arXiv subjects

Yasemin Kara

Publications and source records attributed to Yasemin Kara.

8 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. Finally, we determine the intersection of the constant field of the arithmetic iterated monodromy group with $k(μ_{2^{\infty}})$, providing the first full study of a PCF quadratic map with non-abelian constant field.

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arXiv2026

Non-trivial Solutions of $Aa^p+Bb^p=Cc^3$ over Number Fields

In this paper, we investigate solutions to the Diophantine equation $ A a^p + B b^p = C c^3 $ over number fields using the modular method. Assuming certain standard modularity conjectures, we first establish an asymptotic result for general number fields satisfying an appropriate $S$-unit condition. In particular, we verify that this condition holds for several imaginary quadratic fields. Beyond the asymptotic setting, we also obtain an effective result. Specifically, for the equation $a^p + d b^p = c^3$ over $ K = \mathbb{Q}(\sqrt{-d}) $ with $ d \in \{7, 19, 43, 67\} $, we determine an explicit bound (depending on $ d $) such that no solutions of a certain type exist whenever $ p $ exceeds this bound.

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arXiv2026

Solving equations of signature $(p,p,2)$ with coefficients over number fields

Using the modular method, we study solutions to the Diophantine equation $$Aa^p+Bb^p=Cc^2$$ over number fields. We first prove an asymptotic result for general number fields satisfying an appropriate $S$-unit condition by assuming some standard conjectures in the case of fields that are not totally real. Specifically, we verify that this condition holds for an infinite family of real quadratic fields. Outside the asymptotic setting, we also obtain effective results. In particular, for the equation $$a^p+db^p=c^2$$ over $K= \mathbb{Q}(\sqrt{-d})$ with $d \in \{3, 11, 19, 43, \}$ and $K= \mathbb{Q}(\sqrt d)$ with $d \in \{3, 5, 11, 13, 19, 29\}$, we find explicit bounds (depending on $d$) such that no non-trivial solutions of a certain type exist whenever $p$ exceeds these bounds.

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arXiv2025

Non-trivial Integer Solutions of $x^r+y^r=Dz^p$

In this paper, we use the modular method over totally real fields together with some standard conjectures (the Weak Frey--Mazur Conjecture and the Eichler--Shimura Conjecture) to prove that infinitely many equations of the type $x^r+y^r=Dz^p$ do not have any non-trivial primitive integer solutions, where $r \geq 5$ is a fixed prime, whenever $p$ is large enough. For $r \equiv 3 \pmod 4$, we get the same result with only assuming the Weak Frey--Mazur Conjecture.

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arXiv2023

Iterated Monodromy Group of a PCF Quadratic Non-polynomial Map

We study the postcritically finite non-polynomial map $f(x)=\frac{1}{(x-1)^2}$ over a number field $k$ and prove various results about the geometric $G^{\text{geom}}(f)$ and arithmetic $G^{\text{arith}}(f)$ iterated monodromy groups of $f$. We show that the elements of $G^{\text{geom}}(f)$ are the ones in $G^{\text{arith}}(f)$ that are fixing the roots of unity by assuming a conjecture on the size of $G^{\text{geom}}_n(f)$. Furthermore, we describe exactly for which $a \in k$ the Arboreal Galois group $G_a(f)$ and $G^{\text{arith}}(f)$ are equal.

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arXiv2022

On Ternary Diophantine Equations of Signature $(p,p,3)$ over Number Fields

In this paper, we prove results about solutions of the Diophantine equation $x^p+y^p=z^3$ over various number fields using the modular method. Firstly, by assuming some standard modularity conjecture we prove an asymptotic result for general number fields of narrow class number one satisfying some technical conditions. Secondly, we show that there is an explicit bound such that the equation $x^p+y^p=z^3$ does not have a particular type of solution over $K=\Q(\sqrt{-d})$ where $d=1,7,19,43,67$ whenever $p$ is bigger than this bound. During the course of the proof we prove various results about the irreducibility of Galois representations, image of inertia groups and Bianchi newforms.

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arXiv2020

Construction of Poincaré-type series by generating kernels

Let $Γ\subset \textrm{PSL}_2({\mathbb R})$ be a Fuchsian group of the first kind having a fundamental domain with a finite hyperbolic area, and let $\widetildeΓ$ be its cover in $\textrm{SL}_2({\mathbb R})$. Consider the space of twice continuously differentiable, square-integrable functions on the hyperbolic upper half-plane, which transform in a suitable way with respect to a multiplier system of weight $k\in{\mathbb R}$ under the action of $\widetildeΓ$. The space of such functions admits the action of the hyperbolic Laplacian $Δ_k$ of weight $k$. Following an approach of Jorgenson, von Pippich and Smajlović (where $k=0$), we use the spectral expansion associated to $Δ_k$ to construct a wave distribution and then identify the conditions on its test functions under which it represents automorphic kernels and further gives rise to Poincaré-type series. An advantage of this method is that the resulting series may be naturally meromorphically continued to the whole complex plane. Additionally, we derive sup-norm bounds for the eigenfunctions in the discrete spectrum of $Δ_k$.

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arXiv2019

Asymptotic Generalized Fermat's Last Theorem over Number Fields

Recent work of Freitas and Siksek showed that an asymptotic version of Fermat's Last Theorem holds for many totally real fields. Later this result was extended by Deconinck to generalized Fermat equations of the form $Ax^p +By^p +Cz^p = 0$, where A;B;C are odd integers belonging to a totally real field. Another extension was given by Sengun and Siksek. They showed that the Fermat equation holds asymptotically for imaginary quadratic number fields satisfying usual conjectures about modularity. In this work, combining their techniques we extend their results about the generalized Fermat equations to imaginary quadratic fields. More specifically we prove that the asymptotic generalized Fermat's Last Theorem holds for many quadratic imaginary number fields.

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