On an upper bound for the arithmetic self-intersection number of the dualizing sheaf on arithmetic surfaces

ON AN UPPER BOUND FOR THE ARITHMETIC SELF-INTERSECTION NUMBER OF THE DUALIZING SHEAF ON ARITHMETIC SURFACES ULF K¨ UHN Abstract. We give an upper bound for the arithmetic self-intersec- tion number of the dualizing sheaf on arithmetic surfaces that arise from Belyi morphisms of a particular kind. In particular our result provides an upper bound for the arithmetic self-intersection number of the dualizing sheaf on minimal regular models of many elliptic modular curves associated with congruence subgroups Γ0(N) and Γ(N) of SL2(Z), as well as for the Fermat curves with prime exponent. Introduction 0.1. Arakelov theory is motivated by the analogy between the intersection theory on an algebraic surface fibered over a complete curve S→Band arithmetic intersection theory on a regular model f:X → Spec OKof a curve Xover the ring of integers of a number field K. A major result in the theory of algebraic surfaces is the Bogomolov-Miyaoke-Yau inequality c1(S)2≤3 c2(S). This inequality may for certain fibered surfaces also be reformulated as ω2 S/B ≤(2g−2)(2q−2) + 3 X b∈B δb, here gis the genus of the generic fiber of S→B,qis the genus of the base curve and δbequals the number of singular points in the fiber above b∈B. The question whether in Arakelov theory an analogous inequality bc1(X)2≤abc2(X) of suitable defined arithmetic Chern numbers with some a > 0 holds is still open. Date: November 3, 2005. Key words and phrases. Arakelov theory, arithmetic self-intersection numbers, effective Mordell, modular curves, Fermat curves. MSC 2000: 14G40,11G18,11G30. 1 2 ULF K¨ UHN From now on we assume that the genus gof Xis larger than 1. Then, it is natural to define bc1(X)2by the arithmetic intersection number bc1(X)2=ω2 Ar, where ω2 Ar is the dualizing sheaf ωX=ωX/OK⊗f∗ωOK/Zequipped with the Arakelov metric (see [Ar], p.1177, [MB1] p.75). It is a delicate problem to define the other quantity bc2(X). A possible candidate is as follows bc2(X) =a1(2g−2) log |∆K|Q|+ a2ÃX x∈Xsing log(#k(x)) + X σ:K→C δFal(Xσ(C))!+a3; here a1, a2, a3∈R, ∆K|Qis the absolute discriminant of the number field Kand δFal is a function on the moduli space of compact Riemann surfaces . One should view δFal as the logarithm of a certain distance of being a singular curve. If with the above choices an inequality bc1(X)2≤abc2(X) holds, then via the Kodaira-Parshin construction an effective version of Mordell’s conjecture is implied (cf. [Pa], [MB2]). The latter conjecture is equivalent to the uniform abc-conjecture for number fields [Fr], which in turn has lots of different applications and consequences in number theory. Here we consider inequalities for the arithmetic self-intersection number of the dualizing sheaf of the minimal regular model Xequipped with its natural Arakelov metric of type ω2 Ar ? ≤a1(2g−2) log |∆K|Q|+a2;(*) here a1is assumed to be an absolute constant and a2should be a real valued function depending on g, [K:Q] and the bad reduction of X(see e.g. [La], p. 166, or [Sz], p.244). We emphasize that if (*) holds for all curves Xthat vary in a single family provided by the Kodaira-Parshin construction, then δF al is bounded and (*) would also imply the desired effective Mordell. 