arXiv subjects
Publications and source records attributed to Ayberk Zeytin.
In this work, we offer a historical stroll through the vast topic of binary quadratic forms. We begin with a quick review of their history and then an overview of contemporary algebraic developments on the subject.
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.
InfoMod is a new software and application devoted to the modular group, PSL2(Z). It has algorithms that deals with the classical correspondences among continued fractions, geodesics on the modular surface and binary quadratic forms. In addition the software implements the recently discovered representation of Gauss' indefinite binary quadratic forms and their classes in terms of certain infinite planar graphs (dessins) called çarks. InfoMod illustrates various aspects of these forms, i.e. Gauss' reduction algorithm, the representation problem of forms, ambiguous and reciprocal forms. It can be used as an educational tool, and might be used to explore some new facts about these objects.
In this work, by using Pauli matrices, we introduce four families of polynomials indexed over the positive integers. These polynomials have rational or imaginary rational coefficients. It turns out that two of these families are closely related to classical Lucas and Fibonacci polynomial sequences and hence to Lucas and Fibonacci numbers. We use one of these families to give a geometric interpretation of the 200 years old class number problems of Gauss, which is equivalent to the study of narrow ideal classes in real quadratic number fields.
In this work, we determine all Lattes maps which are Belyi morphisms. It turns out that in the generic case, i.e. when the automorphism group is $\ZZ/2\ZZ$, the corresponding family of Lattes maps are Belyi morphisms if and only if the isogeny is multiplication by two. This family form a continuous family of Belyi maps. Elliptic curves with complex multiplication also determine a family over $\ZZ$ of Belyi morphisms. We give the explicit formulas for the first few Belyi morphisms when the curve has complex multiplication by 3rd root of unity.
We give an overview of the category of subgroups of the modular group, incorporating both the tame part, i.e. finite index subgroups, and the non-tame part, i.e. the rest. We also discuss arithmetic related questions which exist in both the tame part (via Belyi's theorem) and the non-tame part.
We show that the class of every primitive indefinite binary quadratic form is naturally represented by an infinite graph (named çark) with a unique cycle embedded on a conformal annulus. This cycle is called the spine of the çark. Every choice of an edge of a fixed çark specifies an indefinite binary quadratic form in the class represented by the çark. Reduced forms in the class represented by a çark correspond to some distinguished edges on its spine. Gauss reduction is the process of moving the edge in the direction of the spine of the çark. Ambiguous and reciprocal classes are represented by çarks with symmetries. Periodic çarks represent classes of non-primitive forms.