Variation of canonical height for\break Fatou points on ℙ¹

Abstract

Let f : ℙ 1 → ℙ 1 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 ∈ ℙ 1 ⁢ ( k ) satisfying a dynamical stability condition, we prove that the Call–Silverman canonical height for specialization f t at point a t , 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 = D f , a on X so that the function t ↦ h ^ f t ⁢ ( a t ) - h D ⁢ ( 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 ↦ λ ^ f t , v ⁢ ( a t ) differ from a Weil function for D by a continuous function on X ⁢ ( ℂ v ) , at each place v of the number field K. These results were known for polynomial maps f and all points a ∈ ℙ 1 ⁢ ( k ) without the stability hypothesis, [21, 14], and for maps f that are quotients of endomorphisms of elliptic curves E over k and all points a ∈ ℙ 1 ⁢ ( k ) . [32, 29]. Finally, we characterize our stability condition in terms of the geometry of the induced map f ~ : X × ℙ 1 ⇢ X × ℙ 1 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 λ ^ f , γ ⁢ ( a ) can be computed as an intersection number.