0.2. In this article we approach (*) by means of a Belyi uniformization X(Γ) of an algebraic curve X. Recall that a complex curve Xis defined over a number field, if and only if there exist a morphism β:X→P1with only three ramification points, if and only X(C) is isomorphic to a modular curve X(Γ) associated with a finite index subgroup of Γ(1) = Sl2(Z). The later isomorphism is called a Belyi uniformization of degree [Γ(1) : Γ]. A regular model Xof Xassociated with a Belyi uniformization βis an arithmetic surface together with an morphism β:X → P1 OKwhich extends β:X→P1. Since the regular model X(1) of the modular curve X(1) ON AN UPPER BOUND FOR ω23 equals P1 Z, we could obtain such regular models Xof Xby taking a desingularisation of the normalization of X(1) in Xwith respect to the morphism X(Γ) →X(1). In many cases the arithmetic surface X → X (1) can be chosen to be the minimal regular model, but we have to stress the fact that in general the arithmetic surface Xis not the minimal regular model. Theorem I. Let β:X → X(1) be an arithmetic surface associated with a Belyi uniformization X(Γ) of a curve Xdefined over a number field K. Assume that all cusps are K-rational points and that all cuspidal divisors (= divisors on Xwith support in the cusps of degree zero) are torsion. Then there exists an absolute constant κ∈Rindependent of Xsuch that the arithmetic self-intersection number of the dualizing sheaf on Xsatisfies the inequality ω2 Ar ≤(4g−4) ¡log |∆K|Q|+ [K:Q]κ¢+X pbad aplog Nm(p),(0.2.1) where a prime pis said to be bad if the fiber of X(Γ) above pis reducible. The coefficients apare rational numbers, which can be calculated explicitly. In particular, if the fiber of Xabove phas rpirreducible components C(p) j, then ap≤4g(g−1)(rp−1)2max j,k |C(p) j.C(p) k|. Moreover if β:X → X(1) is a Galois cover, then ap≤0. This result is still far from giving an affirmative answer of the conjectured inequality (*), since it is not clear whether a general curve possesses such a particular Belyi uniformization. If Xis not the minimal model Xmin of XK, then our formula (0.2.1) will become additional contributions coming from those primes of OKthat give rise to fibers of Xwhich contain a (−1)-curve. Indeed let π:X → Xmin be the morphism that exists by the minimality of X, then there exist a vertical divisor Wwith support in those fibers which contain a (−1)-curve such that π∗ωXmin =ωX⊗ O(W). Therefore we have ω2 Xmin,Ar =π∗ω2 Xmin,Ar =ω2 X,Ar + 2ωX.O(W) + O(W)2 =ω2 X,Ar +X pbad bplog Nm(p). Notice that, the coefficients ap+bp, which could be seen as a measure of how complicated the minimal regular model Xmin is, may be arbitrarily large compared to the number of singular points. We can apply our result whenever Γ is a congruence subgroup, this is because of the Manin-Drinfeld theorem (see e.g. [El]). In particular if Γ is of 4 ULF K¨ UHN certain kind, then the coefficients apin (0.2.1) could be calculated explicitly by using the descriptions of models for X(Γ) (see e.g. [KM], [DR]). We illustrate this with the following theorems. Theorem II. Let X0(N)be the minimal regular model of the modular curve X0(N), where Nis a square free integer having at least two different prime factors and (N, 6) = 1. Then the arithmetic self-intersection number of its dualizing sheaf equipped with the Arakelov metric is bounded from above by ω2 Ar ≤(4g−4)κ+ (3g+ 1) X p|N p+ 1 p−1log p, where κ∈Ris an absolute constant independent of N. The modular curves X0(N), with square free Nand (6, N) = 1, are defined over Q. We point to the fact that the completely different methods in [AU],[MU],[JK], which depend strongly on the specific arithmetic of Γ0(N), give the slightly better estimate ω2 X0(N),Ar = 3glog(N)(1 + O(log log(N)/log(N)), which is the best possible one. Theorem III. Let X(N)be the minimal regular model of the modular curve X(N), where Nhas at least two different prime divisors. Then the arithmetic self-intersection number of its dualizing sheaf equipped with the Arakelov metric is bounded from above by ω2 Ar ≤(4g−4) ¡log |∆Q(ζN)|Q|+ [Q(ζN) : Q]κ¢ where κ∈Ris an absolute constant independent of N. Other examples of curves where our result could be applied are the Fermat curves xn+yn=zn. Here we consider just the Fermat curves with prime exponents. Theorem IV. Let Xbe the desingularisation of the closure in P2 Z[ζp]of the Fermat curve xp+yp=zpwith prime exponent p(see [Mc]). Then the arithmetic self-intersection number of its dualizing sheaf equipped with the Arakelov metric is bounded from above by ω2 Ar ≤(4g−4)³log |∆Q(ζN)|Q| + [Q(ζN) : Q]κ´+ 4g(g−1)(rp−1)2(p−1) log(p). where κ∈Ris independent of Nand rp≤4 + p(p−1)/2. 