π

∫ ∞−∞

φ′d′

φd′(1/2 + ir)((1/4 + r2)− s(1− s))−wdr.

Here φd′(s) denotes the determinant of the scattering matrix Φd′(s) for thegroup Γ0(d

′M) [22, Chap. 11, Prop. 4.8] (see [20, Sec. 6.3] for the generaltheory of scattering matrices and functional equations of Eisenstein series).

Remark 3.5. The sums involved in the definitions of ζD(w, s) and ζd′(w, s)make sense. Indeed, mD(λ) and md′(λ) vanish outside a discrete subset ofR+ and we assume Re s > 1. Hence, for Rew > 1, the sums are absolutelyconvergent, uniformly convergent on compact subsets of Rew > 1, and thesummations do not depend on the order.


Lemma 3.6. For Rew > 1 and Re s > 1, we have the equality

(3.1) ζD(w, s) = ζpq(w, s) + 4ζ1(w, s)− 2ζp(w, s)− 2ζq(w, s).

Proof. By Lemma 3.3, the corresponding relation for the terms∑

λ≥0 isstraightforward. It remains to show that the integrals contribute to 0 onthe right hand side of (3.1). For this it will be enough to show

φpq(s) = φp(s)2φq(s)

2φ1(s)−4.

This is an elementary application of [22, Chap. 11, Lemma 4.7 (v), Prop.4.8].

Theorem 3.7. The following relation between meromorphic functions holds:

Z(XD0 (M)an, s) =

Z(Y0(pqM)an, s)Z(Y0(M)an, s)4

Z(Y0(pM)an, s)2Z(Y0(qM)an, s)2.

Proof. Because the Selberg zeta functions are meromorphic, we may restrictto s ∈ R, s > 1. In [11, Sec. 2], it is shown that ζD(w, s), ζd′(w, s) extendto meromorphic functions in w ∈ C, regular at w = 0. Loc. cit. and [12]establish the identity

exp(− ∂

∂wζd′(w, s) |w=0) =Z(Y0(d

′M), s)2ψd′(s),

ψd′(s) :=Z∞ d′(s)2Γ(s+ 1/2)−2hd′

· (2s− 1)Ad′eBd′ (2s−1)2+Cd′ (2s−1)+Ed′ .

The terms Z∞ d′(s), hd′ , Ad′ , Bd′ , Cd′ and Ed′ will be made explicit below.An analogous expression holds for ζD(w, s). Together with Lemma 3.6, wederive

(3.2) Z(XD0 (M), s)2 = ϕ(s)

(Z(Y0(pqM), s)Z(Y0(M), s)4

Z(Y0(pM), s)2Z(Y0(qM), s)2

)2

,

where we wrote ϕ(s) = ψpq(s)ψ1(s)4ψ−2

q ψq(s)−2ψD(s)−1. We claim that

ϕ(s) = 1. For this we quote from [11, Thm., p. 445]:

Z∞ d′(s) =

((2π)s

Γ2(s)

Γ(s)

)−χ(Y0(d′M)an)

(χ = Euler characteristic),

hd′ = number of cusps of X0(d′M)an,

Ad′ = hd′ − tr Φd′(1

2),


Bd′ =χ(Y0(d

′M)an)

2,

Cd′ = −hd′ log 2,

Ed′ = −2χ(Y0(d′M)an)(2ζ ′(−1)− log

√2π) + 2hd′ log

√2π − Ad′ log 2.

An analogous expression holds for XD0 (M)an. The recipe [20, Eq. (2.14), p.

45] for the number of cusps readily implies hpq = 2hp = 2hq = 4h1. Thisyields

χ(Y0(pqM)an) + 4χ(Y0(M)an)− 2χ(Y0(pM)an)− 2χ(Y0(qM)an) = −2g + 2,

with

g = g(X0(pqM)an) + 4g(X0(M)an)− 2g(X0(pM)an)− 2g(X0(qM)an),

and g(R) stands for the genus of a surface R. As an outcome of the proofsof propositions 4.2–4.3 below, g = g(XD

0 (M)an).13 Hence χ(XD0 (M)an) =

−2g+2. Also, from [22, Chap. 11, Lemma 4.7 (iv)] results without difficultythe identity tr Φpq(

12) = 2 tr Φp(

12) + 2 tr Φq(

12)− 4 tr Φ1(

12). All in all, we get

ϕ(s) = 1, as was to be shown. To end, for s > 1 the Selberg zeta functionsare positive [22, Chap. 10, Prop. 5.2]. We come up with the conclusion by(3.2).

Remark 3.8. With the appropriate notations, the statements of this sectioncarry over to any d > 1. For instance, Theorem 3.7 generalizes to

Z(XD0 (M)an, s) =

∏d′|d

Z(X0(d′M), s)µ(d′)2ν(d/d

′),

where µ(d′) is the Mobius function and ν(d/d′) stands for the number ofprime factors of d/d′. This generalizes [33, Thm. 9] to Eichler orders of“type” ΓD0 (M).

