No abstract

Given a point ξ on a complex abelian variety A, its abelian logarithm can be expressed as a linear combination of the periods of A with real coefficients, the Betti coordinates of ξ. When (A, ξ) varies in an algebraic family, these coordinates define a system of multivalued real-analytic functions. Computing its rank (in the sense of differential geometry) becomes important when one is interested about how often ξ takes a torsion value (for instance, Manin's theorem of the kernel implies that this coordinate system is constant in a family without fixed part only when ξ is a torsion section).We compute this rank in terms of the rank of a certain contracted form of the Kodaira-Spencer map associated to (A, ξ) (assuming A without fixed part, and Zξ Zariski-dense in A), and deduce some explicit lower bounds in special situations. For instance, we determine this rank in relative dimension ≤ 3, and study in detail the case of jacobians of families of hyperelliptic curves.Our main application, obtained in collaboration with Z. Gao, states that if A → S is a principally polarized abelian scheme of relative dimension g which has no non-trivial endomorphism (on any finite covering), and if the image of S in the moduli space Ag has dimension at least g, then the Betti map of any non-torsion section ξ is generically a submersion, so that ξ −1 Ators is dense in S(C).

Abstract. In the present paper we solve in particular the function field version of a special case of Vojta's conjecture for integral points, namely for the variety obtained by removing a conic and two lines from the projective plane. This will follow from a bound for the degree of a curve on such a surface in terms of its Euler characteristic.This case is special, but significant, because it lies "at the boundary", in the sense that it represents the simplest case of the conjecture which is still open. Also, it was studied in the context of Nevanlinna Theory by M. Green already in the seventies.Our general results immediately imply the degeneracy of solutions of Fermat's type equations z d = P (x m , y n ) for all d ≥ 2 and large enough m, n, also in the case of non-constant coefficients. Such equations fall apparently out of all known treatments.The methods used here refer to derivations, as is usual in function fields, but contain fundamental new points. One of the tools concerns an estimation for the gcd(1 − u, 1 − v) for S-units u, v; this had been developed also in the arithmetic case, but for function fields we may obtain a much more uniform quantitative version.In an Appendix we shall finally point out some other implications of the methods to the problem of torsion-points on curves and related known questions. §1. Introduction and main results.A celebrated conjecture in diophantine geometry, first proposed by Vojta (see [V, Conjecture 3.4.3] and [V, Prop. 4 .1.2]), reads as follows:Let X be a smooth affine variety defined over a number field k,X be a smooth projective variety containing X as an open subset, D =X \ X the divisor at infinity and K a canonical divisor ofX. Suppose that D is a normal crossing divisor. Then if D + K has maximal Kodaira dimension, for every ring of S-integers O S ⊂ k, the set of S-integral points X(O S ) is not Zariski-dense.It is known that the Kodaira dimension of D + K is in fact independent of the smooth compactificatioñ X of X, provided that D has normal crossings (see for instance [KMK]). Following a frequent notation, we will call log Kodaira dimension of X the Kodaira dimension of D + K.A complex analytic analogue of Vojta's Conjecture asks for the degeneracy of entire curves on affine varieties with maximal log Kodaira dimension; more precisely, one expects that, for every holomorphic map f : C → X(C) to such a variety, the image f (C) be contained in a proper closed algebraic subvariety.Finally an (algebraic) function field analogue of Vojta's Conjecture predicts that, given a smooth curvẽ C and a finite subset S ⊂C, there should exist a bound for the degree (in a suitable projective embedding) for the images of non-constant morphismsC \ S → X, where X is again an algebraic variety with maximal log Kodaira dimension.The particular case whereX is the projective plane has been widely studied, in the arithmetic, analytic and algebraic setting. The condition on the Kodaira dimension of D + K is equivalent to the inequality deg(D) ≥ 4. In all settings, it turns out that...

We analyze the integral points on varieties defined by one equation of the form f 1 · · · f r = g , where the f i , g are polynomials in n variables with algebraic coefficients, and g has "small" degree; we shall use a method that we recently introduced in the context of Siegel's Theorem for integral points on curves. Classical, very particular, instances of our equations arise, e.g., in a well-known corollary of Roth's Theorem (the case n = 2, f i linear forms, deg g < r -2) and with the norm-form equations , treated by W. M. Schmidt. Here we shall prove (Thm. 1) that the integral points are not Zariski-dense, provided Σdeg f i > n · max (deg f i ) + deg g and provided the f i , g , satisfy certain (mild) assumptions which are "generically" verified. Our conclusions also cover certain complete-intersection subvarieties of our hypersurface (Thm. 2). Finally, we shall prove (Thm. 3) an analogue of the Schmidt's Subspace Theorem for arbitrary polynomials in place of linear forms.

Abstract. Let a, b be given multiplicatively independent positive integers and let ǫ > 0. In a recent paper written jointly also with Y. Bugeaud we proved the upper bound exp(ǫn) for gcd(a n − 1, b n − 1); shortly afterwards we generalized this to the estimate gcd(u − 1, v − 1) < max(|u|, |v|) ǫ , for multiplicatively independent S-units u, v ∈ Z. In a subsequent analysis of those results it turned out that a perhaps better formulation of them may be obtained in terms of the language of heights of algebraic numbers. In fact, the purposes of the present paper are: to generalize the upper bound for the g.c.d. to pairs of rational functions other than {u − 1, v − 1} and to extend the results to the realm of algebraic numbers, giving at the same time a new formulation of the bounds in terms of height functions and algebraic subgroups of G 2 m .