0.3. The starting point in the proof of the main result Theorem I is the following inequality of arithmetic intersection numbers ω2 Ar ≤(4g−4) log |∆K|Q| − 4g(g−1)O(P)2 Ar, ON AN UPPER BOUND FOR ω25 that holds for all sections Pof X(Γ). Under our assumptions this holds in particular for all cusps Sjof X. Furthermore, our assumptions allow then to express the weighted sum PbjO(Sj)2 Ar, here bjdenotes the width of the cusp Sj, in terms of the arithmetic self-intersection number of O(1) on X(1) and an integral depending on a hermitian metric on O(1) (see Theorem 3.5); for our convenience we have chosen the Fubini-Study metric on O(1). The second step is to bound the integral of the first step, i.e. to control the contribution that measures the difference of the Fubini-Study metric and the canonical metric on X(Γ). We obtain this upper bound of the integral in question (see Theorem 4.4) by means of techniques that were first used by J. Jorgenson and J. Kramer in their study of sup-norm bounds of automorphic forms in [JK2]. 0.4. Plan of paper. In the first and second sections of this paper we present the necessary background material on Arakelov theory on arithmetic surfaces. After this preparatory work we study in the third section the behavior of arithmetic intersection numbers with respect to a finite morphism to P1. The fourth section is devoted to the analytical aspects needed in our bound (0.2.1). In section five we study the geometric aspects used in the bound of the quantities apof (0.2.1). In the last section we finally prove the previously stated theorems II to IV. 1. Intersection numbers of hermitian line bundles In this section we give an overview of Arakelov theory for arithmetic surfaces. For more details we refer to the articles [Ga], [So] and [K¨u]; we use the notation, if not explained in the following, as given in these references. 1.1. Notation. Let Kbe a number field, OKits ring of integers. Let Σ be the set of infinite places of K, i.e., Σ is the set of complex embeddings of Kin C. Let XKbe a smooth, projective curve of genus gbigger than two defined over K. An arithmetic surface Xis a regular scheme of dimension 2 together with a projective flat morphism f:X −→ Spec OK, where OKis the ring of integers of K. Moreover we assume that the generic fiber XKof f is geometrically irreducible, i.e., Xis a regular model for XKover Spec OK. We let X∞be the set of complex points of Xof the scheme defined over Spec Zdefined by X, i.e. X∞=Y σ∈Σ Xσ(C). 6 ULF K¨ UHN Note the complex conjugation F∞acts on X∞. Finally, by abuse of notation we set ZX∞ := X σ∈ΣZXσ(C) 1.2. Remark. Note the assumption g≥2 implies that, after a possible replacement of Kby a finite extension, there exist a unique minimal regular model Xfor XK. It has the property, that for any regular model X0the induced birational morphism X0→ X is in fact a morphism. Moreover Xis a semi-stable arithmetic surface associated to X, i.e., Xis a regular scheme of dimension 2, together with a projective flat morphism π:X −→ Spec OK, which has generic fiber XKand such that the special fibers of πdo not contain a copy of P1with self-intersection (−1). 1.3. We call a line bundle Lon Xequipped with a hermitian metric ha hermitian line bundle and denote it by L= (L, h). The analytic degree of a hermitian line bundle Mis given by deg(M) = 1 [K:Q]ZX∞ c1(M), where the first Chern form c1(M) is outside the zeros of a section mof M given by the (1,1)-form c1(M)¯¯Xσ= ddc−log kmσk2. It equals the geometric degree of Mrestricted to the generic fiber XKof X. Two hermitian line bundles L,Mon Xare isomorphic, if L ⊗ M−1∼ =(OX,|·|). The arithmetic Picard group, denoted by c Pic(X), is the group of isomorphy classes of hermitian line bundles Lon Xthe group structure being given by the tensor product. 