4 Faltings’ heights of jacobians of modular

and Shimura curves

Fix M a square-free positive integer and D an indefinite quaternion algebraover Q of discriminant d prime to M . Assume furthermore d = pq, for twodifferent prime numbers p, q. Consider the curves XD

0 (M) → Spec Z[1/dM ]

13This is a consequence of the Jacquet-Langlands correspondence for holomorphic cuspforms of weight 2.


and X0(d′M)→ Spec Z[1/d′M ], d′ | d. Denote by JD0 (M), J0(d

′M) the cor-responding jacobians. If Q :=

∏l|pqM l(l2 − 1), then we define the group

RD,M = R/∑

l|Q Q log l. The Faltings’ heights of the jacobians JD0 (M),

J0(d′M) are well defined in RD,M .

The aim of this section is to provide a proof of the next theorem.

Theorem 4.1. The following equality of Faltings’ heights of jacobians holdsin the group RD,M :

hF (JD0 (M)) =hF (J0(pqM)) + 4hF (J0(M))

− 2hF (J0(pM))− 2hF (J0(qM)).

The theorem is the conjunction of Proposition 4.2 and Proposition 4.3below.

Let us denote by J0(pqM)pq−new the pq-new quotient of J0(pqM) [24, Sec.2]. This is an abelian scheme over Z[1/pqM ].

Proposition 4.2 (Prasanna). The identity

hF (JD0 (M)) = hF (J0(pqM)pq−new)

holds in RD,M .

Proof. The argument follows the discussion of [34, pp. 962–963], that webriefly review. By the Jacquet-Langlands correspondence [25, Thm. 16.1],[36], [24, Thm 2.3, Cor. 2.4] and Faltings’ isogeny theorem [13, Cor. 2],there exists a non canonical Hecke equivariant isogeny ϕ : J0(pqM)pq−newQ →JD0 (M)Q [24, Prop. 2.2, Cor. 2.4]. Let K = (kerϕ)(Q), l - Q a prime andKl the l-primary part of K. Then Kl is a TpqM [Gal(Q/Q)]-module of finitecardinality, where TpqM is the Hecke algebra on J0(pqM). On the one hand,an irreducible subquotient14 U of Kl can be realized as an irreducible subquo-tient of J0(pqM)[m](Q), for some maximal ideal m of TpqM lying over l [32,Sec. 14, p.112]. On the other hand, l - Q ensures that m is not an Eisensteinprime [38, Thm. 5.1] and U is automatically isomorphic to J0(pqM)[m](Q)[36, Thm 5.2 (b)]. Since the Gal(Q/Q) representation J0(pqM)[m](Q) isautodual and J0(pqM) has good reduction at l, [34, Lemma 5.5] applies.Together with [13, Lemma 5], this completes the proof.

14By irreducible subquotient we mean the quotient of two consecutive factors in aJordan-Holder filtration. These filtrations trivially exist for modules of finite cardinal-ity, such as Kl.


Proposition 4.3. The equality of Faltings’ heights of abelian schemes

hF (J0(pqM)pq−new) =hF (J0(pqM)) + 4hF (J0(M))

− 2hF (J0(pM))− 2hF (J0(qM))

holds in RD,M .

The proof of the proposition requires some preliminary considerations.

Notation 4.4. Let M be a positive integer and p a prime number, primeto M . The degeneracy maps αMp : X0(pM) → X0(M) and βMp : X0(pM) →X0(M) are the morphisms of Z[1/pM ]-schemes defined, after passing to thecoarse moduli schemes, by the assignments

αMp : (E,C) 7−→ (c(E), C[M ]), βMp : (E,C) 7−→ (E/C[p], C/C[p]).

Here E is a generalized elliptic curve over a Z[1/pM ] scheme and C is a finiteflat cyclic subgroup of order pM of Esm, intersecting all the components ofthe geometric fibers of Esm; c(E) is the contraction of E away from C[p] [9,Chap. IV, Sec. 1]. We also define a degeneracy map between jacobians

ιMp : J0(M)2 −→ J0(pM)

(x, y) 7−→ (αMp )∗x+ (βMp )∗y.

If p, q are different prime numbers, prime to M , then we may construct

ιM = ιpMq (ιMp , ιMp ) : J0(M)4 (ιMp ,ι

Mp )

−→ J0(pM)2 ιpMq−→ J0(pqM).

From the definitions of α••, β•• , one checks with ease that interchanging the

roles of p and q leads to the same morphism, i.e. ιM = ιqMp (ιMq , ιMq ). To lighten

notations, we will skip the reference to the superscripts of the degeneracymaps. Finally, the Atkin-Lehner involution wp : X0(pM) → X0(pM) arisesfrom the correspondence

(E,C) 7−→ (E/C[p], (C + E[p])/C[p]).

The identities wpαp = βp and wpβp = αp follow directly from the modulardescriptions, and entail wpιp(x, y) = ιp(y, x).

Lemma 4.5. i. The degeneracy maps ιp and ι are isogenies onto their im-ages.

ii. Let A be an abelian subscheme of J0(M)2. Then ιp |A is an isogenyonto its image. An analogous property is true for ι.


iii. Let A be an abelian subscheme of J0(M)2, invariant under the diag-onal action of TM on J0(M)2. Then the image ιp(A) is invariant under theaction of TpM . An analogous property is true for ι.

iv. Let K be the kernel of the degeneracy map ιp : J0(M)2 → J0(pM).Then K(Q) is a TM [Gal(Q/Q)]-submodule of J0(M)2(Q), of finite cardinal-ity.

v. Let K be a TpM [Gal(Q/Q)]-submodule of J0(pM)2(Q). Then the sub-group (ιp, ιp)

−1(K) is a TM [Gal(Q/Q)]-submodule of J0(M)4(Q).vi. Let K be the kernel of the degeneracy map ι : J0(M)4 → J0(pqM).

