Dynamics of quadratic polynomials and rational points on a curve of genus $4$
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- by Hang Fu and Michael Stoll
- Math. Comp. 93 (2024), 397-410
- DOI: https://doi.org/10.1090/mcom/3883
- Published electronically: July 20, 2023
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Abstract:
Let $f_t(z)=z^2+t$. For any $z\in \mathbb {Q}$, let $S_z$ be the collection of $t\in \mathbb {Q}$ such that $z$ is preperiodic for $f_t$. In this article, assuming a well-known conjecture of Flynn, Poonen, and Schaefer [Duke Math. J. 90 (1997), pp. 435–463], we prove a uniform result regarding the size of $S_z$ over $z\in \mathbb {Q}$. In order to prove it, we need to determine the set of rational points on a specific non-hyperelliptic curve $C$ of genus $4$ defined over $\mathbb {Q}$. We use Chabauty’s method, which requires us to determine the Mordell-Weil rank of the Jacobian $J$ of $C$. We give two proofs that the rank is $1$: an analytic proof, which is conditional on the BSD rank conjecture for $J$ and some standard conjectures on L-series, and an algebraic proof, which is unconditional, but relies on the computation of the class groups of two number fields of degree $12$ and degree $24$, respectively. We finally combine the information obtained from both proofs to provide a numerical verification of the strong BSD conjecture for $J$.References
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Bibliographic Information
- Hang Fu
- Affiliation: Department of Mathematics, National Taiwan University, Taipei, Taiwan
- MR Author ID: 1172768
- ORCID: 0000-0002-1214-4664
- Email: drfuhang@gmail.com
- Michael Stoll
- Affiliation: Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany
- MR Author ID: 325630
- ORCID: 0000-0001-5868-2962
- Email: Michael.Stoll@uni-bayreuth.de
- Received by editor(s): July 17, 2022
- Received by editor(s) in revised form: March 20, 2023
- Published electronically: July 20, 2023
- © Copyright 2023 American Mathematical Society
- Journal: Math. Comp. 93 (2024), 397-410
- MSC (2020): Primary 11G30, 11G40, 14G05, 14G10, 14H45, 37P05
- DOI: https://doi.org/10.1090/mcom/3883
- MathSciNet review: 4654627