1.4. Definition. Let L,Mbe two hermitian line bundles on Xand l, m (resp.) be non-trivial, global sections, whose induced divisors on X∞have no points in common. Then, the arithmetic intersection number L.Mof L and Mis given by L.M:= (l.m)fin + (l.m)∞;(1.4.1) ON AN UPPER BOUND FOR ω27 here (l.m)fin is defined by Serre’s Tor-formula, which for l, m having proper intersection specializes to (l.m)fin =X x∈X log ]¡OX,x±(lx, mx)¢, where lx,mxare the local equations of l,mrespectively at the point x∈ X and (l.m)∞=−(log kmk)£div(l)¤+ZX∞ log klk · c1(M).(1.4.2) Note that by convention −(log kmk)£PnpP¤equals P−nP(log kmk)(P). 1.5. Proposition. (Arakelov, Deligne) The formula (1.4.1) induces a bilinear, symmetric pairing c Pic(X)×c Pic(X)−→ R. ¤ 1.6. Arithmetic Chow groups. Instead of the arithmetic Picard groups c Pic(X) we could also consider the arithmetic Chow groups d CH(X). In the sequel it will be more convenient to use these groups to check some identities on arithmetic intersection numbers, this is why we briefly discuss them here. The objects of these arithmetic Chow groups are equivalence classes of arithmetic cycles represented by pairs (D, gD), where Dis a divisor on Xand gD is Green function for D∞=Qσ∈ΣDσ(C). If Lis a hermitian line bundle, then the first arithmetic Chern class bc1(L)∈d CH1(X,Dpre) can be represented by any of the pairs (div(s),−log ksk2) where sis a section for L. The assignment L 7→ bc1(L) induces a morphism c Pic(X)→d CH1(X) that is compatible with the arithmetic intersection pairings in both groups. In particular we have the formulas (div(s),−log ksk2).(0, g) = 1 2ZX∞ gc1(L),(1.6.1) (div(s),−log ksk2).(D, 0) = (div(s), D)fin,(1.6.2) which are just reformulations of particular cases in (1.4.1). 1.7. We put c Pic(X)Q=c Pic(X)⊗ZQ. Let c Pic0(X)Q⊂c Pic(X)Qdenote the subgroup generated by the hermitian line bundles L= (L,k·k) satisfying deg(L|C(p) l ) = 0 for all irreducible components C(p) lof the fiber f−1(p) above p∈Spec OK, and c1(L) = 0. 8 ULF K¨ UHN We recall that if Dis a divisor on XKwith deg D= 0, then there exist a divisor D, which may have rational coefficients, on Xsatisfying DK=D and a hermitian metric k·kon O(D)∞such that O(D) = (O(D),k · k)∈ c Pic0(X)Q. The divisor Dis unique up to multiples of the fibers of fand the metric k · k is unique up to multiplication by scalars. 1.8. Theorem. (Faltings-Hriljac) Let X → Spec OKbe an arithmetic surface. Let Dbe a divisor on XKwith deg D= 0. Then, for all extensions O(D)of O(D)to c Pic0(X)Q, there is an equality O(D).O(D) = −[K:Q]hD, DiNT ≤0, where hD, DiNT is the N´eron-Tate height pairing of the induced class in the Picard group of XKand O(D).O(D)denotes the arithmetic intersection number. Proof. The statement follows immediately from [Hr] theorem 3.1 and [Hr] proposition 3.3. ¤ 1.9. Definition. Let Xbe an arithmetic surface. Then, X∞=Qσ∈ΣXσ(C) is a finite union of compact Riemann surfaces. By abuse of notation we call a (1,1)-form νon X∞such that ν=Qσ∈Σνσ, where νσis a K¨ahler form on Xσ(C), also a K¨ahler form on X∞. A hermitian line bundle Lis called ν-admissible if c1(L) = ν. 1.10. If the genus of Xis greater than 1, then for each σwe have on Xσ(C) the canonical K¨ahler form νσ Ar(z) = i 2gX j |fσ j|2dz∧dz, where fσ 1(z)dz, ... fσ g(z)dzis an orthonormal basis of H0(Xσ(C),Ω1) equipped with the natural scalar product. We write νAr for the induced K¨ahler form on X∞. 