Rend. Sem. Mat. Univ. Poi. Torino Voi. 53, 3 (1995) Number Theory

C. Gasbarri

SOME TOPICS IN ARAKELOV THEORY OF ARITHMETIC SURFACES

Abstract. We introduce the principal tools in the Arakelov Theory of arithmetic curves and surfaces and we illustrate how this Theory can be used to study questions in Number Theory and Arithmetic Geometry.

1. Introduction

Arakelov Theory is a language which allows to translate some number theoretic questions into the language of Algebraic Geometry.

The principal tool of the Theory comes from Analytic Geometry: we "compactify" objects we are dealing with, with hermitian metrics at infinity (see below).

The Theory, in the one dimensionai case, is a way to look at orders in number fields, as if they were projective algebraic curves. Using this theory, we can prove some classical results in algebraic number theory using the same methods (formally) as in Algebraic Geometry.

In the same direction, using Arakelov Theory, we are able to study (formally) the theory of arithmetic surfaces as if they were smooth projective algebraic surfaces. This means that Arakelov Theory provides us with an intersection theory over arithmetic surfaces which enjoys a lot of "algebraic properties": a Hodge index theorem, a Riemann-Roch theorem, a Nakai-Moishezon theorem etc.

Here we want to introduce the principal tools of the Theory, and we will try to explain how they can be used to attack some interesting problems in arithmetic.

This paper is just a survey, so it contains no originai material; most of the results we will speak about have been found in the last fìfteen years principally by G. Faltings, L. Szpiro and S. Zhang.

The Arakelov Theory has been strongly generalised by the work of H. Gillet and C. Soulé. They constructed an intersection theory for arithmetic varieties of arbitrary dimension and proved very important theorems in this theory: A very general Riemann-

310 C. Gas barri

Roch theorem, an analogue of Bézout theorem (with J.B. Bost) etc.

We would like to thank Steven Galbraith and the Referee for their comments on a first version ofthis paper.

2. Arakelov theory of arithmetic curves

Let K be a number fìeld and OK be its ring of integers. Let S^ be the set of infinite places of K (the set of the archimedean absolute values of K)\ for each a £ Soo, let Ka be the completion of K by a and let '<pa : Ka —• C be a continuous embedding of Ka in C.

DEFINITION. A metrized OK-module E — (E; (•; •)a)aescx> is afinitely generated OK-module E such that, for each a £ S^o, there is an hermitian metric (•; -)a on the Ka— vector space Ea (where Ea = E <S>oK K<j).

lf C = (£; (•; )(r)cre500 is a metrized projective OK-module of rank one, we cali it a metrized line blindi e over OK, and we define its degree by the formula

deg(Z)=ìog(Card(£M))'- £ e^logHI,

where s £ C is a non zero e le me nt and ea is 1 if' <r is real and is 2 if a is complex (see

[26]).

This definition does not depend on the chosen s by the product formula.

We will denote by VÌCC(OK) the group of isometry classes of metrized line bundles

on OK-

It is not difficult to prove that the map

deg:Picc(Or<)—>M

is a group morphism.

If E is a metrized O/v'-module, then we can consider the naturai map

w-E^ 0 Ea = Em. veSoo

The image a(E) is a lattice in £k.The M-vector space E®, is naturally equipped

with a sup norm: for each x — (xa)aes00 £ E^ we define

Ikllsup = sup{(a:;a:)<T}. ' a

then we put:

-J3(A) = { Z < E £ M / I H | S U P < A } ;

Some topics in Arakelov theory of arìthmetic surfaces 311

-Xi(E), the smallest X such that B(\) contains i linearly independent vectors of a{E);

-^max(^) = max{Ai}.

-X(E) = -log(vo\(EM/a(E))) + log(Card(i?Tors)); where ^Tors *s m e t o r s i ° n

submodule of E.

Let

0 -+ M0 —• M1 —• E -> 0

be a projective resolution of E; we defìne then

/max \ /max

det(£) = I / \ Mi 1 <g> ( / \ M0

where V* is the dual of V. We see that, if E is metrized, then det(Z?) is a metrized line bundle on OK \ we define then

-deg(£) = dég(det(£)); this is well defìned (see [19]).

-H°(E) - {x E E / \\a(x)\\sup < 1}. We cali this set "the set of global sections of £".

