Projective Equivalence for the Roots of Unity

Let $\mu_{\infty}\subseteq\mathbb{C}$ be the collection of roots of unity and $\mathcal{C}_{n}:=\{(s_{1},\cdots,s_{n})\in\mu_{\infty}^{n}:s_{i}\neq s_{j}\text{ for any }1\leq i<j\leq n\}$. Two elements $(s_{1},\cdots,s_{n})$ and $(t_{1},\cdots,t_{n})$ of $\mathcal{C}_{n}$ are said to be projectively equivalent if there exists $\gamma\in\text{PGL}(2,\mathbb{C})$ such that $\gamma(s_{i})=t_{i}$ for any $1\leq i\leq n$. In this article, we will give a complete classification for the projectively equivalent pairs. As a consequence, we will show that the maximal length for the nontrivial projectively equivalent pairs is $14$

Torsion of elliptic curves and unlikely intersections

We study effective versions of unlikely intersections of images of torsion points of elliptic curves on the projective line.Comment: 19 page

Pairs of elliptic curves with 2222 common projective torsion points

For $i=1,2$, let $E_{i}$ be an elliptic curve defined over $\overline{\mathbb{Q}}$, $E_{i}[\infty]$ the collection of all torsion points, and $\pi_{i}:E_{i}\to\mathbb{P}^{1}(\overline{\mathbb{Q}})$ a double cover identifying $\pm P\in E_{i}$. In this article, we will prove that there exist infinitely many nontrivial pairs $(E_{1},\pi_{1})$ and $(E_{2},\pi_{2})$ such that $\#\pi_{1}(E_{1}[\infty])\cap\pi_{2}(E_{2}[\infty])\geq22$

Uniform unlikely intersections for unicritical polynomials

Fix $d\geq2$ and let $f_{t}(z)=z^{d}+t$ be the family of polynomials parameterized by $t\in\mathbb{C}$. In this article, we will show that there exists a constant $C(d)$ such that for any $a,b\in\mathbb{C}$ with $a^{d}\neq b^{d}$, the number of $t\in\mathbb{C}$ such that $a$ and $b$ are both preperiodic for $f_{t}$ is at most $C(d)$

Dynamics of quadratic polynomials and rational points on a curve of genus 44

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, 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 $J$