2021/07/13 by Laura DeMarco, DeMarco, Laura, Niki Myrto Mavraki +1
Mathematics · #Algebraic Geometry and Number Theory #Advanced Differential Equations and Dynamical Systems #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2107.05982
Let f: ℙ1→ ℙ1 be a map of degree >1 defined over a function field k = K(X), where K is a number field and X is a projective curve over K. For each point a ∈ ℙ1(k) satisfying a dynamical stability condition, we prove that the Call-Silverman canonical height for specialization ft at point at, for t ∈ X(ℚ) outside a finite set, induces a Weil height on the curve X; i.e., we prove the existence of a ℚ-divisor D = Df,a on X so that the function t↦ hft(at) - hD(t) is bounded on X(ℚ) for any choice of Weil height associated to D. We also prove a local version, that the local canonical heights t↦ λft, v(at) differ from a Weil function for D by a continuous function on X(ℂv), at each place v of the number field K. These results were known for polynomial maps f and all points a ∈ ℙ1(k) without the stability hypothesis, and for maps f that are quotients of endomorphisms of elliptic curves E over k. Finally, we characterize our stability condition in terms of the geometry of the induced map f: X× ℙ1 → X× ℙ1 over K; and we prove the existence of relative Néron models for the pair (f,a), when a is a Fatou point at a place γ of k, where the local canonical height λf,γ(a) can be computed as an intersection number.