Using this dictionary, we can state a list of Theorems. Each of them has an evident analogue in the theory of projective algebraic curves.

THEOREM 2.1 (Riemann-Roch). x(E) - àeg(E) + rk(E)x(0); where O is the metrized line bundle (OR', (1; l)o- = l')«

THEOREM 2.2. If C is a metrized line bundle such that H°(C) / {0} and deg(Z) = 0; then £~~6.

THEOREM 2.3. If C is a metrized line bundle such that deg(£) > ~x(0), then H°(C)^{0}.

The last Theorem is a reformulation of the first Minkowski Theorem on lattices.

For more about these Theorems see [26].

In analogy with algebraic geometry, a metrized line bundle C is said to be ampie if, for each metrized Ojc-module E, there exists a n0 G N such that, for every n > n0, Amaxó# ® £ ") < 1. In Arakelov Theory this can be stated in the following, very geometrical way: "£ is ampie if, for every E and ri suffìciently large, the metrized vector bundle E (g> C is generated by global sections."

As in the theory of algebraic curves we ha ve

312 C. Gasbarri

THEOREM 2.4. A metrizedline bundle C is ampie if and only if deg(C) > 0.

For a proof see [11].

As we stated in the introduction, a lot of classical theorems in Algebraic Number Theory ha ve a very simple proof using this language. We give here an example.

THEOREM 2.6 (Dirichlet). The class group CI(OK) ofOK is finite.

Proof Let C G Cl(0K), provide C with a metric such that deg(£) = -x(0). Then H°(C) ^ {0} and so C~l is an ideal in OK such that NmOK/il(jC-1) < exp(~x(0)). So, the following classical lemma completes the proof.

LEMMA 2.7. Let a > 0, then there exist only a finite number of ideals a of OK

such that NmoK/%((Ì) < a.

With the same strategy we can prove

- The Hermite-Minkowski Theorem: Spec(^) is simply connected;

- The Dirichlet Units Theorem;

- The Hermite Theorem: If A is a number field, d E N and S is a finite set of places of A'; then there are only finitely many L/K such that [L : A'] < d and L is ramified only over S.

For an Arakelovian proof see [26].

As we have seen, the Arakelov Theory of arithmetic curves allows us to give a very geometrie interpretation of classical algebraic number theoretic objeets. Indeed, using Arakelov Theory, you can think about orders in numbér fields as if they were projective algebraic curves.

3. The Arakelov theory of arithmetic surfaces

Let A be a number field, OK be its ring of integers and 5co be the set of infinite places of A\

Let XK be a projective, smooth, geometrically connected curve over Spec(A').

It is well known that there exists a regular minimal model X of XK over Spec(0A-)-

To be more precise, the minimal model is a regular scheme X of dimension two, with a projective fiat morphism / : X —*• Spec((9j<:) which has generic fibre XK and such that the special fibres of / do not contain a copy of P1 with self-intersection (—1). If the genus of XK is bigger than zero, this model is unique.

We cali À' an arithmetic surface.

If the genus of XK is greater than zero, we suppose that the singular fibres of / are reduced fibres with only ordinary singularities (semistables curves) this can always be

Some topics in Arakelov theory of arithmetic surfaces 313

arranged after a suitable base change.

For each a £ Soo, we will denote by Xa the Riemann surface obtained by extension of scalars from À" to C via Ka and <pa.

DEFINITION. A metrized ìnvertible sheaf C = (£; || • ||) over X, is an invertible sheaf C over X such that, for each a 6 5oo, the holomorphic line bundle Ca — C'®a C over the Riemann surface Xa, obtained from C by extension of scalars, is equipped with a C°° hermitian metric || • ||cr.

As before, we denote by Picc(X) the group of isometry classes of metrized line bundles over X.

The Arakelov Theory of arithmetic surfaces is, roughly speaking, an intersection theory between classes of metrized line bundles on X. We will now describe this intersection pairing.

Let C and M be two metrized line bundles on X. The Arakelov intersection (£; M) between C and M is defìned as the following sum, where the two terms will be defined below:

Suppose for the moment, it is possible to choose global sections / and m of C and M respectively, such that the associated Cartier divisors div(/) and div(???) have no common components.

Let div(/) be the Cartier divisor on X associated to the global section / of C (namely

(9x(div(/)) ~ £) and div(??7,) be the Cartier divisor associated to the global section m of

M.

