Rational 6-cycles under iteration of quadratic polynomials. (English) Zbl 1222.11083
Summary: We present a proof, which is conditional on the Birch and Swinnerton-Dyer Conjecture for a specific abelian variety, that there do not exist rational numbers \(x\) and \(c\) such that \(x\) has exact period \(N = 6\) under the iteration \(x\mapsto x^{2} + c\). This extends earlier results by P. Morton [Acta Arith. 87, No. 2, 89–102 (1998; Zbl 1029.12002)] for \(N = 4\) and by E. V. Flynn, B. Poonen and E. Schaefer [Duke Math. J. 90, 435–463 (1997; Zbl 0958.11024)] for \(N = 5\).
MSC:
| 11G30 | Curves of arbitrary genus or genus \(\ne 1\) over global fields |
| 11G40 | \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture |
| 37P05 | Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps |
| 37P35 | Arithmetic properties of periodic points |
Software:
MagmaReferences:
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