1.11. Definition. Let νbe a fixed K¨ahler form on X∞. Given a divisor D=PnPPon Xwe define O(D)ν= (O(D),k·kν), where the metric on the line bundle O(Dσ) on Xσ(C) is such that −log k1Dσk2 ν=Xnpgνσ(z, P ), where gνσ(z, P ) is the unique normalized Green function for Pσ(C) associated with νσ(see definition 2.1). ON AN UPPER BOUND FOR ω29 Observe O(D)νis a ν-admissible line bundle. We note that if gis an algebraic function on X, then in general OX6=O(div(g))ν, although both sides have vanishing Chern forms. 1.12. Theorem.(Arakelov) There is a unique metric k · kAr on ωXsuch that for all sections Pof Xit holds the equality ωAr.O(P)Ar +O(P)2 Ar = log |∆K|Q|. Moreover ωAr = (ωX,k · kAr)is a νAr-admissible line bundle. ¤ For the convenience of the reader we recall the convention ωX=ωX/OK⊗f∗ωOK/Z, where ωOK/Z=∂−1 K|Q. The above result is equivalent to the adjunction formula ωX/OK,Ar.O(P)Ar +O(P)2 Ar = 0, which holds for arithmetic surfaces in the sense of 1.1 (see e.g. [La], p. 101), since by construction ωAr =ωX/OK,Ar ⊗f∗ωOK/Z, where the later metric is the natural one. 1.13. Lemma. Let Pbe a section of X → Spec OK, then there exists a fibral divisor FPsuch that ω2 Ar ≤(4g−4) log |∆K|Q| − 4g(g−1) O(P)2 Ar +O(FP)2, in particular ω2 Ar ≤(4g−4) log |∆K|Q| − 4g(g−1) O(P)2 Ar. Proof. These formulae are well-known for semi-stable arithmetic surfaces (see e.g. [Fa], p. 410), however the essential ingredients namely theorem 1.8 and theorem 1.12 hold also for arithmetic surfaces in the sense of 1.1. For convenience we recall the proof: Let Fpbe a fibral divisor such that ωAr ⊗ O(P)2−2g Ar ⊗ O(FP)∈c Pic0(X)Q.(1.13.1) Note FPhas support in the fibers of bad reduction of X. Then because of the Faltings Hriljac theorem, the definition of c Pic0(X)Qand the adjunction 16 ULF K¨ UHN now. From the equalities Xbj¡div(g1/ej j),−log kg1/ej jk2¢=dbc1(M) = d¡div(p),−log kpk2¢ of classes in d CH1(X)Q, we deduce that there exists an element a∈ OKwith (div(a),−log |a|2) =³dβ∗P−Xbjdiv(g1/ej j),−dlog kpk2+Xbjlog kg1/ej jk2´ =³XbjGj,−dlog kpk2+Xbjlog kg1/ej jk2´. So we get the identities XbjZX∞ log kg1/e jkνAr =dZX∞ log kpkνAr −log Nm(a) 1 dXbjβ∗O(1).O(Gj) = 1 dβ∗O(1).div(a) =deg(β∗O(1)) dlog Nm(a) = log Nm(a). ¤ 3.5. Theorem. Let β:X→P1 Kbe a morphism of degree d. Let PK be K-rational point on P1 Kand write β∗PK=PjbjSj. Assume that all Si are K-rational points and that all divisors PniSiof degree zero are torsion divisors. We denote by β:X → P1 OKan arithmetic surface associated with βand by β∗Pthe horizontal divisor induced by the divisor β∗PK. Let O(1) = (O(1),k · k)∈c Pic(P1)and put M=β∗O(1). Then it holds the inequality ω2 Ar ≤(4g−4) log |∆K|Q| − 4g(g−1) dµO(1)2+ZX∞ log kpk2νAr¶ −4g(g−1) d3X j bjO(Gj)2+1 dX j bjO(Fj)2;(3.5.1) where pis the unique section of Mwhose divisor equals β∗Pand for each cusp Fjand Gjare vertical divisors as in (1.13.1) and (3.2.1). Proof. Let Sj∈ {β−1(PK)}, then for any K-rational point Rof P1 Kthe divisor dSj−β∗Ris a torsion divisor since by assumption d Sj−β∗Pis one. We let gjbe a section of M⊗eand −Gjits vertical divisor as in lemma 3.2. By abuse of notation we write Sjalso for the section induced by Sj. We write Fjfor the vertical divisor determined by Sjby means of formula ON AN UPPER BOUND FOR ω217 (1.13.1). Then we have because of the previous lemmata ³ω2 Ar −(4g−4) log |∆K|Q|−O(Fj)2´/(4g(g−1)) 1.13 ≤ −O(Sj)2 Ar 3.2 =−1 d2ÃO(β∗R)2 Ar +dZX(Γ)∞³log kg1/e jk2−log krk2´νAr + 2O(β∗R).O(Gj) + O(Gj)2! 3.3 =−1 dÃO(1)2+ 2 ZX∞ log kg1/e jkνAr +cµ,νAr 2 +2 dβ∗O(1).O(Gj) + 1 dO(Gj)2! 2.8 ≤ −1 dµO(1)2+ 2 ZX∞ log kg1/e jkνAr +2 dβ∗O(1).O(Gj) + 1 dO(Gj)2¶. We now add these inequalities for all Sj∈ {β−1(P)}weighted with the factor bj/d, which is determined by β∗P=PbjSjand obtain by means of lemma 3.4 the claim. ¤ 4. Upper bounds for the analytic contributions If in theorem 3.5 the morphism βis a Belyi morphism, Pequals the unique cusp ∞of X(1) and the metric on O(1) is the Fubini Study metric, then the contributions −4g(g−1) dµO(1)2+ZX∞ log kpk2νAr¶ which contain analytical data can be bounded from above by means of formula (4.3.1) and theorem 4.4 below. 