Then K(Q) is a TM [Gal(Q/Q)]-submodule of J0(M)4(Q).

Proof. i. By definition of ι, it suffices to see that ιp is an isogeny onto itsimage. First of all, observe that the image of ιp is an abelian subscheme ofJ0(pM); call it A. By [23, Cor. 1.6.2], the morphism ιpQ : J0(M)2

Q → AQis an isogeny.15 Then there exist an integer m and an isogeny jQ : AQ →J0(M)2

Q such that jQ ιpQ = [m], the multiplication by m map. By theuniversal property of Neron models, jQ uniquely extends to a morphismj : A → J0(M)2 of abelian schemes over Z[1/pM ] and j ιp = [m]. Itthen follows that ιp is an isogeny.

ii. Let A be an abelian subscheme of J0(M)2. By i, ιp is finite. Henceιp(A) is an abelian subscheme of J0(pM) of the same relative dimension asA. In particular, ιpQ : AQ → ιp(A)Q is an isogeny. We conclude as in theproof of the first item i.

iii. We begin with ιp. Only the invariance under Tp is not obvious. It isenough to appeal to the formula

(4.1) Tpιp(x, y) = (−y, px+ Tpy)

(see [36, Rmk. 3.9]). The claim is then clear. The corresponding propertyfor ι is handled in a similar manner.

iv. Firstly, K(Q) is a Gal(Q/Q)-submodule of J0(M)2(Q), of finite cardi-nality, because ιp is a finite morphism of abelian schemes over Z[1/pM ]. Weneed to check the invariance under TM . The only non-trivial issue is the in-variance under Tp. Due to the identity wpιp(x, y) = (y, x), K(Q) is closed un-der wp. From this fact and equation (4.1), we see that if (x, y) ∈ K(Q), thenthe elements ((p−1)y, (p−1)x), (−y, px+Tpy), (py+Tpx,−x) lie in K(Q) aswell. Adding the last two elements, we find that ((p−1)y+Tpx, (p−1)x+Tpy)is also in K(Q). It follows (Tpx, Tpy) ∈ K(Q).

v. The proof goes as for iv.vi. Combine iv -v.

15This amounts to an easy computation with q-expansions.


Lemma 4.6. Let m be a positive integer, A an abelian subscheme of J0(M)m

and ϕ : A → B an isogeny of abelian schemes with kernel K. SupposeK(Q) is a TM [Gal(Q/Q)]-submodule of J0(M)m(Q). Then hF (A) = hF (B)in RD,M .

Proof. Let l be a prime number, l - Q. The l-primary part Kl of K(Q) isa TM [Gal(Q/Q)]-module of finite cardinality. We claim that any irreduciblesubquotient of Kl is isomorphic to J0(M)[m](Q), for some maximal ideal m ofTM lying above l. For this, notice that the projection pi of J0(M)m onto itsi-th factor is a Hecke equivariant morphism. Accordingly, pi(Kl) is a finite l-primary TM [Gal(Q/Q)]-module. Furthermore, Kl becomes a TM [Gal(Q/Q)]-submodule of

∏mi=1 pi(Kl) through the morphism (p1, . . . , pm). Consequently,

every irreducible subquotient of Kl is an irreducible subquotient of somepj(Kl). By [32, Sec. 14, p. 112], every irreducible subquotient of pj(Kl) isan irreducible subquotient of J0(M)[m](Q), for some maximal ideal m of TM

lying over l. We come up with the claim because l is not an Eisenstein prime,namely J0(M)[m](Q) is irreducible. We conclude as in Proposition 4.2, i.e.by the autoduality of the Galois representation J0(M)[m](Q), [34, Lemma5.5] and [13, Lemma 5].

We are now in position to establish Proposition 4.3

Proof of Proposition 4.3. We split the proof into several steps. Below, allthe schemes are defined over S = Spec Z[1/pqM ].

Step 1. Denote by A and B the images of the degeneracy maps ιp :J0(M)2 → J0(pM) and ιq : J0(M)2 → J0(qM). Define the quotients QA =J0(pM)/A and QB = J0(qM)/B. By autoduality of the jacobians, there areclosed immersions of abelian schemes

Q∨A → J0(pM), Q∨B → J0(qM).

Here Q∨A and Q∨B denote the dual abelian schemes of QA and QB. Observethat Q∨A and Q∨B are Hecke invariant. Consider the degeneracy maps ιq :

J0(pM)2 → J0(pqM) and ιp : J0(qM)2 → J0(pqM). Finally, put A = ιq(Q∨ 2A )

and B = ιp(Q∨ 2B ). Then, by lemmas 4.5–4.6, there is a chain of equalities in

RD,M

hF (A) = hF (Q∨ 2A ) =2hF (QA)(4.2)

=2hF (J0(pM))− 2hF (A)(4.3)

=2hF (J0(pM))− 4hF (J0(M)).(4.4)


In (4.2) we followed [35]. In (4.3) we applied [39, Prop. 3.3] and observed

that log 2 has zero image in RD,M . The corresponding formulas for B arealso true.

