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2,614 research outputs found

Projective Equivalence for the Roots of Unity

Author: Fu Hang
Publication venue
Publication date: 04/05/2019
Field of study

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})

\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

arXiv.org e-Print Archive

Torsion of elliptic curves and unlikely intersections

Author: Bogomolov Fedor
Fu Hang
Tschinkel Yuri
Publication venue
Publication date: 05/06/2017
Field of study

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

arXiv.org e-Print Archive

Crossref

Uniform unlikely intersections for unicritical polynomials

Author: Fu Hang
Publication venue
Publication date: 25/09/2020
Field of study

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)

arXiv.org e-Print Archive

Pairs of elliptic curves with $22$ common projective torsion points

Author: Fu Hang
Publication venue
Publication date: 20/12/2019
Field of study

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

arXiv.org e-Print Archive

Dynamics of quadratic polynomials and rational points on a curve of genus $4$

Author: Fu Hang
Stoll Michael
Publication venue
Publication date: 24/06/2022
Field of study

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

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