arXiv subjects

Özlem Ejder

Publications and source records attributed to Özlem Ejder.

4 recordsLinked to original sources

arXiv2026

Arboreal Galois Groups of a PCF Map with Strictly Pre-periodic Critical Points

We study the arithmetic and geometric iterated monodromy groups associated to the postcritically finite (PCF) quadratic rational function $f(x)=\frac{2}{(x-1)^2}$ defined over a number field $k$, whose critical points are both strictly pre-periodic. We give explicit recursive descriptions of the topological generators of the geometric iterated monodromy group of $f$ and show that the arithmetic iterated monodromy group has Hausdorff dimension zero. We describe an explicit criterion to determine the values $a \in k$ for which the associated arboreal Galois group achieves its maximum possible size. In particular, we show that maximality of the arboreal Galois group can already be verified at level four, which is computationally accessible. We also determine the normalizer of the geometric IMG of $f$ and show that the arithmetic IMG equals the normalizer if $ζ_8$ is not contained in $k$. Finally, we show that when $k=\mathbb{Q}$, the constant field contains $\mathbb{Q}(μ_{2^{\infty}})$, providing the first full study of a PCF quadratic map with non-abelian constant field.

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arXiv2026

Settled Elements in Arboreal Galois Groups of Quadratic PCF Polynomials

Let $f(x) \in K(x)$ be a quadratic polynomial where $K$ is a field of characteristic not equal to $2$. The associated arboreal Galois representation of the absolute Galois group of $K$ acts on a regular rooted binary tree. Boston and Jones conjectured that, for $f \in \mathbb{Z}[x]$, the image of this representation contains a dense set of settled elements. Roughly speaking, a cycle of an automorphism $τ$ of the tree is called stable if its length strictly increases at each subsequent level, and $τ$ is called settled if the proportion of vertices contained in stable cycles goes to $1$ as the level goes to infinity. In this article, we prove that the arithmetic iterated monodromy groups of postcritically finite quadratic polynomials in $K[x]$ with periodic postcritical orbits are densely settled. In the number field case, by a result of Benedetto--Ghioca--Juul--Tucker \cite{BGJT2025s}, it follows that for infinitely many $a \in K$, the associated arboreal Galois representations are densely settled. In particular, our results apply to the arithmetic IMG of the Basilica map $f(x)=x^2-1$.

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arXiv2025

Rational isolated $j$-invariants from $X_1(\ell^n)$ and $X_0(\ell^n)$

Let $\ell$ and $n$ be positive integers with $\ell$ prime. The modular curves $X_1(\ell^n)$ and $X_0(\ell^n)$ are algebraic curves over $\mathbb{Q}$ whose non-cuspidal points parameterize elliptic curves with a distinguished point of order $\ell^n$ or a distinguished cyclic subgroup of order $\ell^n$, respectively. We wish to understand isolated points on these curves, which are roughly those not belonging to an infinite parameterized family of points having the same degree. Our first main result is that there are precisely 15 $j$-invariants in $\mathbb{Q}$ which arise as the image of an isolated point $x\in X_1(\ell^n)$ under the natural map $j:X_1(\ell^n) \rightarrow X_1(1)$. This completes a prior partial classification of Ejder. We also identify the 19 rational $j$-invariants which correspond to isolated points on $X_0(\ell^n)$.

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arXiv2025

Galois theory of quadratic rational functions with periodic critical points

Given a number field $k$, and a quadratic rational function $f(x) \in k(x)$, the associated arboreal representation of the absolute Galois group of $k$ is a subgroup of the automorphism group of a regular rooted binary tree. Boston and Jones conjectured that the image of such a representation for $f \in \mathbb{Z}[x]$ contains a dense set of settled elements. An automorphism is settled if the number of its orbits on the $n\text{th}$ level of the tree remains small as $n$ goes to infinity. In this article, we exhibit many quadratic rational functions whose associated Arboreal Galois groups are not densely settled. These examples arise from quadratic rational functions whose critical points lie in a single periodic orbit. To prove our results, we present a detailed study of the iterated monodromy groups (IMG) of $f$, which also allows us to provide a negative answer to Jones and Levy's question regarding settled pairs. Furthermore, we study the iterated extension $k(f^{-\infty}(t))$ generated by adjoining to $k(t)$ all roots of $f^n(x) = t$ for $n \geq 1$ for a parameter $t$. We call the intersection of $k(f^{-\infty}(t))$ with $\bar{k}$, the field of constants associated with $f$. When one of the two critical points of $f$ is the image of the other, we show that the field of constants is contained in the cyclotomic extension of $k$ generated by all $2$-power roots of unity. In particular, we prove the conjecture of Ejder, Kara, and Ozman regarding the rational function $\frac{1}{(x-1)^2}$.

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