Step 2. Define C = A + B, the image of A× B under the addition mor-phism J0(pqM)2 → J0(pqM). The map A × B → C is a TpqM -equivariant

isogeny. Indeed, Lemma 4.5 iii ensures that A and B are Hecke invariant.The isogeny property is easily established over C, by identification of theinduced map on tangent spaces, via q-expansions. This implies the isogenyproperty over Q, and then the same argument as for Lemma 4.5 i leads tothe conclusion. The kernel of A × B → C is isomorphic to A ∩ B, em-bedded into J0(pqM)2 through the map x 7→ (x,−x). Because K(Q) is aTpqM [Gal(Q/Q)]-submodule of J0(pqM)2(Q), Lemma 4.6 applies. We find

hF (C) =hF (A) + hF (B)(4.5)

=2hF (J0(pM)) + 2hF (J0(qM))− 8hF (J0(M))(4.6)

in RD,M . In the derivation of (4.6) we took into account (4.2)–(4.4) and the

corresponding equations for B. If QC = J0(pqM)/C, then in RD,M

hF (QC) =hF (J0(pqM))− hF (C)(4.7)

=hF (J0(pqM)) + 8hF (J0(M))(4.8)

− 2hF (J0(pM))− 2hF (J0(qM)).(4.9)

Step 3. Let E be the image of the degeneracy map ι : J0(M)4 → J0(pqM).By Lemma 4.5 i–iii, J0(M)4 → E is an isogeny and E is TpqM -invariant. Let

E be the image of E by the quotient map J0(pqM)→ QC . One checks that

E → E is an isogeny. If K = ker(E → E), then K(Q) is TpqM [Gal(Q/Q)]-

invariant. For this, we notice that E → E is defined over Z[1/pqM ], K =C ∩ E and use the TpqM invariance of E and C. Lemma 4.5 vi and Lemma4.6 then provide the relations in RD,M

(4.10) hF (E) = hF (E) = 4hF (J0(M)).

Step 4. We remark that J0(pqM)pq−new = QC/E. Recalling that log 2has zero image in RD,M and applying [39, Prop. 3.3], we infer

(4.11) hF (J0(pqM)pq−new) = hF (QC)− hF (E)

in RD,M . The proposition results from (4.7)–(4.11).

Remark 4.7. The proofs of propositions 4.2–4.3 show that g(XD0 (M)) =

g(X0(pqM)) + 4g(X0(M)) − 2g(X0(pM)) − 2g(X0(qM)), where g denotesthe genus of a Riemann surface.


5 Proof of the main theorem

5.1 Preliminary reductions

Lemma 5.1. It is enough to prove the statement of Theorem 1.1 when M isprime, M ≡ 11 mod 12 and computing in R/

∑l|p(p2−1)q(q2−1)M(M2−1) Q log l

instead of R/∑

l|3p(p2−1)q(q2−1)M .

Proof. Assume the statement of the theorem known for primes M of theform M ≡ 11 mod 12, and computing in R/

∑l|p(p2−1)q(q2−1)M(M2−1) Q log l.

We must deduce the statement in its general form. Suppose first of all thatM ≥ 5 is an integer prime to pq, without prime factors N ≡ 11 mod 12 in11, 47, 83, 107. Chose distinct N1, N2 ∈ 11, 47, 83, 107 prime to pq. Thecommon prime factors of N1(N

21 − 1) and N2(N

22 − 1) are exactly 2 and 3.

Observe that

SpecZ[1/(p(p2 − 1)q(q2 − 1)N1(N21 − 1))]

∪ Spec Z[1/(p(p2 − 1)q(q2 − 1)N2(N22 − 1))]

covers Spec Z[1/(3p(p2 − 1)q(q2 − 1))]. We conclude considering the etalecovers

X(D)1 (MNi)

xxppppppppppp

&&NNNNNNNNNNN

X(D)1 (Ni) X

(D)1 (M)

defined over Spec Z[1/pqNiM ], i = 1, 2, and applying the functoriality prop-erties of arithmetic intersection numbers and geometric degrees. Secondly,suppose that M ≥ 5 is prime to pq and divisible by N1 ∈ 11, 47, 83, 107.Chose N2 ∈ 11, 47, 83, 107 prime to pqN1. Then

SpecZ[1/(p(p2 − 1)q(q2 − 1)N1(N21 − 1))]

∪ Spec Z[1/(p(p2 − 1)q(q2 − 1)N2(N22 − 1))]

covers Spec Z[1/3p(p2−1)q(q2−1)]. We finish by considering the etale covers

X(D)1 (N1N2)

wwppppppppppp

''NNNNNNNNNNNX

(D)1 (M)

xxrrrrrrrrrr

X(D)1 (N2) X

(D)1 (N1)

each defined over the appropriate Zariski open subset of Spec Z.


Proposition 5.2. Let N be a prime number with N ≡ 11 mod 12. Let Mbe a square-free positive integer divisible by N and S := Spec Z[1/M ]. Thenthe canonical morphism of S-schemes X1(M)→ X0(M) is etale.