4.1. Modular curves over C.Let H:= {τ=x+iy ∈C|Im τ=y > 0} denote the upper half plane. The full modular group SL2(Z) acts properly discontinuously on Hvia fractional linear transformations: τ7→ aτ +b cτ +d;τ∈H,³a b c d ´∈SL2(Z). Define Γ(1) := PSL2(Z) = SL2(Z)/{±1}. For any subgroup Γ of finite index in Γ(1) the quotient Γ \Hhas the structure of a Riemann surface, which can be compactified by adding finitely many cusps. We denote this 18 ULF K¨ UHN compact Riemann surface by X(Γ); in fact, we have X(Γ) = Γ\(H∪P1(Q)). Following [Se], p. 71, we call X(Γ) a (general) modular curve. On X(Γ) there may be also finitely many elliptic points, which correspond to τ∈H with a non-trivial isotropy subgroup in Γ. The compact Riemann surface X(1) := X(Γ(1)) is known to be isomorphic to P1(C), the isomorphism being induced by the classical j-invariant. For a subgroup Γ of finite index in Γ(1) we finally have a finite, holomorphic map πΓ:X(Γ) −→ X(1) of degree deg πΓ= [Γ(1) : Γ]. This morphism of compact Riemann surfaces is ramified only above the two elliptic points Pi,Pρand the cusp S∞of X(1). For more details, cf. [Sh]. 4.2. Hyperbolic metric on X(Γ).The volume form µassociated to the hyperbolic metric on His given by the (1,1)-form µ=dx ∧dy y2=i 2·dτ ∧d¯τ (Im τ)2. By abuse of notation we denote the (1,1)-form induced by µon X(Γ) also by µ. It is singular and its singularities occur at the elliptic fixed points and at the cusps of X(Γ) (see e.g. [K¨u], p. 222). We write µΓ=1 [Γ(1) : Γ] 3 πµ for the log-log singular K¨ahler form on X(Γ) determined by µ; i.e. we have the formula RX(Γ) µΓ= 1. 4.3. Fubini-Study metric. A modular form fof weight kis a holomorphic function f:H→C, such that f(γ(τ)) = (cτ +d)kf(τ) for all ¡a b c d ¢∈Γ and such that fis holomorphic in the cusps. Important examples are the modular forms ∆ and E3 4=j∆ of weight 12. The vector space of modular forms of weight kwith 12|kis naturally identified with the vector space H0(X(Γ),M⊗e), where M⊗e=π∗ ΓO(∞)⊗eand e=k/12. The Fubini-Study metric on M⊗eis for the modular form f corresponding to a section fgiven by kfk2=|f|2 (|E4|6+|∆|2)e. The metrized line bundle M= (M,k·k) is a hermitian line bundle. It is well known, that on X(1) = P1 Z M2=O(1)2=1 2.(4.3.1) ON AN UPPER BOUND FOR ω219 4.4. Theorem. Let f:X(Γ) →X(1) be a Belyi uniformization of degree d. There exists a constant κsuch that ZX(Γ) −log k∆kνAr ≤d gκ, here log k∆kdenotes the Fubini-Study metric of the weight 12 cusp form ∆ for Γ(1) and νAr is the canonical K¨ahler form. Proof. It is shown by Jorgenson and Kramer in [JK2] that there exists a constant κonly depending on X(1) such that for all X(Γ) of genus g≥2 it holds the estimate νAr ≤κ[Γ(1) : Γ] gµΓ.(4.4.1) However, the logarithmic singularities of log |∆|2and the singularities of µΓ near the cusps prevent us to apply the inequality (4.4.1) directly. For a small εwe set X(Γ)ε=X(Γ) \f∗Bε(∞). Since 0 ≤νAr ≤κ1 d gµΓwe have ZX(Γ)ε −log k∆k2νAr ≤ZX(Γ)ε£−log k∆k2¤+νAr ≤κ1 d gZX(Γ)ε£−log k∆k2¤+µΓ ≤κ1 d gZX(1)ε£−log k∆k2¤+µΓ(1) ≤d gκ2(ε), here [−log k∆k2]+= (| − log k∆k2| − log k∆k2)/2 is the positive part of −log k∆k2. It remains to bound the integral over the closure of f∗Bε(∞). We write f∗Bε(∞)=∪SjBε(Sj). Observe that near the cusp S∞we have −log k∆k2= 4πy +O(1). We have ZBε(Sj) O(1)νAr ≤κ1 d gZBε(Sj) |O(1)|µΓ ≤κ1 d g bj dZBε(∞) |O(1)|µΓ(1) ≤bj gκ3(ε) 20 ULF K¨ UHN and with the same considerations as in [JK2], p. 1274, we get ZBε(Sj) 4πf∗(y)νAr ≤κ1 bj gκ4(ε) Taking into account that the sum of the width of cusps equal dwe derive ZBε(Sj) −log k∆k2νAr ≤κ1 d gκ5(ε) ¤ 5. Upper bounds for the geometric contribution Here we give general bounds for the geometric contributions X pbad aplog Nm(p) := −4g(g−1) d3X j bjO(Gj)2+1 dX j bjO(Fj)2 in the inequality provided by theorem 3.5, which in turn completes the proof of theorem I. 5.1. Proposition. Keep the