arXiv subjects

Özge Ülkem

Publications and source records attributed to Özge Ülkem.

6 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.

math.NT↗

arXiv2026

A family of Hecke eigenforms for Drinfeld modular forms of arbitrary rank

We aim to provide an infinite family of cuspidal Drinfeld-Hecke eigenforms given in terms of a determinant of twisted Eisenstein series. Our main tool is the theory of vectorial Drinfeld modular forms, previously introduced by Pellarin [20] and extensively studied by Pellarin [21] as well as in his joint work with Perkins [22] in the rank two setting. We will develop this theory in the present paper for the arbitrary rank setting by using a particular representation.

math.NT↗

arXiv2025

Intersections of Hecke correspondences on the modular varieties of $\mathcal{D}$-elliptic sheaves

This paper studies the intersections of Hecke correspondences on the modular varieties of $\mathcal{D}$ -elliptic sheaves in the higher-rank setting, where $\mathcal{D}$ is a "maximal order" in a central division algebra $D$ over a global function field $k$. Assuming that $\dim_k(D) = r^2$, where $r$ is a prime distinct from the characteristic of $k$, we express the intersection numbers of Hecke correspondences as suitable combinations of modified Hurwitz class numbers of "imaginary orders". This result establishes a higher-rank analogue of the classical class number relation.

math.NT↗

arXiv2024

Mapping class groupoids and Thompson's groups

We introduce a groupoid ${\mathbf{ΠMG}}}$, called the fundamental modular groupoid, which is a variant of Penner's mapping class groupoid. We study how it relates to the surface mapping class groups and Thompson's group $\mathsf T$. We also introduce larger groupoid $\mathbf{ΩMG}$, which is related to outer automorphisms of free groups and Thompson's group $\mathsf V$ in a similar manner.

math.GT↗

arXiv2023

On perfect powers that are sums of cubes of a nine term arithmetic progression

We study the equation $(x-4r)^3 + (x-3r)^3 + (x-2r)^3+(x-r)^3 + x^3 + (x+r)^3+(x+2r)^3 + (x+3r)^3 + (x+4r)^3 = y^p$, which is a natural continuation of previous works carried out by A. Argáez-García and the fourth author (perfect powers that are sums of cubes of a three, five and seven term arithmetic progression). Under the assumptions $0 < r \leq 10^6$, $p \geq 5 $ a prime and $\gcd(x, r) = 1$, we show that solutions must satisfy $xy=0$. Moreover, we study the equation for prime exponents $2$ and $3$ in greater detail. Under the assumptions $r>0$ a positive integer and $\gcd(x, r) = 1$ we show that there are infinitely many solutions for $p=2$ and $p=3$ via explicit constructions using integral points on elliptic curves. We use an amalgamation of methods in computational and algebraic number theory to overcome the increased computational challenge. Most notable is a significant computational efficiency obtained through appealing to Bilu, Hanrot and Voutier's Primitive Divisor Theorem and the method of Chabauty, as well as employing a Thue equation solver earlier on.

math.NT↗

arXiv2023

Power values of power sums: a survey

Research on power values of power sums has gained much attention of late, partially due to the explosion of refinements in multiple advanced tools in (computational) Number Theory in recent years. In this survey, we present the key tools and techniques employed thus far in the (explicit) resolution of Diophantine problems, as well as an overview of existing results. We also state some open problems that naturally arise in the process.

math.NT↗