Proof. Let f denote X1(M) → X0(M). The relative curves X1(M) andX0(M) are smooth over S and f is finite surjective. Therefore f is flat [19,Exp. I, Cor. 5.9], [21, Chap. III, Prop. 9.3]. It is enough to show that f isunramified. We follow Mazur [32, Chap. II, Sec. 2].

Let k be an algebraically closed field of characteristic prime to M . Let(E,C) be a generalized elliptic curve together with a cyclic subgroup C oforder M , over k. Let Autk(E,C) denote the stabilizer of C in Autk(E). Wemay prove Autk(E,C) is at most ±1. If E is a singular curve, then itis automatically a Neron polygon with n | M sides [9, Prop. 1.15]. WriteM = ln. The Γ0(M) structure is the cyclic subgroup C = µl×Z/nZ of Esm =Gm × Z/nZ. Let ϕ be an automorphism of E, compatible with the groupstructure on Esm. The restriction of ϕ to Gm × 0 is either the identity orthe inversion map (z, 0) 7→ (z−1, 0). Assume the first case. One easily checksthat the restriction of ϕ to Gm × a is of the form ϕ(z, a) 7→ (λaz, a), forsome λa ∈ µl (ϕ preserves C). Then (1, 0) = ϕ((1, 0)) = ϕ((1, na)) = (λna , 0),so that λa ∈ µn. Because M is square-free, µl ∩ µn = 1. Therefore ϕ is theidentity map. For the second case, introduce the involution ι of E such thatι(z, a) = (z−1,−a) for (z, a) ∈ Gm×Z/nZ. Clearly ι(C) = C. The conditionon ϕ is ιϕ |Gm×0= id. We have just shown this implies ιϕ is the identity,hence ϕ = ι. Let us now suppose E is an elliptic curve. Notice that C[N ]is a cyclic subgroup of order N of C and Autk(E,C) ⊂ Autk(E,C[N ]). Itsuffices to check Autk(E,C[N ]) ⊂ ±1. This is the case treated by Mazurin loc. cit. The condition N ≡ 11 mod 12 ensures the required property(see [32, Chap. II, Sec. 2, Table I]).

Recall the definition of the reduced divisor of cusps cusps in X0(M) (resp.X1(M)) [26, Chap. 8, Sec. 8.6, Eq. 8.6.3.2], i.e. (X0(M) \ Y0(M))red, whereY0(M) is the open subscheme coarsely parametrizing elliptic curves. If Mis square-free and divisible by a prime N ≡ 11 mod 12, then the hermitianmetric ‖dτ‖P = 4π Im τ on ωH induces pre-log-log hermitian metrics on bothωX1(M)/Z[1/M ](cusps) and ωX0(M)/Z[1/M ](cusps), with singularities along thecusps cf. §3.1. We write ωXi(M)/Z[1/M ](cusps)P to refer to these hermitianline bundles, i = 0, 1.

Corollary 5.3. Let m,M be square-free positive integers with m | M andboth divisible by a prime number N ≡ 11 mod 12. Write S = Spec Z[1/M ].


We have equalities of arithmetic intersection numbers

(5.1)(ωX1(M)/S(cusps)P )2

degωX1(M)/S(cusps)=

(ωX0(M)/S(cusps)P )2

degωX0(M)/S(cusps).

and

(5.2)(ωX0(M)/S(cusps)P )2

degωX0(M)/S(cusps)=

(ωX0(m)S/S(cusps)P )2

degωX0(m)S/S(cusps)

in the group R/∑

l|M Q log l.

Proof. For the first equality (5.1), let f : X1(M) → X0(M) be the canon-ical morphism. By the proposition, the morphism f is etale. We have anisomorphism

f ∗(ωX0(M)/S(cusps)) ' ωX1(M)/S(cusps).

This isomorphism becomes an isometry for the metrics ‖ · ‖P , at the archi-median places. The corollary follows by the functoriality properties of thearithmetic intersection numbers and geometric degrees of line bundles.

The relation (5.2) follows from (5.1) and the functoriality properties ofarithmetic intersection numbers, once we observe the morphism X1(M) →X1(m)S is etale, too.

Proposition 5.4. Let M be a positive integer, with a prime factor N ≡ 11mod 12. Let D be an indefinite quaternion algebra over Q of discriminantd prime to M . Define S = Spec Z[1/dM ]. Then the natural morphism ofS-schemes XD

1 (M)→ XD0 (M) is etale.

Proof. The relative curves XD1 (M) and XD

0 (M) are smooth over S and f :XD

1 (M) → XD0 (M) is finite surjective. Therefore f is flat. It remains to

show that f is unramified. The proof is adapted from Buzzard [6, Lemma2.2].

