Arakelov geometry and its related topics

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The main results of this project are (1) Distribution of rational points over a finitely generated field, and (2) The Picard group of the moduli space of stable curves and its cone.(1) We considered the problems over an abelian variety and proved a generalization of Lang conjecture and Bogomolov conjecture. Let K be a finitely generated field over Q and A an abelian variety over K.Then, using a good height function due to the head investigator, we can define a height pairing < , > : A (F) × A (F) -> R, which is an extension of Neron-Tate height pairing over a number field (note that F is the algebraic closure of K). For x_1, ... , x_r ∈A (F), we denote det(<x_i, x_j>) by δ (x_1, ..., x_r) . Let Γ be a finite rank subgroup of A (F), and X a subvariety of A.Moreover, let {x_1, ... , x_n} be a Q-basis of Γ. Then, we obtained that if the set { x ∈ X(F)|δ(x_1, ..., x_n, x)≦ε }is Zariski dense in X for every positive number ε , then X is a translation of an abelian subvariety by an element of Γ_ {div}.(2) Let X be a normal complete variety and U a Zariski open set of X.A Q-line bundle L on X is said to be nef over U if, for any complete curve C passing through U, the intersection number of L with C is non-negative. Let M_g be the moduli space of stable curves of genus g, and M_g^1 the Zariski open set consisting of stable curves with one node at most. Then, in terms of tautological classes on M_g, we determine the necessary and sufficient condition to guarantee that a Q-line bundle on M_g is nef over M_g^1. Using this, we can see that the cone generated by curves passing through M_g^1 is a rational polyhedra, and we can also describe extremal rays of the cone in a concrete way. We believe that this is a first step toward a Fulton conjecture concerning the Mori cone of M_g, n.