Su-Ion Ih

Height Uniformity for Algebraic Points on Curves

We recall the main result of L. Caporaso, J. Harris, and B. Mazurs 1997 paper of ‘Uniformity of rational points.’ It says that the Lang conjecture on the distribution of rational points on varieties of general type implies the uniformity for the numbers of rational points on curves of genus at least 2. In this paper we will investigate its analogue for their heights under the assumption of the Vojta conjecture. Basically, we will show that the Vojta conjecture gives a naturally expected simple uniformity for their heights.

A FINITENESS PROPERTY FOR PREPERIODIC POINTS OF CHEBYSHEV POLYNOMIALS

Let K be a number field with algebraic closure

Integral division points on curves

\overline K

ALGEBRAIC POINTS ON THE PROJECTIVE LINE

, let S be a finite set of places of K containing the Archimedean places, and let φ be a Chebyshev polynomial. We prove that if

Height uniformity for integral points on elliptic curves

\alpha \in \overline K

INTEGRAL POINTS ON THE CHEBYSHEV DYNAMICAL SYSTEMS

is not preperiodic, then there are only finitely many preperiodic points

A finiteness property of torsion points

\beta \in \overline K

A nondensity property of preperiodic points on Chebyshev dynamical systems

which are S-integral with respect to α.

Equidistribution of small subvarieties of an abelian variety

Let k be a number field with algebraic closure k, and let S be a finite set of primes of k containing all the infinite ones. Let E/k be an elliptic curve, Γ 0 be a finitely generated subgroup of E(k), and Γ ⊆ E(k) be the division group attached to Γ0. Fix an effective divisor D of E with support containing either (i) at least two points whose difference is not torsion, or (ii) at least one point not in Γ . We prove that the set of “integral division points on E(k),” i.e., the set of points of Γ which are S-integral on E relative to D, is finite. We also prove the Gm-analogue of this theorem, thereby establishing the 1-dimensional case of a general conjecture we pose on integral division points on semi-abelian varieties.

A nondensity property of preperiodic points on the projective plane

Schanuels formula describes the distribution of rational poi- nts on projective space. In this paper we will extend it to algebraic points of bounded degree in the case of P1. The estimate formula will also give an explicit error term which is quite small relative to the leading term. It will also lead to a quasi-asymptotic formula for the number of points of bounded degree on P1 according as the height bound goes to ∞.