1. Introduction
Let X be an algebraic variety defined over a number field F. We will say that rational points are potentially dense if there exists a finite extension K/F such that the set of K-rational points X(K) is Zariski dense in X. The main problem is to relate this property to geometric invariants of X. Hypothetically, on varieties of general type rational points are not potentially dense. In this paper we are interested in smooth projective varieties such that neither they nor their unramified coverings admit a dominant map onto varieties of general type. For these varieties it seems plausible to expect that rational points are potentially dense (see [2]).
Abstract. We investigate analytic properties of height zeta functions of toric varieties. Using the height zeta functions, we prove an asymptotic formula for the number of rational points of bounded height with respect to an arbitrary line bundle whose first Chern class is contained in the interior of the cone of effective divisors.Supported by Deutsche Forschungsgemeinschaft. Leibniz Fellow at ENS, Paris.
Let K be a number field and X a smooth projective algebraic surface defined over K. We say that X admits a structure of an elliptic fibration if there exists a regular map ϕ : X → B onto a smooth (irreducible) curve B whose fibers are connected curves such that the generic fiber is a smooth curve of genus 1. We denote by X b the fiber over b ∈ B. We will say that X admits a structure of a Jacobian elliptic fibration if there exists a zero section e : B →
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