Özlem Ejder
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View article: Rational isolated $j$-invariants from $X_1(\ell^n)$ and $X_0(\ell^n)$
Rational isolated $j$-invariants from $X_1(\ell^n)$ and $X_0(\ell^n)$ Open
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…
View article: Galois theory of quadratic rational functions with periodic critical points
Galois theory of quadratic rational functions with periodic critical points Open
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 Jon…
View article: Iterated Monodromy Group of a PCF Quadratic Non-polynomial Map
Iterated Monodromy Group of a PCF Quadratic Non-polynomial Map Open
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 $f$…
View article: Arithmetic Monodromy Groups of Dynamical Belyi maps
Arithmetic Monodromy Groups of Dynamical Belyi maps Open
We consider a large family of dynamical Belyi maps of arbitrary degree and study the arithmetic monodromy groups attached to the iterates of such maps. Building on the results of Bouw-Ejder-Karemaker on the geometric monodromy groups of th…
View article: Isolated Points on $X_1(\ell^n)$ with rational $j$-invariant}
Isolated Points on $X_1(\ell^n)$ with rational $j$-invariant} Open
Let $\ell$ be a prime and let $n\geq 1$. In this note we show that if there is a non-cuspidal, non-CM isolated point $x$ with a rational $j$-invariant on the modular curve $X_1(\ell^n)$, then $\ell=37$ and the $j$-invariant of $x$ is eithe…
View article: Modular symbols for Fermat curves
Modular symbols for Fermat curves Open
Let $F_n$ denote the Fermat curve given by $x^n+y^n=z^n$ and let $\mu _n$ denote the Galois module of $n$th roots of unity. It is known that the integral homology group $H_1(F_n,\mathbb {Z})$ is a cyclic $\mathbb {Z}[\mu _n\times \mu _n]$ …
View article: Dynamical Belyi maps
Dynamical Belyi maps Open
We study the dynamical properties of a large class of rational maps with exactly three ramification points. By constructing families of such maps, we obtain infinitely many conservative maps of degree $d$; this answers a question of Silver…
View article: Monodromy of Fermat Surfaces and Modular Symbols for Fermat curves
Monodromy of Fermat Surfaces and Modular Symbols for Fermat curves Open
Let $F_n$ denote the Fermat curve given by $x^n+y^n=z^n$ and let $μ_n$ denote the Galois module of $n$th roots of unity. It is known that the integral homology group $H_1(F_n,\Z)$ is a cyclic $\Z[μ_n\times μ_n]$ module. In this paper, we p…
View article: Torsion groups of elliptic curves over quadratic cyclotomic fields in elementary abelian 2-extensions
Torsion groups of elliptic curves over quadratic cyclotomic fields in elementary abelian 2-extensions Open
Let K denote the quadratic field $\mathbb{Q}(\sqrt{d})$ where d=$-1$ or $-3$. Let E be an elliptic curve defined over K. In this paper, we analyze the torsion subgroups of E in the maximal elementary abelian $2$-extension of $K$.