Let k be an algebraically closed field of characteristic prime to dM .Since XD

1 (M) is a fine moduli scheme, we may show that for a false el-liptic curve with a ΓD0 (M) structure over k, (A/k, i, α), the automorphismgroup Autk(A, i, α) is at most ±1. Giving a ΓD0 (M) structure α on (A/k, i)amounts to give an equivalence class of isomorphisms β : OD ⊗Z Z/MZ ∼→A[M ], for the left action of the Borel subgroup

B =

(∗ ∗0 ∗

)⊂ GL2(Z/MZ)

∼−→γ

(OD ⊗Z Z/MZ)×,


on the set of full level M structures on (A/k, i).16 Let θ be an automorphismof (A/k, i) preserving the class of β. Then

θ β = β Rg× for some g× = γ

(a b0 c

), a, c ∈ (Z/MZ)×

and Rg× : x 7→ xg× on OD⊗Z Z/MZ. Put ψ = θ− 1. According to Buzzard,we reduce to the case when ψ is an isogeny. Consider the transpose isogeniesθt, ψt : A→ A.17 The following facts are shown in loc. cit.:

i. θtθ = 1 and therefore θtβ = βR(g×)−1 ;

ii. ψtψ is the multiplication by an integer n ≥ 1;

iii. θ + θt is the multiplication by m = 2− n, with |m| ≤ 2, and therefore1 ≤ n ≤ 4;

iv. θ satisfies the equation θ2 −mθ + 1 in Endk(A).

We discuss on the possibilities for θ, depending on n ∈ 1, . . . , 4.

Let P = γ

(0 00 1

)∈ OD ⊗Z Z/MZ and Q = β(P ). Combining the

items i–iii we getmβ = βRg× + βR(g×)−1 .

Applied to P , this leads to

mQ = cQ+ c−1Q.

Therefore (m − c − c−1)Q = 0. Because β is an isomorphism, we inferm = c+ c−1 in Z/MZ. We rewrite this equation as c2 + (n− 2)c+ 1 = 0 inZ/MZ. Projecting to Z/NZ, it will be enough to study the equation

(5.3) c2 + (n− 2)c+ 1 = 0 in Z/NZ.

Case n = 1. Equation (5.3) reads c2 − c + 1 = 0 in Z/NZ. Since4 ∈ (Z/NZ)×, this is equivalent to

(5.4) (2c− 1)2 = −3 in Z/NZ.16Under the identification γ, an element g ∈ B becomes an element g× ∈ (OD ⊗Z

Z/MZ)×. The left action of g is obtained by composing on the right by the automorphismRg× : x 7→ xg× of OD⊗Z Z/MZ. Namely, by sending a full level M structure β to the fulllevel M structure β Rg× .

17For an isogeny h : A → A, the transpose ht is constructed from the dual isogenyh∨ : A∨ → A∨ and the principal polarization on A obtained after fixing t ∈ OD witht2 = −d. See [6, Sec. 1] for details.


Observe that (5.4) has no solution in Z/NZ. Indeed, because N ≡ 11mod 12, −1 is not a square modulo N . Therefore

(−1N

)= −1. On the

other hand, N ≡ 2 mod 3, and so(N3

)= −1. By the quadratic reciprocity

law(

3N

)= −(−1)(N−1)/2. The congruence (N − 1)/2 ≡ 5 mod 6 implies(

3N

)= 1. We conclude

(−3N

)= −1, so that −3 is not a square modulo N .

Case n = 2. Equation (5.3) reduces to c2 = −1 in Z/NZ. This isimpossible because N ≡ 11 mod 12.

Case n = 3. Now (5.3) becomes c2 + c + 1 = 0 in Z/NZ. In particularc 6= 1, since otherwise N = 3. This is excluded by the condition N ≡ 11mod 12. The equation is thus equivalent to c3 = 1 in Z/NZ. A nontrivialsolution to this equation would imply N ≡ 1 mod 3, in contradiction withN ≡ 11 mod 12.

Case n = 4. By the property iv recalled above, the morphism φ := θ+ 1satisfies φ2 = 0. Define E to be (kerφ)red. Then E is an abelian subvarietyof A. Observe that E 6= 0 since φ2 = 0. Suppose E has dimension 1.Namely, E is an elliptic curve. The action of OD on A induces an action byendomorphisms on E: a map of unitary rings OD → Endk(E). Tensoring byQ we find a morphism of unitary ringsD → Endk(E)⊗ZQ. But Endk(E)⊗ZQis either Q, or an imaginary quadratic extension of Q or a definite quaternionalgebra over Q. Such maps don’t exist. We conclude E = A, θ = −1 andAutk(A, i, α) = ±1.

By the proposition, if M is divisible by a prime N ≡ 11 mod 12, bothωXD

1 (M)/S and ωXD0 (M)/S come equipped with smooth hermitian metrics de-

noted ‖ · ‖P , coming from ‖dτ‖P = 4π Im τ on ωH cf. §3.1. Let ωXDi (M)/S,P

denote the corresponding hermitian line bundles.

Corollary 5.5. Let M be a positive integer, with a prime factor N ≡ 11mod 12. Let D be an indefinite quaternion algebra over Q of discriminantd prime to M . Write S = Spec Z[1/dM ]. Then we have an equality ofnormalized arithmetic intersection numbers

(ωXD1 (M)/S,P )2

degωXD1 (M)/S

=(ωXD

0 (M)/S,P )2

degωXD0 (M)/S

in R/∑l|dM

Q log l.

Proof. The canonical map f : XD1 (M) → XD

0 (M) is etale and there is anisomorphism

f ∗ωXD1 (M)/S ' ωXD

0 (M)/S.

The isomorphism is an isometry for the metrics ‖ · ‖P , at the archimedianplace. This proves the statement.


