DAO for curves

TL;DR

This paper proves the DAO conjecture for families of rational maps parameterized by algebraic curves, extending it to a Bogomolov type generalization, advancing understanding in arithmetic dynamics.

Contribution

It establishes the DAO conjecture for curves and generalizes it to a Bogomolov type setting, providing new insights in arithmetic dynamics and rational maps.

Findings

Proved DAO conjecture for rational maps on algebraic curves

Extended DAO to a Bogomolov type generalization

Enhanced understanding of dynamics on algebraic curves

Abstract

We prove the Dynamical Andr\'e-Oort (DAO) conjecture proposed by Baker and DeMarco for families of rational maps parameterized by an algebraic curve. In fact, we prove a stronger result, which is a Bogomolov type generalization of DAO for curves.