- For each closed point x e X, let Ox-x be the locai ring at x; let fx be a locai function for / and gx a locai function for m (in a neighbourhood of x)\ the ring Ox xI(t .„ \ is a finite artinian ring. So we defìne the finite part of the intersection as

(C;M)fin = 52 log(Cavd(Ox.x/{fx.gx))) xex

where the sum turns out to be a finite sum.

- (£; M)oo is called the infinite part of the intersection and it is defìned as follows: Let o- G Soo ar,d let div(/CT) be the image of div(/) on the Riemann Surface Xa. We denote

div(^) = 52naPa-ot

On the other hand, ma is a section of the metrized line bundle Ma (where we denote by || • ||a the norm on Ma).

314 C. Gasbarri

We write log^ra^H)^) for the sum Y/ana\og(\\ma\\l7(Pa)) (this makes sense because div(/) and div(m) have no eommon component). Let us denote by ci{Mo) the first Chern form of Ma\ away from the zeroes of ma, it is the 1-1 form -dd° log Hm Hcr (it depends on Ma,on \\-\\a, but not on raa). So we set

The intersection number (£; M) does not depend on the choice of / and m.

For general C and M it suffices to remark that there exist invertible sheaves M% and

Tj (i,j = 1,2) such that Mi has disjoint global sections with Tj (ì,j = 1,2) and

(1) C-Fi^Ff-1 , M~Mi®Mf-\

For each cr E 5oo, we provide Mi-a and Tj]a with metrics in such a way that the equivalences in (1) are isometries. Then we define (£;M) by linearity.

It is possible to prove that this definition does not depend on the choices we made. The first Theorem is

THEOREM 3.1. The intersection Pairing

(•;•): Picc(A:) (8) Picc(X)—^M

is a bilinear, symmetric pairing.

For a proof see [8] or [23].

REMARK. The most important of the Arakelov ideas (see [2]) is the introduction of the infinite part of the intersection; without this infinite part we can not define the degree of the intersection; so the intersection of two line bundles is just a class of zero-cycles.

It is clear that our intersection theory is just between metrized line bundles and not between divisore on X; for, in general, if D is a divisor on X there is no canonical metric on Ox(D).

There are several ways to solve this problem, we describe the classical one (which was proposed by Arakelov) when the genus of XK is greater than zero.

Let a G S'oo and let Xa the Riemann surface obtained from X by extension of scalare to Spec(C) via a.

There is a canonical metric dfi^ on Xa, the so called Arakelov metric, which is the following:

Let Q,a be the canonical sheaf on Xat and let u>i;... \u9 be an orthonormal basis

Some topics in Arakelov theory of arithmetic surfaces 315

of H0(Xa;Qo) with respect to the hermitian produet

(2) (a;^) = i / « A ^ .

The metric dpp^x is defined by the formula:

< Ar = 2^Y1UJ A^i"-

Let £ = (£; || • ||) be a metrized line bundle on Xa, we say that the metric || • || is admissible, and that C is an admissible metrized line bundle if

c\(C) = 27ndeg(;C)c/jUAr.

Every line bundle on X a has an admissible metric and any two admissible metrics on such a line bundle are proportional.

Now, let D be a divisor on X; for each a G 5'co, let Da be the divisor on Xa

obtained from D by extension of scalars. We can endow Oxv(Da) with the unique admissible metric || • H such that, if 1# is the canonical section of Ox„(Da), then

log||li>||ad/*Ar = °-

We will denote by Ox{D) the line bundle Ox{D) endowed, for each a £ S'co, with such a metric.

Sometimes in Arakelov Theory it is very useful to consider compactified divisors.

DEFINITION . A compactified divisor is a formai sum

where D is a divisor on X and À 6 1. A compactified divisor D is said effective if D is an effective divisor and \a > 0.

If D = D -f 52ff€5 XÓX'G ls a compactified divisor, we can associate to D an

admissible line bundle Ox(D) in the following way:

OxtD) = Ox(D)®MOx(K) a

where Ox(^a) is the trivial line bundle Ox, endowed with the following metric

" 1 " T " \exp(-A a ) ifr = (7.

