Examples of Quadratic Polynomials over $\mathbb{Q}$ with Surjective Arboreal Galois Representations

TL;DR

This paper constructs infinite families of quadratic polynomials over rationals with preperiodic points, demonstrating that their arboreal Galois representations are surjective, thus expanding known examples in this area.

Contribution

It introduces new infinite families of quadratic polynomials with surjective arboreal Galois representations, focusing on preperiodic points over rationals.

Findings

Provided infinitely many examples of quadratic polynomials with surjective arboreal Galois representations.

Linked preperiodic points to the surjectivity of associated Galois representations.

Expanded the class of known polynomials with this property.

Abstract

We explore families of pairs of quadratic polynomials $f(x)=x^2+c\in \mathbb{Q}$ and $a\in \mathbb{Q}$ with $a$ being a strictly preperiodic point of $f$ to provide infinitely many new examples for which the associated arboreal Galois representations are surjective.