(Both the published version and the arXiv version are available online.)

For any elements b,c of a number field K, let G(b,c) denote the backwards orbit of b under the map g_c : C→C given by g_c(x) = x²+c. We prove an upper bound on the number of elements of G(b,c) whose degree over K is at most some constant B. This bound depends only on b, [K : Q], and B, and is valid for all b outside an explicit finite set. We also show that, for any N>3 and any b in K outside a finite set, there are only finitely many pairs of complex numbers (y,c) for which [K(y,c) : K] < 2^N-3 and the value of the N-th iterate of g_c(x) at x=y is b. Moreover, the bound 2^N-3 in this result is optimal.

Our proofs involve the curve X(N,b) whose affine points are the pairs (y,c) where the value of the N-th iterate of g_c(x) at x=y is b. In case K=Q and B=1, we combine a height argument with the Mordell conjecture (Faltings' theorem) in order to prove the first result for all b in Q. To obtain the result for general K and B, we use a consequence of Vojta's inequality on arithmetic discriminants instead of Faltings' theorem; this requires some additional arguments adapting Vojta's result to our situation.

Our second result should be compared with a conjecture of Abramovich and Harris, which says that a curve C over a number field K admits a rational map of degree at most d to a curve of genus 0 or 1 if and only if there is a finite extension L of K for which infinitely many algebraic points P on C satisfy [L(P) : L] ≤ d. This has been proved when d is small, but was disproved in general by Debarre and Fahlaoui. We show that the conjecture holds for the curves X(N,b), for all b outside an explicit finite set. (We exclude the values b for which X(N,b) is singular; although our arguments fail for such b, we do not know whether the Abramovich–Harris conjecture remains valid for them.)

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