The author does not know how to thank Professor Yasuo Morita for his patient and warm encouragement. Without it, the author could not be able to write up the thesis. He thanks Professor Atsushi Sato for having made some comments on the drafts of my paper and preprint.Thanks also go to Professor Tadao Oda and other staffs of the institute for many things and a comfortable environment.1991 Mathematics Subject Classification. Primary 11G30; Secondary 11D41, 14G05, 14H25.
Author addresses:Mathematical Institute, Tohoku University, Sendai 980-77, Japan
IntroductionThe logarithmic absolute height function is a fundamental tool when we investigate the distribution of rational points on a projective variety V over the rational number field Q.LetQ be an algebraic closure of Q. A (logarithmic absolute) height function is a real-valued function on the set V (Q) = Hom(SpecQ, V ) of Q-valued points on the variety V . It is defined up to a bounded function for the pair V and a line bundle L on V . We denote one of the representatives by h V (L, ·), or simply, by h(L, ·). The set of rational points, orwhere v runs through the set of rational prime numbers and the infinite prime, and | · | v is the standard absolute value on Q defined by v. This is a height function attached to the hyperplane section sheaf O(1) on Taking an embedding ι : V → P N , we obtainA height function is additive with respect to the tensor operation on the group Pic V of line bundles on the projective variety V . For L and of rational points on the abelian variety A is finitely generated, which was substantially proved in [16] and [25]. Let R be the rank of A(Q). The quadratic form attached to the ample line bundle N on A determines an R-dimensional Euclidean space structure on R ⊗ A(Q) and we seeIf we start from polynomial equations with coefficients in the rational integer ring Z, which define a quasi-projective variety over Q, we naturally
where r(T, a) is the rank of the group of rational points on the Jacobian variety of the plane curve overMumford where O C (−(2g − 2)P ) is the line bundle on C one of whose rational sections has −(2g − 2)P as the corresponding divisor, and Pic • is the functor which associates the group of isomorphism classes of line bun-height function attached to the invertible sheaf Ω. We call it the canonical height function on the curve C. is naturally regarded as a subset of the set of rational points on the planeWe denote by J a the Jacobian variety of C a . Corollary 0.5. If a is n-th power-free and |a| > a 1 , then where g is the genus of C and
phrase "in a neighborhood" means that the function of v on the left side of the equation becomes bounded on the image of C(Q).For P ∈ C(Q), we see by the theoremThis leads to a new proof of a fact which is usually an application of the Riemann-Hurwitz formula.Corollary 0.8. Notation and Terminology 0.11. For the objects X, Y, Z and the We denote respectively by Z, Q, and R the ring of rational integers, the field of rational numbers, and the field of real numbers. A finite extensio...