To conclude the preliminary reductions, we remark that the isometries

ωXD1 (M),L2

∼→ ωXD1 (M)/S,P , ω⊗2

X1(M),L2

∼→ ωX1(M)/S(cusps)P

explained in the introduction, together with Lemma 5.1, Corollary 5.3 andCorollary 5.5, reduce Theorem 1.1 to the validity of the following statement.

Theorem 5.6. Let N be a prime number, N ≡ 11 mod 12, and D anindefinite quaternion algebra over Q, of discriminant d prime to N . Assumed = pq is the product of two prime numbers. Let S = Spec Z[1/pqN ]. Thenthe equality of normalized arithmetic intersection numbers

(ωXD0 (N)/S,P )2

degωXD0 (N)/S

=(ωX0(N)/S(cusps)P )2

degωX0(N)/S(cusps)

holds in the group R/∑

l|p(p2−1)q(q2−1)N(N2−1).

5.2 Proof of Theorem 5.6

We are now prepared to establish Theorem 5.6, and therefore Theorem 1.1.The proof is an application of the arithmetic Riemann-Roch formula forpointed stable curves [14, Thm. A] and the results in the previous sections.The reader is referred to loc. cit. for the statement of the formula andprecise definitions. Especially, the arithmetic psi line bundle, endowed withthe Wolpert metric [14, Sec. 1, Def. 2.1], has still not been used in thepresent article and is to be reviewed.

Let M be a square-free positive integer divisible by a prime N ≡ 11mod 12. The cusps of X0(M) over S = Spec Z[1/M ] are given by disjointssections σ1, . . . , σn : S → X0(M).18 The line bundle ψ := ⊗jσ∗jωX0(M)/S

is endowed with the –tensor product– Wolpert metric. We write ψW =(ψ, ‖ · ‖W ).

Proposition 5.7. Let M be a square-free positive integer, with a prime factorN ≡ 11 mod 12. We have

degψW = 0 in R/∑l|M

Q log l.

Proof. Introduce S ′ = Spec Z[ζM , 1/M ], where ζM is a primitive M -th rootof unity. Let X0(M)S′ denote the base change of X0(M) to S ′, and σ′j,

18The sections S → X0(M) are induced by the standard m-sided Neron polygons overS, with m |M , together with their natural Γ0(M)-level structures.


j = 1, . . . , n the base change of the cusp sections σj. The morphism f ′ :X1(M)S′ → X0(M)S′ is etale and the cusps of X1(M)S′ are given by sec-tions S ′ → X1(M)S′ [26, Thm. 10.9.1, Thm. 10.9.4]. The isomorphismf ′∗(ωX0(N)S′/S

′(cusps)) ' ωX1(N)S′/S′(cusps) is an isometry for the metric

structures ‖ · ‖P . Therefore, if τ is any cusp section of X1(M)S′ lying over aσ′j, we have an isometry

(τ ∗ωX1(M)S′/S′ , ‖ · ‖W ) ' (σ′∗j ωX0(M)S′/S

′ , ‖ · ‖W ).

By the functoriality compatibilities of the arithmetic degree, we reduce toshow deg(τ ∗ωX1(M)S′/S

′ , ‖ · ‖W ) = 0 in R/∑

l|M Q log l. We next work outthis case.

Let τ1, . . . , τm be the cusp sections of X1(M)S′ . For every integer k ≥ 0,there is a canonical adjunction isomorphism

(5.5) τ ∗1 (ω⊗(k+1)X1(M)S′/S

′(kτ1 + . . .+ kτm)) ' τ ∗1ωX1(M)S′/S′ .

We endow τ ∗1 (ω⊗(k+1)X1(M)S′/S

′(kτ1 + . . .+kτm)) with the hermitian structure such

that (5.5) becomes an isometry. We still denote this metric by ‖ · ‖W . Sincethe line bundle ωX1(M)S′/S

′(cusps) is relatively ample, for k large enoughthere exists

θ ∈ H0(X1(M)S′ , ω⊗(k+1)X1(M)S′/S

′(k cusps))

θ 6∈ H0(X1(M)S′ , ω⊗(k+1)X1(M)S′/S

′((k − 1)τ1 + kτ2 + . . .+ kτm)).

Equation (5.5) entails the identity

deg(τ ∗1ωX1(M)S′/S′ , ‖ · ‖W ) = deg[(div(τ ∗1 θ),− log ‖τ ∗1 θ‖2W )].

We proceed to identify τ ∗1 θ and ‖τ ∗1 θ‖W . The completion of X1(M)S′ alongthe section τ1 is isomorphic to the formal scheme Spf(Z[ζM , 1/M ][[q]]), forsome indeterminate q [26, Thm 10.9.1, Thm.19.9.4]. The pull-back of the

invertible sheaf ω⊗(k+1)X1(M)S′/S

′(k cusps) to Spf(Z[ζM , 1/M ][[q]]) is trivialized by

the section (dq)⊗(k+1)/qk. The pull-back of θ can be developed as

∑n≥0

anqn (dq)⊗(k+1)

qk∈ Z[ζM , 1/M ][[q]], a0 6= 0.

Then τ ∗1 θ gets identified with the non-trivial global section a0 of the trivial


line bundle OZ[ζM ,1/M ]. By the very definition of the Wolpert metric19

log ‖τ ∗1 θ‖W =∑

σ:Q(ζM )→C

log |σa0|.

