Clayton Petsche

Global discrepancy and small points on elliptic curves

Let E be an elliptic curve defined over a number field k. In this paper, we define the “global discrepancy” of a finite set Z ⊂ E(k) which in a precise sense measures how far the set is from being adelically equidistributed. We prove an upper bound for the global discrepancy of Z in terms of the average canonical height of points in Z. We deduce from this inequality a number of consequences. For example, we give a new and simple proof of the Szpiro-Ullmo-Zhang equidistribution theorem for elliptic curves. We also prove a non-archimedean version of the Szpiro-Ullmo-Zhang theorem which takes place on the Berkovich analytic space associated to E. We then prove some quantitative ‘non-equidistribution’ theorems for totally real or totally padic small points. The results for totally real points imply similar bounds for points defined over the maximal cyclotomic extension of a totally real field.

A dynamical pairing between two rational maps

This is the publisher’s final pdf. The published article is copyrighted by American Mathematical Society and can be found at: http://www.ams.org/home/page.

A QUANTITATIVE VERSION OF BILU'S EQUIDISTRIBUTION THEOREM

We use Fourier-analytic methods to give a new proof of Bilus theorem on the complex equidistribution of small points on the one-dimensional algebraic torus. Our approach yields a quantitative bound on the error term in terms of the height and the degree.

S-integral preperiodic points for dynamical systems over number fields

Given a rational map

On quadratic rational maps with prescribed good reduction

\phi: {\mathbb P}^1\to {\mathbb P}^1

Critically Separable Rational Maps in Families

defined over a number field

On the distribution of orbits in affine varieties

K

Non-Archimedean Hénon maps, attractors, and horseshoes

, we prove a finiteness result for

Energy integrals and small points for the Arakelov height

\phi

Isotriviality is equivalent to potential good reduction for endomorphisms of PN over function fields

-preperiodic points which are