316 C. Gasbarri

REMARE. If M is a metrized line bundle, then it is easy to see that

(Ox(D);M) = (OxjD); M) + deg(A< *) E , V '

On the other hand, if C is an admissible line bundle and s is a "meromorphic" section of C, we cali the compactified divisor

d i v ( s ) - I 3 f / log\\S(T\\adpM) Xa

(where sa is the divisor on Xa obtained from s by extension of scalars) a compactified divisor associated to C

REMARK. In the language of compactified divisors, for each a E SQO the Riemann surface Xa can be seen as a vertical divisor of the family f : X -+ SPQC(OK), over the arithmetic curve Spec(C7/c); the only difference between the X^'s and the other irreducible divisors is that, in a compactified divisor, their coefficients can be real numbers, and not just integers.

We denote by H^r(X; C) the set of effective, compactified divisors associated to C.

REMARK. If D is a compactified divisor on X, then D is a compactified divisor associated to Ox{D).

REMARK. {base change) Let K' be an extension of K and let QK> be its ring of integers; let XKI be the curve XK ®K Spec(A'') over K' and let X' be the minimal regular model of XK> over SPQC(OK')- There exists a canonical morphism g : X' —» X. Let D and D' be two compactified divisors on X then g*(D) and g*{D') are compactified divisors on X' and we have the formula

We can put a canonical metric on the relative dualizing sheaf U>X/OK:- f°r e a c n

e G Soo and for each P E Xa(C) we ask the residue map

QXAP)\P-^^

to be an isometry.

REMARK. The line bundle Qx9{P) ys metrized as described above.

For every a E Soo, this metric on (OJX/OK)a is admissible and we denote by u>x/oK

the relative dualizing sheaf endòwed with these metrics.

One of the most important theorems in the Arakelov Theory of arithmetic surfaces is an analogue of the Riemann-Roch Theorem which is stated in Theorem 3.3.

Let C be a line bundle on X; we define then

DetR/*(£) = det(Jff°(X;£))(g)det(^1(X;£))*.

Some topics in Arakelov theory of arithmetic surfaces 317

Given an admissible line bundle C on X, the following very important Theorem provides a canonical metric on DetR/*(£):

THEOREM 3.2 (Faltings). There exists a unique way to assign to each admissible line bundle C on X, a metric on DetR/* (C) such that, for each cr E S'^ the following properties hold:

a) If £ is the trivial bundle with the trivial metric, then the metric on DetR/*(£) is

induced by the canonical hermitian product (1) on H°(X(7\UJXIT)'

b) If(p : C —> A4 is an isometry then

DetR/;(v?) : DetR/*(£) -—+ DetR/;(A?)

is an isometry.

e) If x E Xa(€.) then the canonical isomorphism

DetR/*(£,(*)) - ^ (DetRAfZ)), 0Ca(x)\x

is an isometry. (See [9]; [19]).

We denote DetR,f*(£) the metrized (9/c-module DetR/*(£) equipped with this metric and we define

x(£) = deg(DetR/*(Z)).

THEOREM 3.3 (Faltings, Riemann-Roch). If C is an admissible line bundle on X then

(See [9]; [19]). As you can see, formally, this Theorem is the same as in the "geometrie" case.

As in the geometrie case, this Theorem has a very important consequence:

COROLLARY 3.4. Let C be an admissible line bundle such that (C;C) > 0 and deg(CK) > 0, then, for n > 0, H°Ar(X;C) ± {0}.

As we announced in the introduction, arithmetic surfaces (equipped with the Arakelov intersection pairing restricted to admissible line bundles) have a lot of properties' which recali the intersection theory on classical algebraic surfaces. We will illustrate this analogy by quoting three theorems.

Firstly we have an Adjunction formula:

Let K'/K be a finite extension. Let X/^'be the curve over Spec(A^) obtained from XK by extension of scalars. Let X' be the regular minimal model of XK> over Spec(C?A-/). Let P E XKI(K'), since X' is proper over Spec((9/c')' t n e r e *s a unique way to extend

318 C. Gas barri

P to a section <pP : Spec((9/<:>) -». À"' of / . The image <£>p(Spec((9/v'/)) = #p will be a divisor over X'; it is called the divisor associateci to P.

THEOREM 3.5 (Faltings). Let P e X(K) be a rational point, and let Ep be the unique divisor on X associated to P then

(U^Z; Ox(EP)) = -(OX(EP); Ox(EP)).

See [9].

