# Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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## Dynamics of quadratic polynomials and rational points on a curve of genus $4$HTML articles powered by AMS MathViewer

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

## 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