2024/10/29 by Yap, Jit Wu · 1 citation
Engineering · #Advanced Numerical Analysis Techniques
paper · pdf · doi:10.48550/arxiv.2410.21679
Let X be a smooth projective variety defined over a number field K and let φ: X → X a polarized endomorphism of degree d ≥ 2. Let \widehathφ be the canonical height associated to φ on X(K). Given a generic sequence of points (xn) with \widehathφ(xn) → 0 and a place v ∈ MK, Yuan [Yua08] has shown that the conjugates of xn equidistribute to the canonical measure μφ,v. When v is archimedean, we will prove a quantitative version of Yuan's result. We give two applications of our result to polarized endomorphisms φ of smooth projective surfaces that are defined over a number field K. The first is an exponential rate of convergence for periodic points of period n to the equilibrium measure and the second is an exponential lower bound on the degree of the extension containing all periodic points of period n. When X is an abelian variety, we also give an upper bound on the smallest degree of a hypersurface that contains all points x ∈ X(K) satisfying [K(x):K] ≤ D and \widehathX(x) ≤ (c)/(D8) for some fixed constant c > 0 where \widehathX is the Neron--Tate height for X.