Secondly we have an analogue of Hodge Index Theorem:

THEOREM 3.6 (Faltings). The signature of the Arakelov intersection pairing restricted to admissible line bundles is '(+;—;•••; —). See [9].

REMARK. In order to define the signature of the Arakelov intersection pairing, we

need to define an appropriate admissiblé group of admissible line bundles modulo numerical

equivalence. See [19].

Finally an analogue of the Nakai-Moishezon Theorem has recently been proved:

THEOREM 3.7 (Zhang). Let C be an admissible line bundle such that:

a) the self-intersection of £ is positive: (C]C) > 0;

b) far every effective compactified divisor D, (£; Ox{D)) > 0.

Then, far n ^$> 0, the Z-module H°(X;C ) is generated by H^r(X; C ).

See [31].

There are a lot of very important relations between Arakelov Theory and the anthmetic and the geometry of arithmetic surfaces; the following two statements show the Arakelov interpretation of two classical arithmetic tools.

The first statement relates Arakelov Theory with the theory of heights:

THEOREM 3.8. Let C be an admissible line bundle, then the function

hc : X(K) —» M

is a height function associated to C.

The second Theorem relates the Arakelov Theory with the Néron-Tate bilinear pairing on the Jacobian JK of XK •

We recali that a vertical divisor V on X is a divisor such that f\supp(v) '• Supp(V) —» Spec(Ox) is not surjective.

Some topics in Arakelov theory of arithmetic surfaces 319

THEOREM 3.9 (Faltings-Hriljac). ìf D and D' are two divisors on X sudi that

a) deg(DA) = deg(Dk) = 0;

b) far eveiy irreducible vertical divisor V we have deg(Ox(D)\v) = Q.

Then

(Ox(D);Ox(D')) = -(D;D')m

where, on the righi, (•; *)NT denote the Néron-Tate pairing on JK <8> JK-

(See [9], [16], [19] and [11]).

As we can see, Arakelov Theory provides a new numerical invariant of an arithmetic surface, namely the self-intersection of the dualizing sheaf ( w j / o K ; w ^ K ) .

As in the geometrie case we have the following:

THEOREM 3.10 (Faltings). Let X be a semistable arithmetic surface with

g(XK) > % then (u>x/oK^x/oK) > 0

(See [9])

The self-intersection of the relative dualizing sheaf has a very important meaning in arithmetic:

Let f \ X —* Spec(Ox) be an arithmetic surface with Q(XK) > 1. Let JK be the Jacobian of XK and let

in '• XK —• JK

P ^ cl(fìXK _ {2g - 2)OxK (P))

be the canonical morphism. There is the following classical Conjecture of Bogomolov:

CONJECTURE (Bogomolov). The image J^I{XK)(K) is discrete far the Néron-Tate topology. An equivalent statement is: there is no sequence of points {Pn} C XK(K) such that

] i m hm(hjPn)) =Q

»-«, [K(Pn):Q] where /^NTO is the canonical Néron-Tate height on JK<

This Conjecture is motivated by a partial, very important, result of Raynaud:

THEOREM 3.11 (Raynaud). Let (<//c)Tors(^0 be the torsion subgroup of JK(K).

Then

(JK)Tovs(K)r\Jn(XK(K))

is a finite set.

See [22].

320 C. Gasbarri

The positivity of the self-intersection of the relative dualizing sheaf is strictly related to the Bogomolov Conjecture:

THEOREM 3.12 (Szpiro). Let f :,X —* Spec((9jv-) be a smooth arithmetic surface (Le. f is a smooth morphìsm) sudi that g{Xx) > 1. We have the strict inequality (UX/OK '^X/OK) > 0 ìf and only if the Bogomolov Conjecture is true for XK..

In order to illustrate the power of Arakelov Theory, we sketch the proof of one implication:

Suppose (^X/O^'Ì^X/OK) = 0;-'we will construct a sequence of points {Pn} C

XK(K) such that limn_oo ^(ply.®] = °-

Using the Faltings-Hriljac Theorem (Thm. 3.9), the Adjunction Formula and, if necessary, making base changes, we have:

2/»NT(Ìn.(P)) = - (uxj^ - (2.7 ~ 2)Ox(EP) ; uT^^T- (2</ - 2)Ox(EP))

= -uxj^2 - (2g - 2)2 (Ox(Ep); Ox(EP))

+ 2(2g-2)(&ZjB-'ìOx{Epj)

= (2g - 2)2 (uxj^; ÒxjÈpj) + 2(2g - 2) (u>xjòZ\ Ox(EP))

= 2g(2g-2)(ùJxJ^:;Ox(Ep)).

