Let n 5 be an integer. We provide an effective method for finding all elliptic curves in short Weierstrass form E/Q with j (E) ∈ {0, 1728} and all P ∈ E(Q) such that the nth term in the elliptic divisibility sequence defined by P over E fails to have a primitive divisor. In particular, we improve recent results of Everest, Mclaren, and Ward on the Zsigmondy bounds of elliptic divisibility sequences associated with congruent number curves.

Let ϕ(z) ∈ Q(z) be a rational function of degree d 2 with ϕ(0) = 0 and such that ϕ does not vanish to order d at 0. Let α ∈ Q have infinite orbit under iteration of ϕ and write ϕ n (α) = A n /B n as a fraction in lowest terms. We prove that for all but finitely many n 0, the numerator A n has a primitive divisor, i.e., there is a prime p such that p | A n and p H A i for all i < n. More generally, we prove an analogous result when ϕ is defined over a number field and 0 is a preperiodic point for ϕ.

A rational function of degree at least 2 with coefficients in an algebraically closed field is postcritically finite (PCF) if and only if all of its critical points have finite forward orbit under iteration. We show that the collection of PCF rational functions is a set of bounded height in the moduli space of rational functions over the complex numbers, once the well-understood family known as flexible Lattès maps is excluded. As a consequence, there are only finitely many conjugacy classes of non-Lattès PCF rational maps of a given degree defined over any given number field. The key ingredient of the proof is a nonarchimedean version of Fatou's classical result that every attracting cycle of a rational function over C attracts a critical point.

Let P d denote the moduli space of polynomials of degree d, up to affine conjugacy. We show that the set of points in P d (C) corresponding to post-critically finite polynomials is a set of algebraic points of bounded height. It follows that for any B, the set of conjugacy classes of post-critically finite polynomials of degree d with coefficients of algebraic degree at most B is a finite and effectively computable set. As an example, we exhibit a complete list of representatives of the conjugacy classes of monic post-critically finite cubic polynomials in Q [z]. The proof of the main result comes down to finding a relation between the natural height on P d , and Silverman's critical height.

Abstract. For any elements a, c of a number field K, let Γ(a, c) denote the backwards orbit of a under the map fc : C → C given by fc(x) = x 2 + c. We prove an upper bound on the number of elements of Γ(a, c) whose degree over K is at most some constant B. This bound depends only on a, [K : Q], and B, and is valid for all a outside an explicit finite set. We also show that, for any N ≥ 4 and any a ∈ K outside a finite set, there are only finitely many pairs (y 0 , c) ∈ C 2 for which [K(y 0 , c) : K] < 2 N−3 and the value of the N th iterate of fc(x) at x = y 0 is a. Moreover, the bound 2 N−3 in this result is optimal.