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Gonality of dynatomic curves and strong uniform boundedness of preperiodic points. (English) Zbl 1445.37080

Let \(d\geq 2\) be an integer and \(k\) be a field containing the \(d\)-th roots of unity with \(\mathrm{char}(k)\nmid d\). The first main result of this paper shows that the dynatomic curves over \(k\), parametrizing the preperiodic points of the polynomial \(z^d+c\) are geometrically irreducible, and their gonality tends to infinity. In fact, over a field of \(\mathrm{char} (k)=0\), the dynatomic curves are explicitly described and they are irreducible. Using the Castelnuovo-Severi inequality, the authors prove that the gonality of a special dynatomic curve goes to \(\infty\). Then, using the fact that any other dynatomic curves dominates the special curve, they prove that the gonality of dynatomic curves sorted in any order tends to \(\infty\). The proof of the same statement over fields of positive charachteristic is based on some methods of counting points over a finite field.
The second important result uses the initial results on growth of gonality and provide some strong uniform boundedness theorems for the number of preperiodic points and the number of preperiodic points with bounded period (over function fields). These results are function-field analogues of Marel’s theorem [P. Morton and J. H. Silverman, Int. Math. Res. Not. 1994, No. 2, 97–109 (1994; Zbl 0819.11045)] and of the Morton-Silverman conjecture [L. Merel, Invent. Math. 124, No. 1–3, 437–449 (1996; Zbl 0936.11037)].

MSC:

37P35 Arithmetic properties of periodic points
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
14H51 Special divisors on curves (gonality, Brill-Noether theory)
37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps
37P15 Dynamical systems over global ground fields
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