Dynamics of quadratic polynomials and rational points on a curve of genus \(4\). (English) Zbl 1536.11098
Let \(K\) be a number field of degree \(D\) and let \(f : \mathbb{P}^n \rightarrow \mathbb{P}^n\) be a morphism of degree \(d \geq 2 \) and defined over \(K\). A point \(P \in \mathbb{P}^n(K)\) is said to preperiodic for \(f\) if its forward orbit \(\{f^n(P) : n \geq 0\}\) is finite. Let \(f_t(z) = z^2 + t\). For any \(z\in \mathbb{Q}\), we consider the set \(S_z = \{t \in\mathbb{Q}: z\) is preperiodic for \(f_t\}\). Flynn, Poonen, and Schaefer [E. V. Flynn et al., Duke Math. J. 90, No. 3, 435–463 (1997; Zbl 0958.11024)] proposed the following conjecture: If \(N \geq 4\), then there is no quadratic polynomial \(f(z)\in \mathbb{Q}[z]\) with a rational point of exact order \(N\).
In this paper, it is assumed that the above conjecture is true and then it is proved that the following hold:
The proof of this result is reduced to the determination of the rational points on five specific algebraic curves. The rational points of the four curves can be found at the \(L\)-functions and modular forms database (http://www.lmfdb.org) The main difficulty is to find the rational points on the genus 4 curve \(C\) defined by the equation \[x^3y^2 + x^2y^3 -x^3y -xy^3 - x^2y -xy^2 + x^2 + 2xy + y^2 - x -y = 0.\] It is proved that \[C(\mathbb{Q}) = \{(0, 0), (0, 1), (0, \infty), (1, 0), (1, 1), (1, \infty), (\infty, 0), (\infty, 1), (\infty,\infty)\}.\] Two proofs are given: one that is conditional on the Birch and Swinnerton-Dyer rank conjecture for the Jacobian variety \(J\) of \(C\) and one that is unconditional and based on the determination of the size of the 2-Selmer group of \(J\). The necessary computations have been performed using Magma algebra system.
In this paper, it is assumed that the above conjecture is true and then it is proved that the following hold:
- (1)
- \(\#S_z = 3\) if and only if \(z= 0\) or \(\pm 1/2\).
- (2)
- \(\#S_z = 5\) if and only if \[z = \pm \frac{a^3-a-1}{2a(a+1)}, \ \ \ \text{for some } \ a\in \mathbb{Q} \setminus \{-1,-1/2,0,2\}. \]
- (3)
- \(\#S_z = 6\) if and only if \[z = \pm \frac{2a}{a^2-1}, \ \ \ \text{for some } \ a\in \mathbb{Q} \setminus \{0,\pm 1/5,\pm 1/3,\pm 1, \pm3,\pm5\}\] or \[z = \pm \frac{a^2+1}{a^2-1}, \ \ \ \text{for some } \ a\in \mathbb{Q} \setminus \{0,\pm 1/3,\pm 1, \pm 3\}.\]
- (4)
- \(\#S_z = 7\) if and only if \(z = \pm 5/12, \pm 3/4\), or \(\pm 5/4\).
- (5)
- \(\# S_z = 4\), otherwise.
The proof of this result is reduced to the determination of the rational points on five specific algebraic curves. The rational points of the four curves can be found at the \(L\)-functions and modular forms database (http://www.lmfdb.org) The main difficulty is to find the rational points on the genus 4 curve \(C\) defined by the equation \[x^3y^2 + x^2y^3 -x^3y -xy^3 - x^2y -xy^2 + x^2 + 2xy + y^2 - x -y = 0.\] It is proved that \[C(\mathbb{Q}) = \{(0, 0), (0, 1), (0, \infty), (1, 0), (1, 1), (1, \infty), (\infty, 0), (\infty, 1), (\infty,\infty)\}.\] Two proofs are given: one that is conditional on the Birch and Swinnerton-Dyer rank conjecture for the Jacobian variety \(J\) of \(C\) and one that is unconditional and based on the determination of the size of the 2-Selmer group of \(J\). The necessary computations have been performed using Magma algebra system.
Reviewer: Dimitros Poulakis (Thessaloniki)
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 |
| 14G05 | Rational points |
| 14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |
| 14H45 | Special algebraic curves and curves of low genus |
| 37P05 | Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps |
Citations:
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