notation as in 3.5. For each prime pof bad reduction, we let X(Fp) = Prp j=1 mjC(p) jand set sp= maxi,j |C(p) i.C(p) j|. Then we have X pbad aplog Nm(p)≤4g(g−1) X pbad (rp−1)2splog Nm(p). Proof. Since O(Fj)2≤0 for all j, it suffices to bound the terms involving O(Gj)2. After possibly renumbering the irreducible components and adding full fibers, we may assume 0 6=G(p) j=Pr k=2 nkC(p) kwith all nk≥0 and n1= 0. Then we have the upper bound deg ³M|C(p) 1´=³dSj+G(p) j´.C(p) 1= rp X k=2 nkC(p) k.C(p) 1≤d. Since the natural numbers C(p) k.C(p) 1are semi-positive whenever k6= 1 and since at least one of it doesn’t vanish, we derive that all nk≤d. So we obtain −O ³G(p) j´2=−  rp X j,k=2 nknjC(p) k.C(p) j log Nm(p) ≤µ(rp−1)2d2max j,k ¯¯¯C(p) k.C(p) j¯¯¯¶log Nm(p). Now summing up over all Sjand over all bad primes yield the claim. ¤ ON AN UPPER BOUND FOR ω221 5.2. Proposition. Keep the notation as in theorem 3.5.If ω⊗d Ar ⊗M⊗(2−2g)∈ c Pic0(X), then we have X pbad aplog Nm(p)≤0 . Proof. Because of lemma 3.2 we have M∼ =O(dSj)⊗ O(−Gj). We deduce using the assumption that ωAr ⊗ O(Sj)2−2g Ar ⊗ O(−2−2g dGj)∈c Pic0(X)Q, which by means of formula (1.13.1) implies that F2 j= (2g−2)2/d2O(Gj)2. Therefore −4g(g−1) d3X j bjO(Gj)2+1 dX j bjF2 j=4 d3X j bjO(Gj)2≤0, which we wanted to to show. ¤ As the following lemma shows, the assumptions of the above proposition are always satisfied if β:X → P1is a Galois cover. 5.3. Lemma. Let f:X → Y be a Galois cover of arithmetic surfaces, then there exists a line bundle Lon Ywith: ωdeg(f) X∼ =f∗L. Proof. By assumption Y=X/G for a finite group G. Let s∈H0(X, ωX), then also sσ∈H0(X, ωX) for all σ∈Gand therefore g=Y σ∈G sσ∈H0(X, ω⊗|G| X). However, since gσ=gfor all σ∈G, we have O(div(g)) = f∗Lfor some line bundle Lon Y. Hence ω⊗|G| X∼ =O(div(g)) ∼ =f∗L, from which we deduce the first claim by using the fact |G|= deg(f). ¤ 5.4. Proof of Theorem I. Applying the upper bounds for the analytical given by means of formula (4.3.1) and theorem 4.4 and those for the geometric contribution given above in the inequality of theorem 3.5 yield our main result. ¤ 22 ULF K¨ UHN 6. Explicit calculations A first application of Theorem I is the following general result for modular curves. 6.1. Theorem. Let X(Γ) be the modular curve associated with a congruence subgroup of Γ(1). Assume the level of Γequals N. Then there exists a constant κonly depending on X(1) such that the arithmetic selfintersection number of the dualizing sheaf on the minimal regular model X of X(Γ) satisfies the inequality ω2 Ar ≤(4g−4) ¡log |∆K|Q|+ [K:Q]κ¢+X p|N aplog p,(6.1.1) where the coefficients apare rational numbers. Proof. Recall that a modular curve of level Nand its cusps are defined over a certain subfield Kof Q(ζN). We also remind to the theorem of Manin and Drinfeld that on modular curves X(Γ) associated with congruence subgroups of Γ(1) all divisors with support in the cusps of degree zero are torsion divisors in Pic0(X(Γ)); hence the assumptions of theorem 3.5 are satisfied with the set of cusps of X(Γ). Two of the quantities in (3.5.1) are determined in (4.3.1) and theorem 4.4, which in turn provides that on regular models X(Γ) as in 3.1 we have the formula ω2 Ar ≤(4g−4) ¡log |∆K|Q|+ [K:Q]κ¢ (1) +1 dX j bjO(Fj)2−4g(g−1) d3X j bjO(Gj)2;(6.1.2) where κ∈Ris a positive constant independent of Γ and for each cusp Sj of width bjthe vertical divisors Fjand Gjare as in (1.13.1) and (3.2.1). As we have seen, the claim now follows immediately from the fact that the fibers above the primes p-Nare smooth (see e.g. [KM]). ¤ 6.2. Modular curves X0(N).We now aim to give explicit formulas for the coefficients apin (6.1.1) for the modular curves X0(N). These curves have a model over Qand its complex valued points correspond to the compact Riemann surfaces Γ0(N)\¡H∪P1(Q)¢, where Γ0(N) = ©¡a b c d ¢∈Γ(1) |c≡0 mod Nª. The index in Γ(1) is given by the formula d= [Γ(1) : Γ0(N)] = Y p|N (p+ 1). From now on Nwill be a square free integer with (6, N) = 1 having at least two different prime factors, because then the cusps are also rational ON AN UPPER BOUND FOR ω223 points [Og]. The minimal model X0(N) over Spec Zhas been determined by Deligne and Rapoport to be as follows: The curve X0(N) is smooth over Z[1/N]. If p|N, then the scheme X0(N)⊗Z/pZis reduced and singular over Z/pZ. We write N=pM =pq1·...