So we must find a sequence of points Pn £ XK(K) such that

1™ r^/T-, N , - . ' = 0-n-oo [K(Pn) : Q]

Let e > 0 and a e SQQ. Consider the admissible line bundle C = wx/oK ® <9xv(e) ori-X. We have deg(£/c) = 2</ - 2 > 0 and (C\Z) = 2(2g - 2)e > 0; so we can apply Corollary 3.4. For m > 0 we have that HQ

Ar(X;C®m) ^ {0}. Let D +' £ , X°x° b e a n

effective divisor associated to C ' = uX/oK®m ® C?;c(me); after base change we can suppose that £) is a sum of sections D = X^=i ~" ^JV ^° w e ^ a v e

m(2sf-2)€= ( c J 3 ^ T ; O x ( ^ + X l A - y Y ^ )

™(2</-2)

= E (^7^^x(^Fl))+^^(2(/-2) i=l <?

> m(2g - 2)inf (^x~I^\Ox(EPi)) •

Some topics in Arakelov theory of arithmetic surfaces 321

Therefore we nave found a point P G XK(K) such that 'fypy^ } < e Varying e gives the desired sequence. It is very important to observe that Nakai-Moishezon and Raynaud Theorems ensure that the sequence is not stationary and that the image under jo of the points in the sequence is not contained in the torsion subgroup of JK(E)\

For a complete proof see [Sz5].

In the direction of the Bogomolov Conjecture there are some partial results.

THEOREM 3.13 (Burnol, Zhang). Let f : X —• Spec(0K) be an arithmetic

surface with semistable reduction then (UX/OK'^X/OK) > 0 iff is notasmoothmorphism.

(See [7] and [31]).

4. Conclusion

As we have seen in this small survey, Arakelov Theory of Arithmetic Surfaces is a fascinating language which allows us to think geometrically about problems in Arithmetic.

There are various ways in which this Theory may be developed.

On one hand this Theory gives a way to make conjectures in Arithmetics. One may simply consider a "classical" theorem in the Theory of Projective Algebraic Surfaces, and then try to translate it into the Theory of Arithmetic Surfaces via Arakelov Theory.

On the other hand it is interesting to use Arakelov Theory in order to find new theorems in Arithmetics. In this direction there are some very interesting developments both in the Theory of Arithmetic Surfaces with elliptic generic fibre (see [29], [30]), and in the Arakelovian study of modular curves (see [1]).

Conjectures regarding upper bounds for the self-intersection of the dualizing sheaf are very important. For instance we know that, by a classical construction (called the "Kodaira-Parshin construction") they imply an effective proof of the Mordell Conjecture (by which we mean a proof giving an explicit bound for the height of the A'-rational points).

Arakelov Theory of Arithmetic Varieties of higher dimension (see [3], [14]) is beyond the scope of this paper; but we note that in recent years, a lot of important results have been found and a lot of arithmetic applications are potentially possible (and each day more reachable). In particular using this Theory, a Theory of height for varieties and sub-varieties of higher dimension has been developed and studied (see [4]; [5]; [6]).

The last result we want to mention, is a Theorem which is in the spirit of Theorem 3.9. Indeed it gives a generalisation of that Theorem to stable vector bundles of high rank.

The statement is intentionally vague, sirice a rigorous statement needs the introduction of too many objects, and this is, once again, beyond the scope of this paper.

322 C. Gasbarri

THEOREM 4 .1 . Let XK be a smooth projective curve over K. Suppose that

the genus g{Xx) > 1 and that the minimal regular model X of XK has good reduction

everywhere (this means that the morphism f : X -* Spec(Otf) is smooth). Let r > 1 and

ci be two coprirne integers, and let TK be a line bundle on XK such that deg(^i^) = d.

Then, using Arakelov Theory, we can define a canonical height on the moduli space

of stable vector bundles over XK of rank r and determinant isomorphic to TK-

This height can be explicitly computed in terms of Arakelov Theory.

See[13].

REFERENCES

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Carlo GASBARRI Université de Rennes 1 Département de Mathématiques Campus de Beaulieu 35000 Rennes, France.