Variation of the canonical height for a family of polynomials

Abstract

A theorem of Tate asserts that, for an elliptic surface E → X defined over a number field k, and a section P : X → E, there exists a divisorwhereĥ Et is the Néron-Tate height on the fibre above t. We prove the analogous statement for a one-parameter family of polynomial dynamical systems. Moreover, we compare, at each place of k, the local canonical height with the local contribution to h D , and show that the difference is analytic near the support of D, a result which is analogous to results of Silverman in the elliptic surface context.

Canonical heights for Hénon maps

“…Consequently, if η ∈ Div(C) has degree one, then (D + + D − ) −ĥ ϕ (P )η has degree zero, and so it follows immediately from Theorem 1.1 and from Proposition 5.4 of [14, p. 115] that h ϕt (P t ) =ĥ ϕ (P )h η (t) + ε(t), (1.1) where ε(t) = O(1) if C = P 1 , and ε(t) = O(h η (t) 1/2 ) in general. Theorem 1.1 result is analogous to a result of Ingram [11], which strengthened more general estimates of Call and Silverman [4] in the case of polynomials of one variable. One application of an estimate of the form (1.1) is in determining, which specializations of a one-parameter family land in periodic cycles (note that, since Hénon maps are automorphisms, orbits are either periodic or infinite in both directions).…”

“…In Section 3, we prove Theorem 1.2, and in Section 4 we prove Theorem 1.4; the proofs are separate, but rely on similar ideas. We treat Theorem 1.1 in Section 5, relying heavily on material from [11], and in Section 6 we turn our attention to the proofs of Theorems 1.3 and 1.8. Finally, Section 7 is devoted to describing the computations need to verify Proposition 1.6, and here we also undertake an initial investigation of the curves parametrizing quadratic Hénon maps with a marked point of period N .…”

Integrality and rigidity for postcritically finite polynomials

“…We imagine that the corresponding archimedean computation is folklore (cf. [8,Lemma 5]). The now classical argument in [2] relies on basic estimates from univalent function theory.…”

Unlikely Intersection for Two-Parameter Families of Polynomials

“…Their result generalizes a result of Silverman [Sil83] on heights of families of abelian varieties. In a recent paper [Ing13], Ingram improves the error term to O(1) when T is a curve, X is P 1 , and the family of endomorphisms Φ is totally ramified at infinity (i.e., Φ is a polynomial mapping). This result is an analogue of Tate's theorem [Tat83] in the setting of arithmetic dynamics.…”

“…There are only a few results in the literature when the error term in (2.1.3) is known to be O(1). Besides Tate's [Tat83] and Silverman's [Sil83] in the context of elliptic curves and more generally abelian varieties (see also the further improvements of Silverman [Sil94a,Sil94b] in the case of elliptic curves), there are only a few known results, all valid for 1-parameter families (see [Ing13,Ing,GHT15,GM13]). To our knowledge, Theorem 2.1 is the first result in the literature where one improves the error term in (2.1.3) to O(1) for a higher dimensional parameter family of endomorphisms of P m .…”