·qνand set Q=Qν i=1(qi+ 1) and define u=(1 if p≡7 or 11 mod 12 and all qi≡1 mod 4, i = 1 . . . ν 0 else v=(1 if p≡5 or 11 mod 12 and all qi≡1 mod 3, i = 1 . . . ν 0 else The fiber X0(N)⊗Z/pZis the union of two copies C0,C∞of X0(M)⊗Z/pZ crossing transversely in certain super singular points and some chains of projective lines connecting the remaining supersingular points with those of the other copy. More precisely, if u= 1 then there are 2νprojective lines Fi, if v= 1 then there are 2νpairs Gi,Hiof projective lines and if u=v= 0 there are no such projective lines. All theses projective lines have self-intersection number −2. The intersection behavior between these irreducible components is given by the figure 1 below. C∞ C0 H1 G1 G2ν F1F2ν H2ν Figure 1. 24 ULF K¨ UHN In figure 1 all intersections are transversal, i.e., all intersection multiplicities are equal to 1. Finally we mention the formula C0.C∞=dp−1 12(p+ 1) −2ν³u 2+v 3´. 6.3. Lemma. With the notation as in theorem 3.5 we have on X0(N)the formulae −4g(g−1) d3X j bjO(Gj)2+1 dX Sj bjO(Fj)2≤(3g+ 1) X p|N p+ 1 p−1log p, Proof. We can consider the above inequality prime by prime. Let Sjbe any cusp, then we set div(g1/e j)¯¯X0(N)⊗Z/pZ=d Sj+αÃC0+X ν βνFν+X µ γµGµ+X ρ δρHρ!. The relations div(g1/e j)¯¯Fν= div(g1/e j)¯¯Gµ= div(g1/e j)¯¯Hρ= 0 imply βµ= 1/2, γν= 2/3 and δρ= 1/3, for all possible µ,νand ρ. We put F0=C0+1 2X ν Fν+2 3X µ Gµ+1 3X ρ Hρ F∞=X0(N)⊗Z/pZ− F0=C∞+1 2X ν Fν+1 3X µ Gµ+2 3X ρ Hρ; where if some of the components Fν, Gµ, Hρdo not exist, then these are set to be the zero divisors. The other relation to be satisfied is f= div(g1/e j)¯¯C∞= (d Sj+αF0).C∞=d Sj.C∞+αd p−1 12(p+ 1). Hence, since |f−d(Sj.C∞)| ≤ d, we derive |α|=|f−d(Sj.C∞)| dp−1 p+1 1 12 ≤12p+ 1 p−1. From (F0+F∞)2= 0, we deduce F2 0=−F0.F∞=−F0.C∞=−dp−1 12(p+ 1), which in turn implies that −(G(p) j)2=−α2F2 0≤12dp+ 1 p−1. ON AN UPPER BOUND FOR ω225 Now summing up over all cusps Sjyields the bound −X Sj bjO(G(p) j)2≤X Sj bj·12d2p+ 1 p−1log(p)≤12d2p+ 1 p−1log(p) For the formula −X Sj bjO(F(p) j)2= 12(g−1)2p+ 1 p−1log p, which can be proven in a similar way, we refer to [AU], Proposition 4.2.1, p.63. Now we conclude prime by prime and obtain −4g(g−1) d3X j bjO(Gj)2+1 dX Sj bjO(Fj)2 ≤48g(g−1) −12(g−1)2 dX p|N p+ 1 p−1log p ≤(3g+ 1) X p|N p+ 1 p−1log p, where in the last inequality we used g−1≤1 12 d.¤ We are now in position to state 6.4. Theorem. Let X0(N)be the minimal regular model of the modular curve X0(N), where Nis a square free integer with (N, 6) = 1. Then the arithmetic self-intersection number of its dualizing sheaf equipped with the Arakelov metric is bounded from above by ω2 Ar ≤(4g−4)κ+ (3g+ 1) X p|N p+ 1 p−1log p, where κ∈Ris an absolute constant independent of N. 6.5. Modular curves X(N).We now consider the modular curves X(N), where Nhas at least two different prime divisors. These curves have a model over Q(ζN), where ζNis a primitive N-th root of unity and its complex valued points correspond to the compact Riemann surfaces Γ(N)\¡H∪P1(Q)¢, where Γ(N) = ©¡a b c d ¢∈Γ(1) ¯¯¡a b c d ¢≡(1 0 0 1 ) mod Nª. 6.6. Theorem. Let X(N)be the minimal regular model of the modular curve X(N), where Nhas at least two different prime divisors. Then the arithmetic self-intersection number of its dualizing sheaf equipped with the Arakelov metric is bounded from above by ω2 Ar ≤(4g−4) log |∆Q(ζN)|Q|+ (g−1)[Q(ζN) : Q]κ