These are notes for my eight lectures given at the C.I.M.E. session on “Vector bundles on curves. New directions ” held at Cetraro (Italy) in June 1995. The work presented here was done in collaboration with M.S. Narasimhan and A. Ramanathan and appeared in [KNR]. These notes differ from [KNR] in that we have

Abstract. Given a dynamical system associated to a rational function ϕ(T) on P 1 of degree at least 2 with coefficients in a number field k, we show that for each place v of k, there is a unique probability measure µϕ,v on the Berkovich space P 1 Berk,v /Cv such that if {zn} is a sequence of points in P 1 (k) whose ϕ-canonical heights tend to zero, then the zn’s and their Galois conjugates are equidistributed with respect to µϕ,v. In the archimedean case, µϕ,v coincides with the well-known canonical measure associated to ϕ. This theorem generalizes a result of Baker-Hsia [BH] when ϕ(z) is a polynomial. The proof uses a polynomial lift F (x, y) = (F1(x, y), F2(x, y)) of ϕ to construct a twovariable Arakelov-Green’s function gϕ,v(x, y) for each v. The measure µϕ,v is obtained by taking the Berkovich space Laplacian of gϕ,v(x, y), using a theory developed in [RB]. The other ingredients in the proof are (i) a potential-theoretic energy minimization principle which says that � � gϕ,v(x, y) dν(x)dν(y) is uniquely minimized over all probability measures ν on P 1 Berk,v when ν = µϕ,v, and (ii) a formula for homogeneous transfinite diameter of the v-adic filled Julia set KF,v ⊂ C 2 v in terms of the resultant Res(F) of F1 and F2. The resultant formula, which generalizes a formula of DeMarco [DeM], is proved using results

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We study families of quantum eld theories of free bosons on a compact Riemann surface of genus g. For the case g > 0 these theories are parameterized by holomorphic line bundles of degree g 1, and for the case g = 0 | by smooth closed Jordan curves on the complex plane. In both cases we dene a notion of -function as a partition function of the theory and evaluate it explicitly. For the case g > 0 the -function is an analytic torsion [3], and for the case g = 0 | the regularized energy of a certain natural pseudo-measure on the interior domain of a closed curve. For these cases we rigorously prove the Ward identities for the current correlation functions and determine them explicitly. For the case g > 0 these functions coincide with those obtained in [21, 36] using bosonization. For the case g = 0 the -function we have dened coincides with the -function introduced in [29, 44, 24] as a dispersionless limit of the Sato's -function for the two-dimensional Toda hierarchy. As a corollary of the Ward identities, we obtain recent results [44, 24] on relations between conformal maps of exterior domains and -functions. For this case we also dene a Hermitian metric on the space of all contours of given area. As another corollary of the Ward identities we prove that the introduced metric is Kahler and the logarithm of the -function is its Kahler potential. Contents

Let K be a discrete valuation field with ring of integers OK.Letf: X → Y beafinite morphism of curves over K. In this article, we study some possible relationships between the models over OK of X and of Y. Three such relationships are listed below. Consider a Galois cover f: X → Y of degree prime to the characteristic of the residue field, with branch locus B. We show that if Y has semi-stable reduction over K, thenX achieves semi-stable reduction over some explicit tame extension of K(B). When K is strictly henselian, we determine the minimal extension L/K with the property that XL has semi-stable reduction. Let f: X → Y be a finite morphism, with g(Y) � 2. We show that if X has a stable model X over OK, then Y has a stable model Y over OK, and the morphism f extends to a morphism X → Y. Finally, given any finite morphism f: X → Y, is it possible to choose suitable regular models X and Y of X and Y over OK such that f extends to a finite morphism X → Y? As was shown by Abhyankar, the answer is negative in general. We present counterexamples in rather general situations, with f a cyclic cover of any order � 4. On the other hand, we prove, without any hypotheses on the residual characteristic, that this extension problem has a positive solution when f is cyclic of order 2 or 3.

This paper addresses questions involving the sharpness of Vojta’s conjecture and Vojta’s inequality for algebraic points on curves over number fields. It is shown that one may choose the approximation term mS(D, −) in such a way that Vojta’s inequality is sharp in Theorem 2.3. Partial results are obtained for the more difficult problem of showing that Vojta’s conjecture is sharp when the approximation term is not included (that is, when D = 0). In Theorem 3.7, it is demonstrated that Vojta’s conjecture is the best possible with D = 0 for quadratic points on hyperelliptic curves. It is also shown, in Theorem 4.8, that Vojta’s conjecture is sharp with D = 0 on a curve C over a number field when an analogous statement holds for the curve obtained by extending the base field of C to a certain function field.

Abstract. Let E be an elliptic curve defined over a number field k. In this paper, we define the “global discrepancy ” of a finite set Z ⊂ E(k) which in a precise sense measures how far the set is from being adelically equidistributed. We prove an upper bound for the global discrepancy of Z in terms of the average canonical height of points in Z. We deduce from this inequality a number of consequences. For example, we give a new and simple proof of the Szpiro-Ullmo-Zhang equidistribution theorem for elliptic curves. We also prove a non-archimedean version of the Szpiro-Ullmo-Zhang theorem which takes place on the Berkovich analytic space associated to E. We then prove some quantitative ‘non-equidistribution ’ theorems for totally real or totally padic small points. The results for totally real points imply similar bounds for points defined over the maximal cyclotomic extension of a totally real field. 1.

In a previous paper, we have defined arithmetic extension groups in the context of Arakelov geometry. In the present one, we introduce an arithmetic analogue of the Atiyah extension that defines an element — the arithmetic Atiyah class — in a suitable arithmetic extension group. Namely, if E is a hermitian vector bundle on an arithmetic scheme X, its arithmetic Atiyah class bat X/Z(E) lies in the group dExt 1 X(E,E ⊗ Ω1), and is an obstruction to the algebraicity over X of the X/Z unitary connection on the vector bundle EC over the complex manifold X(C) that is compatible with its holomorphic structure. In the first sections of this article, we develop the basic properties of the arithmetic Atiyah class which can be used to define characteristic classes in arithmetic Hodge cohomology. Then we study the vanishing of the first Chern class ĉH 1 (L) of a hermitian line bundle L in the arithmetic Hodge cohomology group dExt 1

We derive closed formulas for the Arakelov-Green function and the Faltings delta-invariant of a compact Riemann surface.

Introduction Fermat's last theorem states that for n 3 the equation X n + Y n = Z n (1.1) has no integer solutions with X;Y; Z 1. Inspiring generations of work in number theory, its proof was finally achieved by Wiles. A qualitative result, Finite Fermat, was obtained earlier by Faltings; it says the Fermat equation has only a finite number of solutions (for each given n, up to rescaling). This paper is an appreciation of some of the topological intuitions behind number theory. It aims to trace a logical path from the classification of surface diffeomorphisms to the proof of Finite Fermat. The route we take is the following. x2. The isotopy classes of surface diffeomorphisms f : S ! S form the mapping class group Mod(S). Thurs