We conclude by the product formula:

deg[(div(τ ∗1 θ),− log ‖τ ∗1 θ‖2W )] =∑v-∞v-M

log |a0|v

−∑

σ:Q(ζM )→C

log |σa0| = 0

in R/∑

l|M Q log l.

Proof of Theorem 5.6. Consider the smooth relative curves πD : XD0 (N) →

S and πd′ : X0(d′N)→ S, d′ | d, with S = Spec Z[1/dN ]. By the arithmetic

Riemann-Roch formula for pointed stable curves [14, Thm. A], there are

equalities in CH1(S)

(5.6) 12 c1(λ(ωXD0 (N)/S)Q) = πD∗ (c1(ωXD

0 (N)/S,hyp)2) + c1(O(C(gD, 0))),

12 c1(λ(ωX0(d′N)/S)Q)− c1(ψW ) =πd′ ∗(c1(ωX0(d′N)/S(cusps)hyp)2)

+ c1(O(C(gd′ , hd′))).(5.7)

We clarify the notations:

• ‖ · ‖Q is a Quillen type metric [14, Def. 2].

• The subscript hyp stands for the hyperbolic metric structure. We onlyneed to know ‖ · ‖2hyp = e−α‖ · ‖2P , for some universal constant α. Thenwe have

(5.8) πD∗ (c1(Lhyp))2) = πD∗ (c1(LP ))2) + [0, α degL],

where L denotes either ωXD0 (N)/S or ωX0(d′N)/S(cusps).

• The symbols gD and gd′ denote the genus of XD0 (N) and X0(d

′N),respectively, and hd′ the number of cusps of the latter.

19The compatibility of the algebraic and the transcendent –Fourier– developments fol-lows from the Kodaira-Spencer isomorphism ω⊗2

X1(M)S′

∼→ ωX1(M)S′/S′(cusps) [26, Thm10.13.11], the theory of the Tate curve [9, Chap. VII, Sec.1] and the comparison isomor-phisms in [9, Chap. VII, Sec. 4].


• O(C(g, n)) is the trivial line bundle on S, endowed with the metricC(g, n)| · |, where | · | is the absolute value on C and

C(g, n) = exp

[(2g − 2 + n)

(ζ ′(−1)

ζ(−1)+

1

2

)].

By construction, the hermitian determinant line bundles λ(ωX/S)Q, X =XD

0 (N) or X0(d′N) satisfy

deg λ(ωX/S)Q =1

2logZ ′(Y, 1) + hF (Jac(X))

+1

2logE(g, n)− 1

2log(2g − 2 + n).

(5.9)

Here (g, n) = (gD, 0) and Y = XD0 (N)an if X = XD

0 (N), while (g, n) =(gd′ , hd′) and Y = Y0(d

′N)an if X = X0(d′N). The constant E(g, n) is defined

by

E(g, n) =2(g+n−2)/3π−n/2

· exp

[(2g − 2 + n)

(2ζ ′(−1)− 1

4+

1

2log(2π)

)].

As in the proof of Theorem 3.7, one checks the relations

degωXD0 (N)/S = degωX0(pqN)/S(cusps) + 4 degωX0(N)/S(cusps)

− 2 degωX0(pN)/S(cusps)− 2 degωX0(qN)/S(cusps)(5.10)

and

logC(gD, 0) = logC(gpq, hpq) + 4 logC(g1, h1)

− 2 logC(gp, hp)− 2 logC(gq, hq), resp. for E(g, n).(5.11)

Because for d′ | d the morphism Y0(dN)an → Y0(d′N)an is unramified20 of

degree∏

l|d/d′(l + 1) [20, Chap. 2, Eq. 2.11], one also has

degωX0(pqN)/S(cusps) =(p+ 1) degωX0(qN)/S(cusps)

=(q + 1) degωX0(pN)/S(cusps)

=(p+ 1)(q + 1) degωX0(N)/S(cusps).

(5.12)

An easy manipulation with (5.10) and (5.12) yields

(5.13)(2gpq − 2 + hpq)(2g1 − 2 + h1)

4

(2gp − 2 + hp)2(2gq − 2 + hq)2=

2gD − 2

(p2 − 1)(q2 − 1).

20Recall this is ensured by the condition N ≡ 11 mod 12.


We now apply the arithmetic degree map deg : CH1(S) → R/

∑l|pqN Q log l

to (5.6)–(5.7), and take into account the equations (5.8)–(5.11), (5.13) to-gether with Theorem 3.7, Theorem 4.1 and Proposition 5.7. We infer anequality

(ωXD0 (pqN)/S,P )2 =(ωX0(pqN)/S(cusps)P )2 + 4(ωX0(N)/S(cusps)P )2

− 2(ωX0(pN)/S(cusps)P )2 − 2(ωX0(qN)/S(cusps)P )2

(5.14)

in the group R/∑

l|p(p2−1)q(q2−1)N(N2−1) Q log l. Corollary 5.3 and equation

(5.10) lead to the conclusion.

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Gerard Freixas i MontpletCNRS - Institut de Mathematiques de Jussieu4, Place Jussieu75005 ParisFRANCEe-mail: [email protected]

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